RICE UNIVERSITY

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A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE

APPROVED, THESIS COMMITTEE

HOUSTON, TEXAS ABSTRACT

Data-driven modeling to infer the function of viral replication in a counting-based decision

by

Seth Coleman

Cells use gene regulatory networks, sets of genes connected through a web of bio- chemical interactions, to select a developmental pathway based on signals from their environment. These processes, called cell-fate decisions, are ubiquitous in biology. Yet efforts to study cell-fate decisions are often stymied by the inherent complexity of organisms. Simple model systems provide attractive alternative platforms to study cell-fate decisions and gain insights which may be broadly applicable. Infection of E. coli by the virus lambda is one such model system. The outcome of this viral infection is dependent on the number of initially coinfecting viruses (multiplicity of infection, or MOI), which the viral regulatory network appears to ‘count’. Yet precisely how the viral regulatory network responds to MOI is still unclear, as is how the system is able to achieve sensitivity to MOI despite viral replication, which quickly obfuscates initial viral copy number. In this thesis, I used mathematical modeling of the network dynamics, calibrated by experimental measurements of viral replication and gene ex- pression during infection, to demonstrate how the network responds to MOI and to show that viral replication actually facilitates, rather than hinders, a counting-based decision. This work provides an example of how complex behaviors can emerge from the interplay between gene/network copy number and gene expression, whose coupling iii cannot be ignored in developing a predictive description of cellular decision-making. iv

I was born not knowing and have had only a little time to change that here and there.

Richard Feynman Acknowledgements

I would like to thank both of my advisers, Ido Golding and Oleg Igoshin. Ido’s passion for biology was what originally drew me to this project. His tireless work ethic and uncompromising standards have provided a sterling example to aspire to. Oleg’s insightful critiques over the years have helped hone my ability to formulate, present and analyze scientific arguments. His patient mentoring has enabled me to tackle complex problems like the one in this thesis. I am grateful to both for their time, the opportunity they provided me, and the many lessons they’ve imparted. To my friends at Rice University, thank you for keeping me sane throughout the years. Through trivia nights, D&D sessions, and pilgrimages through the Houston heat to Valhalla, your companionship has made this journey far more enjoyable than it had any right to be. I would like to thank my wonderful parents, Danny and Carla, without whom none of this would have been possible. I’m constantly amazed at how kind and giving you both are, and you have provided me both the strength and ability to pursue this dream. I want to thank my in-laws, the Bisseys, Caskeys, Randalls, and Turners. I have found in you a second home, and your company has brightened these last few years. I look forward to many more adventures with you all. My supervisors in Germany, Christian Gross and Carsten Klempt, also deserve my sincere thanks. You both set me on this path, and I have done my best to carry your lessons forward. You are both exceptional mentors. I’d also like to thank Jim Schemmer. The initial seed of this quest was planted by you. I still keep a copy of Euclid’s Elements in my library. vi

Lastly and most importantly, I want to thank my wife Alaina for putting up with all of this. You have shared my hopes, frustrations, and worries throughout this journey. Thank you for being my rock. Contents

Abstract ii List of Illustrations xi List of Tables xxx

1 Introduction 1 1.1 Cells make information-based decisions ...... 1 1.2 Phage lambda is a model system for studying cell-fate decisions . . .2 1.3 The lambda regulatory network drives a counting-based decision through unclear mechanisms ...... 3 1.4 Aims of this work ...... 4

2 Decision making in temperate phages 6 2.1 Introduction ...... 7 2.2 The post-infection decision of phage lambda ...... 9 2.3 Counting by infecting phages ...... 14 2.4 The view from the single cell ...... 19 2.5 The decision to remain dormant ...... 22 2.6 Conclusion ...... 28

3 Bacteriophage self-counting in the presence of viral repli- cation 31 3.1 Introduction ...... 32 3.2 Results ...... 33 viii

3.2.1 In the absence of viral replication, gene expression does not diverge into lytic and lysogenic trajectories ...... 33 3.2.2 Modeling network dynamics reveals that viral replication is required for a lysis-to-lysogeny transition ...... 37

3.2.3 CII activation of PRE defines a time window for the network’s response to MOI ...... 40 3.2.4 Changes in viral copy number outside the CII activity window do not alter the decision ...... 42 3.2.5 Phage replication enables the lytic choice and lowers the MOI required for lysogeny ...... 44 3.3 Discussion ...... 47

4 Future Directions 50 4.1 Extending the model to capture stochasticity in gene expression . . . 50 4.1.1 A stochastic reformulation of the model deviates from deterministic predictions ...... 50 4.1.2 Estimated single-cell lysogenization frequencies fail to capture observed trends ...... 56 4.1.3 Possible explanations for deviations in predicted single-cell behavior ...... 57 4.2 A Q-based lytic decision yields alternative predictions of infection outcome at high MOI ...... 58 4.2.1 The role of Q during infection ...... 58 4.2.2 A toy module of Q regulation predicts only lysogeny at high MOI...... 59 4.2.3 Experimental measurements of Q mRNA dynamics are needed to extend this analysis ...... 66 4.3 Viral replication may enable subcellular decision-making ...... 67 ix

4.3.1 The subcellular decision-making hypothesis ...... 67 4.3.2 Control of replication by initially infecting viruses is a candidate voting mechanism ...... 69

A Supplementary Information for Chapter 3 72 A.1 Experimental methods ...... 72 A.1.1 Growth media and conditions ...... 72 A.1.2 Bacterial strains and plasmids ...... 73 A.1.3 Phage construction ...... 73 A.1.4 Phage infection ...... 77 A.1.5 Single-molecule fluorescence in situ hybridization (smFISH) . 80 A.1.6 Microscopy ...... 80 A.1.7 Cell segmentation and spot recognition ...... 81 A.1.8 Data analysis following cell segmentation and spot recognition 83 A.1.9 DNA extraction and quantitative PCR (qPCR) ...... 84 A.2 Theoretical Methods ...... 85 A.2.1 Overview of the Governing Differential Equations ...... 85 A.2.2 Formulation of Regulatory Functions ...... 87 A.2.3 Infection Simulation Methods ...... 92 A.2.4 Parameter Fitting ...... 93 A.2.5 Decision Thresholds ...... 97 A.2.6 Modeling Bulk Lysogenization ...... 98 A.2.7 Theoretical Methods Tables ...... 99 A.3 Supplementary Figures ...... 106 A.4 Supplementary Tables ...... 122

B Detailed description of the stochastic model 133 B.1 Conversion of the deterministic model ...... 133 B.2 Reactions ...... 134 x

B.3 Infection simulation methods ...... 135 B.4 Decision Thresholds ...... 137

C Detailed description of the Q module 138 C.1 Formulation of CII inhibition of Q ...... 138 C.2 Decision Thresholds ...... 139

Bibliography 140 Illustrations

2.1 The post-infection decision of bacteriophage lambda. Following infection of an E. coli cell, a binary choice is made between lysis, defined by rampant viral replication and cell death, and lysogeny, in which the viral DNA is integrated into the host’s genome to become a dormant prophage. The lysogenic state is stably maintained, but a switch back to lysis can be induced by cellular damage. Adapted with permission from Golding, I., 2016. Single-cell studies of phage λ: Hidden treasures under Occam’s rug. Annual Review of Virology 3 (1), 453–472, permission conveyed through Copyright Clearance Center, Inc...... 8 2.2 Key lambda genes and host factors involved in the lysis/lysogeny decision. Regulatory interactions between the various nodes rely on diverse molecular mechanisms, at the level of phage DNA, mRNA and proteins...... 11 2.3 The cascade of transcriptional events during the lambda lysis/lysogeny decision. Regulatory elements and their interactions are depicted on the relevant region of the lambda genome. The expression pattern of early genes is qualitatively similar irrespective of the eventual fate. After a decision has been reached, different sets of genes are expressed to execute the lysis/lysogeny choice. 12 xii

2.4 The dependence of lysogenization on the multiplicity of infection: bulk data. A known number of E. coli bacteria is infected with varying concentrations of lambda phage, and the number of resulting lysogens is measured using selection for an antibiotic marker that was engineered into the viral genome. The experimental trend is reproduced by a simple mathematical model, where infection by a single phage leads to lysis, whereas simultaneous infection by two or more phages results in lysogeny...... 15 2.5 The lysis/lysogeny decision at the single-cell level. (A) Images from a live-cell movie following the fate of two E. coli cells, infected by fluorescently-labeled lambda phages. The upper cell, infected by a single phage, proceeds to produce new viral particles and undergo lysis. The lower cell, co-infected by three phages, enters lysogeny, as

indicated by a fluorescent reporter for PRE activity, and proceeds to divide normally. (B) The fraction of cells undergoing lysogeny as a function of the multiplicity of infection, as measured from 41,000 infection events. In contrast to the original modeling of the bulk data (Fig. 2.4 above), the single-cell curve rises gradually, suggesting that the MOI dependence is probabilistic rather than deterministic. (C) Incorporating the effect of intracellular viral concentration (MOI divided by cell volume) captures the experimental data and yields a decision curve that is markedly more step-like. Adapted from Zeng, L., Skinner, S.O., Zong, C., et al., 2010. Decision making at a subcellular level determines the outcome of bacteriophage infection. Cell 141 (4), 682–691, Copyright 2010, with permission from Elsevier. 20 xiii

2.6 Detecting the transcriptional activity of individual lambda phages. (A) Each phage is detected through the binding of fluorescently-tagged ParB proteins to the parS sequence, engineered into the lambda genome. mRNA molecules transcribed by the phage are simultaneously detected using single-molecule fluorescence in situ hybridization (smFISH). (B) Four lambda genomes (cyan) inside a single infected cell, at 10 min after infection. Individual phages vary in their transcriptional activity, with one transcribing cro (green) and two others producing cI (red)...... 23 2.7 The maintenance of lambda lysogeny. (A) The lysogenic state is maintained by a regulatory circuit consisting of CI and Cro,

expressed from the PRM and PR promoters, respectively. CI and Cro

compete for binding at six operator sites (OR1–3 and OL1–3), to

determine which promoters (PRM, or PR and PL) are active, and thus decide whether lysogeny is maintained or, instead, lytic genes induction takes place. Two examples of binding configurations are shown. On the left, CI dimers bind to four operator sites, resulting in

DNA looping that ensures repression of PR and PL during lysogeny.

On the right, binding of Cro to OR3 represses transcription of CI from

PRM and allows lytic genes to be expressed. (B) The regulatory interactions between CI and Cro form a bistable switch. The system can alter its state in response to a large perturbation, such as depletion of CI by RecA, leading to lytic induction, but is immune to small perturbations. Adapted with permission from Golding, I., 2016. Single-cell studies of phage λ: Hidden treasures under Occam’s rug. Annual Review of Virology 3 (1), 453–472, permission conveyed through Copyright Clearance Center, Inc...... 25 xiv

2.8 A theoretical biophysical model captures PRM regulation by

CI. The regulatory curve relating CI concentration in the cell to PRM activity can be predicted from a thermodynamic description of the

possible binding configurations of CI at the OR1–3 and OL1–3 operator sites. The theoretically predicted curve agrees with the experimental measurement of the regulatory relation. The average amount of CI protein present in each lysogenic cell can be estimated by requiring that CI production is exactly balanced by CI elimination (via dilution, due to cell growth and division). Adapted from Sep´ulveda, L.A., Xu, H., Zhang, J., Wang, M., Golding, I., 2016. Measurement of gene regulation in individual cells reveals rapid switching between promoter states. Science 351, 1218–1222, reprinted with permission from AAAS; and from Golding, I., 2011. Decision making in living cells: Lessons from a simple system. Annual Review of Biophysics 40 (1), 63–80 with permission conveyed through Copyright Clearance Center, Inc...... 27

3.1 The lambda decision circuit measures the multiplicity of infection (MOI) even as viral copy number is changing. A higher multiplicity of infection (MOI) increases the propensity to lysogenize. Here, infection by a single lambda phage (top) results in lysis, whereas coinfection by two phages (bottom) leads to lysogeny. In choosing cell fate, the infecting phage must respond to the initial number of viral genomes in the cell but ignore the subsequent increase in number due to viral replication...... 34 xv

3.2 A simplified model of the decision network captures the kinetics of mRNA and viral copy number following infection. (A) Top, the three-gene circuit at the heart of the lysis/lysogeny decision. Bottom, the corresponding segment of the

lambda genome. Upon viral entry, PR expresses both cro and (following a leaky terminator) cII. CII then activates cI expression

from PRE. CI and Cro repress PR and PL, as well as phage

replication. In a lysogen, CI regulates its own expression from PRM. (B) Images of a single E. coli cell at 10 minutes following infection by λ cI 857 Pam80 P1parS. Phage genomes are labeled using ParB-parS, and the mRNA for cI, cro, and cII using smFISH. Yellow dashed line indicates the cell boundary. (C) The numbers of cI, cro, and cII mRNA per cell, at different times following infection by λ cI 857 Pam80 P1parS, at MOI = 1—5. Markers and error bars indicate experimental mean ± SEM per sample (see Table A.9 for detailed sample sizes). Solid lines indicate model fit. (D) Viral copy number, measured using qPCR, following infection at MOI = 1 by P+ (λ cI 857 Sam7) and P- (λ cI 857 Pam80 P1parS) phages. Markers and error bars indicate experimental mean ± standard deviation due to qPCR calibration uncertainty. Lines indicate model prediction. See Experimental Methods and Theoretical Methods for detailed experimental procedures, image and data analysis, and modeling. . . 36 xvi

3.3 Phage replication is required for an MOI-driven lysis-to-lysogeny transition. (A) Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by P- (non-replicating) phage at varying MOI. Protein concentrations were normalized by the lytic and lysogenic thresholds. (B) Same as panel A, for the case of infection by a replicating (P+) phage. See Theoretical Methods for detailed information...... 39

3.4 CII activation of PRE defines a time window for the

network’s response to MOI. (A) Model-predicted PRE promoter activity (per phage) following infection by P+ phage at MOI = 1—5. Gray shading indicates the MOI-averaged CII activity window,

defined as the period during which PRE activity is greater than 10% of its maximum. (B) Cellular CI concentration (normalized by the threshold concentration for lysogeny, dashed red line) following infection by P+ phage at MOI = 1—5. (C) The strength of cro repression by CI, calculated as the magnitude of the CI repression

term in the PR transcription rate, following infection by P+ phage at MOI = 1—5. (D) The strength of repression of replication by CI, calculated as the magnitude of the CI repression term in the viral replication rate, following infection by P+ phage at MOI = 1—5. See Theoretical Methods for detailed information...... 41 xvii

3.5 Changes in viral copy number outside the CII activity window do not alter the decision. (A) Model-predicted system trajectories, in the plane of Cro and CI concentration, during the first 60 minutes following infection by P+ phages, at 4 different scenarios: Infection by a single phage (solid line, light blue), infection by a single

phage followed by a second phage at time τPRE/3 (with τPRE the end

of the CII activity window; dashed line, dark blue) and τPRE (dotted line, dark blue), and simultaneous infection by two phages (solid line,

dark blue). (B) PRE activity from the second arriving phages, for the cases modeled in panel A. The shaded grey region indicates the MOI-averaged CII activity window estimated for synchronized infections. (C) Viral copy number over time, for the cases modeled in panel A. See Theoretical Methods for detailed information. . . . . 43 xviii

3.6 Replication is required for the lytic outcome and lowers the

MOI required for lysogeny. (A) Model-predicted PR activity per phage during P- (solid blue line) and P+ (dashed blue line) infection at MOI = 1. (B) Cellular Cro concentration, normalized by the lytic threshold, for the cases modeled in panel A. (C) Cellular CI concentration, normalized by the lysogenic threshold, during P- infection at MOI = 2 and 4 (solid light and dark blue lines, respectively), and during P+ infection at MOI = 2 (dashed line). (D) The fraction of cells undergoing lysogeny as a function of average MOI, during bulk infection with P- (circles), P+ (squares), and phages with prolonged CII lifetime (P+ phages infecting hflKC- hosts; triangles). The experimental data was fitted to a model (black lines) where virus-cell encounters follow Poisson statistics, and infection at MOI ≥ MOI* results in lysogeny [1]. The hflKC- strains are either ∆hflK or ∆hflC (see Table A.4). (E) Predicted infection outcome as a function of MOI and viral replication rate (normalized by the fitted replication rate for P+ phage). See Experimental Methods and Theoretical Methods for detailed experimental procedures, data analysis, and modeling...... 45

4.1 The stochastic model recaptures the deterministic model’s trajectories during P- infection. The numbers of (A) cI, cro, and cII mRNA and (B) CI, Cro, and CII protein per cell, during the first 60 minutes following simulated infection by P- phages, at MOI = 1—5. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3)...... 52 xix

4.2 The stochastic model yields outcomes consistent with the deterministic model during P- infection. Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by P- (nonreplicating) phage at varying MOI. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3). Protein concentrations were normalized by the lytic and lysogenic thresholds...... 53 4.3 The stochastic model strongly deviates from the deterministic model’s trajectories during P+ infection. The numbers of (A) cI, cro, and cII mRNA, (B) CI, Cro, and CII protein, and (C) viral copy number per cell, during the first 60 minutes following simulated infection by P+ phages, at MOI = 1—5. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3). . . . 54 4.4 The stochastic model generates outcomes that are inconsistent with the deterministic model during P+ infection. Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by P+ phage at varying MOI. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3). Protein concentrations were normalized by the lytic and lysogenic thresholds...... 55 xx

4.5 The stochastic model is unable to replicate experimentally observed lysogenization frequencies. The measured single-cell frequencies of lysogeny during infection with P- (red markers) and P+ (blue markers) phages, at MOI = 1–4 (data from [2]). Model-predicted frequencies (solid lines) are calculated using (A) decision thresholds used in Chapter 3, and (B) fitted thresholds (see Appendix B, Section B.4)...... 57 4.6 CI-Q trajectories support a lysis-to-lysogeny transition. Model-predicted trajectories, in the CI-Q plane, during infection by P- (left) and P+ phages, for three values of the CII-Q interaction

threshold (KQ): (A) KQ = KPRE (the default value), (B) KQ =

0.5KPRE, and (C) KQ = 2KPRE. Lytic and lysogenic threshold ranges are indicated by the green and red bands, respectively. . . . . 61 4.7 A CI-Q formulation of the decision exhibits only lysogenic outcomes at high MOI. Predicted infection outcome as a function of MOI and viral replication rate (normalized by the fitted replication rate for P+ phage)...... 63 xxi

4.8 MOI-dependent CII inhibition of Q translation generates the observed inverse MOI-dependence of Q during P+ infection. (A) Strength of CII inhibition of Q translation during P+ infection, calculated as the magnitude of the CII-dependent term in

the regulatory function fQ (see Appendix C). Gray shading indicates the MOI-averaged CII activity window, defined as the

period during which PRE activity is greater than 10% of its maximum. (B) Cellular Q concentration during P+ infection at MOI = 1–5. (C) Strength of CII inhibition of Q translation when this interaction occurs during a fixed time window at each MOI (green curve), defined by the MOI-averaged turn-on and turn-off times for

CII activation of PRE (see Appendix A, Figure A.10). (D) Same as panel B, for the case where CII inhibition of Q occurs during the fixed time window (green shaded region) shown in panel C...... 64 4.9 MOI-dependent CII inhibition of Q translation generates the observed inverse MOI-dependence of Q during P- infection. (A) Strength of CII inhibition of Q translation during P- infection, calculated as the magnitude of the CII-dependent term in

the regulatory function fQ (see Appendix C). Gray shading indicates the MOI-averaged CII activity window, defined as the

period during which PRE activity is greater than 10% of its maximum. (B) Cellular Q concentration during P- infection at MOI = 1–5. (C) Strength of CII inhibition of Q translation when this interaction occurs during a fixed time window at each MOI (green curve), defined by the MOI-averaged turn-on and turn-off times for

CII activation of PRE (see Appendix A, Figure A.15). (D) Same as panel B, for the case where CII inhibition of Q occurs during the fixed time window (green shaded region) shown in panel C...... 65 xxii

4.10 The subcellular voting hypothesis. Coinfecting phages make independent decisions to lyse the cell (green) or transition to dormancy (red). Only if all phages vote for dormancy does the infection result in lysogeny...... 68

A.1 The distribution of single-cell MOI within a population. The distribution of single-cell MOI following infection by λ cI 857 Pam80 P1parS. The infection procedure is described in Experimental Methods Section 4.2, and the identification of single-cell MOI in Experimental Methods Section 8.2. Markers and error bars indicate mean ± SEM from samples taken at 1, 2 and 5 minutes following infection (see Table A.9 for detailed sample sizes). The mean MOI calculated from these samples was 1.9 ± 0.5 (mean ± SEM). Red line: Poisson distribution of the same mean, reflecting the assumption of random encounters between phages and bacteria (Kourilsky, P., Mol Gen Genet, 1973)...... 106 A.2 The estimated number of lambda prophage copies in lysogenic cells. (A) Lysogenic strain MG1655 carrying prophage λ cI 857 ind− P1parS and plasmid pALA3047 (expressing CFP-ParB). Cells were grown at 30◦C in LBMM supplemented with 10 µM IPTG. Individual prophages are labeled using CFP-ParB. The imaging procedure is described in Experimental Methods Section 6. (B) Newborn lysogenic cells (“short cells”, defined as the 5–20 percentiles of cell lengths, N = 350) contain two lambda genome copies as expected (Bremer, H. and Churchward, G., J Theor Biol, 1977), while cells about to divide (“long cells”, the 80–95 percentiles of cell length, N = 349) contain four copies. Error bars indicate SEM...... 107 xxiii

A.3 The transcription kinetics of cI, cro and cII across biological replicates. The numbers of cI, cro, and cII mRNA per cell (mean ± SEM), at different times following infection by λ cI 857 Pam80 P1parS. The infection procedure is described in Experimental Methods Section 4.2, and the mRNA quantification in Experimental Methods Section 8.1. (A) Results from dataset 1 (see Table A.9 for detailed sample sizes). The mean MOI was 2.1 ± 0.3 (SEM). (B) Results from dataset 2 (see Table A.9 for detailed sample sizes). The mean MOI was 2.1 ± 0.2 (SEM). Solid lines are splines, used to guide the eye...... 108 A.4 MOI scaling of cI, cro and cII trajectories. (A) The numbers of cI, cro, and cII mRNA per cell (mean ± SEM), at different times following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1—5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. Solid lines indicate splines, used to guide the eye. (B) The values in panel A, scaled by MOI, with  equal 1.16, 0.475, 0.441 for cI, cro and cII, respectively. The optimal value of  for each gene was obtained by minimizing the sum of squared deviation between mRNA values at different MOI at all time points. Solid lines indicate a single spline over all scaled mRNA numbers...... 109 xxiv

A.5 Estimated CI-Cro trajectories following infection. The estimated trajectories in the plane of Cro and CI concentration, during the first 60 minutes following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1—5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. We first calculated the concentration of cI and cro mRNA, [mRNA], by dividing the number of molecules in each cell by the cell volume (approximated as a spherocylinder, with dimensions obtained from the automated segmentation, see textbfExperimental Methods Section 7.1). [mRNA] was then used to estimate the protein concentration of the corresponding species using the relation: d[protein]/dt = translation rate × [mRNA] – decay rate × [protein], with the rates of translation and decay taken from the literature (Zong, C., et al., Mol Syst Biol, 2010; Reinitz, J. and Vaisnys J.R., J Theor Biol, 1990)...... 110 A.6 Model trajectories from ensembles of fits capture mRNA kinetics in biological replicates of P- infection. The numbers of cI, cro, and cII mRNA per cell, at different times following infection at MOI = 1—5 by P- phage (λ cI 857 Pam80 P1parS; see also Figure 2C, main text). (A) Results from dataset 1. (B) Results from dataset 2. Markers and error bars indicate experimental mean ± SEM of each sample. Solid lines indicate ensembles of model fits that yield consistent predictions, obtained from minimizing the objective function described in Theoretical Methods Section 4. See Table A.9 for samples sizes...... 111 xxv

A.7 Model trajectories from an ensemble of fits capture the dynamics of viral copy number during P+ infection. Viral copy number, measured using qPCR, following infection at MOI = 1 by P+ phage (λ cI 857 Sam7 ; see also Figure 2D, main text). Markers and error bars indicate experimental mean ± standard deviation due to qPCR calibration uncertainty. Solid lines indicate an ensemble of model fits obtained from minimizing the objective function described in Theoretical Methods Section 4, using the P- mRNA measurements from dataset 2 (Figure A.6)...... 112 A.8 Model trajectories from an ensemble of fits capture cI and cII mRNA kinetics during infections with various lambda genotypes. The numbers of cII mRNA (normalized by maximum cII mRNA count during P- infection) and cI mRNA (normalized by maximum cI mRNA count during cro-P- infection) per cell at different times following infection at MOI = 1 with various lambda genotypes (data from Shao, Q. et al., iScience, 2018). (A) cII mRNA following infection with P-, cI-, and cro- phages. (B) cII (top) and cI (bottom) mRNA following infection with WT (cI +cro+P+) and cro-P- phages. Solid lines indicate an ensemble of model fits obtained from minimizing the objective function described in Theoretical Methods Section 4, using the P- mRNA measurements from dataset 2 (Figure A.6)...... 113 xxvi

A.9 CII-activated cI expression from PRE is required for a lysis-to-lysogeny transition, while cI autoactivation is not. (A) The model-predicted fraction of total cI mRNA expressed from

PRE (purple) and PRM (red) during the first 60 minutes of infection with P+ phages over a range of MOI. The majority of cI expression

comes from PRE for all MOI simulated. (B) Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by phages in which cI

autoactivation of PRM has been removed. A lysis-to-lysogeny transition is achieved even in the absence of CI-activated

transcription from PRM. (C) Same as panel B, for the case of

infection by a phage in which CII activation of PRE has been

removed. In the absence of cI transcription from PRE, a transition to lysogeny is not observed. Protein concentrations were normalized by the lytic and lysogenic thresholds...... 114

A.10 Single phage PRE activity during P+ infection depends only weakly on MOI. Model-predicted CII-activated cI transcription

from PRE, at the single phage level, shows weak MOI-dependence of

its (A) amplitude, (B) turn-on time, (C), turn-off time (τ PRE), and (D) turn-on duration. The turn-on (turn-off) time is defined as the

first (last) time that PRE activity is greater than or equal to 10% of its maximum value, while turn-on duration is the difference between these times...... 115 xxvii

A.11 Measured kinetics of nascent cI mRNA. (A) The number of nascent cI mRNA per phage genome following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1—5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. Nascent mRNA was quantified based on colocalization of the smFISH and ParB signals, following the method of Wang, M., et al., Nat Microbiol, 2019. The turn-on and turn-off of cI transcription were both fitted to a Hill function with coefficient h = 10 (solid lines). (B) The maximum number of nascent cI mRNA per phage genome as a function of MOI. (C-D) The time of turn-on and of turn-off of cI transcription, estimated using the midpoint of the fitted Hill curves, as a function of MOI. (E) The duration of cI transcription pulse, estimated using the time interval between activation and repression time, as a function of MOI...... 116

A.12 Coinfection delays of τ PRE still result in lysis even when the number of coinfecting viruses is greater than 2. (A) Following infection with a single phage, ∆MOI additional phages infect at t =

τd. (B) Model-predicted infection outcome as a function of

coinfection delay τd and ∆MOI following MOI = 1 infection by P+ phage. Even when ∆MOI is 3-fold larger than MOI*=2, the critical MOI at which the system transitions to lysogeny during simultaneous coinfection (Figures 3.3B and 3.6D, main text), only coinfection

delays below τ PRE (the time when PRE activity falls below 10% of its maximum possible value; see main text) result in lysogeny...... 117 xxviii

A.13 Delayed infection does not result in a second pulse of PRE activity. Model-predicted CII concentration during infection by P+ phages for 4 scenarios: Infection by a single phage (light blue solid line), simultaneous infection by two phages (dark blue solid line), infection by a single phage followed by a second phage at time

τ PRE/3 (dark blue dashed line), and infection by a single phage

followed by a second phage at time τ PRE (dark blue dotted line). The addition of a second phage after a delay (dark blue dashed and dotted lines) does not result in a second pulse of CII. CII

concentration is normalized by the PRE activation threshold, and the MOI-averaged CII activity window (defined as the time span during

which PRE activity is at least 10% of its maximum possible value) is indicated by the gray shading...... 118 A.14 Measured kinetics of nascent cro mRNA. The number of nascent cro mRNA per phage genome following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1–5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. Nascent mRNA was quantified based on colocalization of the smFISH and ParB signals, following the methods of Wang, M., et al., Nat Microbiol, 2019. Solid lines are splines, used to guide the eye...... 119 xxix

A.15 CII activation of PRE during infection by P- phage is only

weakly MOI-dependent. (A) Model-predicted activity of the PRE promoter following infection by P- phage at MOI = 1-–5. Similar to

PRE activity following infection by P+ phage (Figure A.10; also see

Figure 3.4A, main text), PRE activity during P- infection does not show strong MOI-dependence in (B) amplitude, (C) turn-on time,

(D), turn-off time (τ PRE), or (E) turn-on duration. The turn-on

(turn-off) time is defined as the first (last) time that PRE activity is greater than or equal to 10% of its maximum value, while turn-on duration is the difference between these times...... 120 A.16 Modulation of CII’s degradation rate can generate lysogenic outcomes at MOI = 1. Model-predicted infection outcome as a function of MOI and CII degradation rate (normalized by the fitted wild-type (hflKC +) degradation rate). Perturbations which sufficiently decrease the CII degradation rate (hflKC- mutants, black dashed line) result in lysogeny even at MOI = 1...... 121 Tables

A.1 Description of constraints used in fitting ...... 100 A.2 Internal particle swarm optimization parameters changed from MATLAB’s built-in defaults...... 101

3 A.3 Fitted parameters used for main text figures (with V0 = 1µm ). . . . 102 A.4 Bacterial strains used in this study ...... 122 A.5 Phage strains used in this study ...... 123 A.6 Plasmids used in this study ...... 124 A.7 Primers used in this study ...... 125 A.8 DNA oligos used for smFISH ...... 127 A.9 Sample sizes for single-cell infection experiments ...... 131 1

Chapter 1

Introduction

1.1 Cells make information-based decisions

Living organisms use cues from their environment to inform their behavior [3]. A text- book example of this is the seasonal migration of birds [4]. But sensory information guides behavior at the microscopic scale too, in cells [3]. Both uni- and multicellular organisms inhabit environments that can be highly variable [5], and cells have evolved an array of behaviors to suit their environment [6, 7], such as detecting and utilizing alternative carbon sources [8], and responding to stresses, such as temperature [9]. These behaviors are orchestrated by gene regulatory networks [10]—sets of genes con- nected through a web of biochemical interactions. Gene regulatory networks sense, relay, and process information about the environment to guide the cell’s behavior [3]. When the cell’s regulatory network uses cues from the environment to select be- tween mutually exclusive developmental pathways, the process is called a cell-fate decision [11]. Such decisions are ubiquitous in biology, occurring in both uni- [12, 13] and multicellular [14, 15] organisms. As a result of this pervasiveness, cell-fate deci- sions are an active area of research in systems ranging from cancer in humans [16] to sporulation in B. subtilis [13]. Yet efforts to study these decisions are often stymied by the biological complexity of cells [17]. Gene regulatory networks can contain thou- sands of genes [18], and the interactions between genes can occur through a diverse array of mechanisms [19]. While high-throughput DNA [20] and RNA [21] sequenc- ing techniques enable fast reading of entire genomes and efficient measurement of gene expression dynamics, mapping the regulatory interactions of networks remains challenging [22]. Simple, well-studied model systems provide attractive alternative 2 platforms in which to investigate biological processes, such as cell-fate decisions, and hopefully gain insights which may be broadly applicable [23, 11, 24]. One such model system for studying cell-fate decisions is the lysis/lysogeny decision of bacteriophage (virus) lambda [11].

1.2 Phage lambda is a model system for studying cell-fate decisions

Bacteriophage lambda is a virus which infects E. coli, and has long served as a model system in molecular biology [11]. Lambda has been used to obtain insights into multiple fundamental biological processes, such as transcriptional regulation, DNA replication, and homologous recombination [19]. The post-infection cell-fate decision is arguably the most well-known aspect of lambda biology, wherein the infected cell selects one of two developmental pathways: rapid viral replication leading to cell death (lysis), or viral dormancy (lysogeny) [12]. Despite the relative simplicity of this system, the lambda decision serves as a paradigm for cell-fate decisions driven by a gene regulatory network [11]. This paradigmatic status is at least in part due lambda’s lengthy tenure in the field—the lysis/lysogeny decision has been studied for nearly 70 years [25]. These decades of investigation have yielded a near complete understanding of the players and interactions in the decision [19], as well as the identification of environmental and host- specific inputs which impact the decision, such as temperature, the metabolic state of the cell, and the number of coinfecting viruses [19]. Our relative understanding of this system has made it common practice to frame other cell-fate decisions in terms of the lambda decision [5, 26, 27]. And lambda is an ideal candidate for finding insights with broad applicability, as it shares multiple features with cell-fate decisions in other systems [11]. 3

1.3 The lambda regulatory network drives a counting-based decision through unclear mechanisms

In spite of the breadth of work performed on lambda, our understanding of the ly- sis/lysogeny decision is still far from complete. Basic questions persist, many of them centered around the impact of viral copy number on the decision [27]. It’s been under- stood since the earliest works on lambda that the decision outcome is strongly biased towards lysogeny as the number of coinfecting viruses (also called the multiplicity of infection, or MOI) increases [25, 1]. Yet despite hundreds of papers published on the lysis/lysogeny decision [28], we still lack clarity on the mechanisms behind this ‘counting’ behavior, even at the basic level of identifying which genes in the lambda network respond to viral copy number [2, 29, 27]. Additionally, it’s unclear how the viral network can count MOI when viral copy number is continuously increasing due to viral replication [12]. The principal reason for our lingering lack of understanding is the insufficient resolution of previous experimental approaches used to study the lambda decision. While experiments have identified correlations between environmental inputs and decision outcome [12], and individual regulatory interactions have been isolated and described in some detail [30], studies have generally struggled to demonstrate how the network processes these inputs. In the particular case of MOI, experiments have been unable to measure the network dynamics during infection at the single-cell level as a function of the number of coinfecting phages. At best, previous attempts have been constrained to measuring responses in bulk experiments with no control over viral coinfection at the single cell level [29], or have only been able to measure gene expression at the single-cell level at MOI=1 [2]. In the absence of precise measurements of the network dynamics and their MOI- dependence, a multitude of hypotheses have been put forward explaining how the decision circuit counts, often based upon mathematical models [31, 32, 33, 34, 2, 35, 36, 37, 38, 39, 40]. Owing to the poor resolution of previous studies that measured 4 gene expression during infection, attempts to model the lambda decision have (with one exception—[34]) been forced to choose between several undesirable options when parameterizing. They must either: (i) arbitrarily choose values from ranges estimated from general literature on cellular biology, (ii) glean parameter values from a mosaic of experiments on lambda performed in different conditions, or (iii) fit to measurements of outcome frequencies, which do little to directly constrain network dynamics. These pitfalls raise questions about the validity of model predictions, as the output of models of network dynamics can be strongly parameter dependent [41].

1.4 Aims of this work

In the following chapters, I will describe how we gained insights into the mecha- nisms behind counting in the lambda cell-fate decision by combining novel single- cell resolution experiments and mathematical modeling. I will begin by reviewing decision-making in temperate phages (Chapter 2), to identify both what the open questions in the field are and what particular questions in the lambda decision are still unresolved. This review illuminates the breadth of behaviors observed in phage decision-making, but also illustrates that features of the lambda decision are shared by other phages. In Chapter 3 I will describe how we 1) elucidated how the lambda network mea- sures viral copy number, and 2) identified how a counting-based decision is achieved in the presence of active viral replication. To resolve these gaps, I designed a model of the network dynamics during the lambda decision, and calibrated this model with single-cell measurements from the Golding lab of gene expression during infection at known MOI. While a recent work used measurements of network dynamics to con- strain modeling [2], their measurements were only performed at MOI = 1. To our knowledge, our work represents the first model of the lambda decision calibrated by MOI-gated measurements of the mRNA kinetics of key players in the lambda decision for a range of MOI. These measurements of mRNA kinetics, combined with qPCR 5 measurements of viral replication and previously published data [2, 42, 43, 44, 30, 12], provide strong constraints for model behavior. I used this calibrated model to demon- strate that the lambda network senses MOI in an early window during infection, and that viral replication actually facilitates, rather than hinders, a counting-based tran- sition from lysis to lysogeny. In Chapter 4, I discuss extensions and future directions for this work. First, I have been working to extend our analysis to the level of individual cells by refor- mulating the model to incorporate the intrinsic noise of biochemical reactions at low copy number. These efforts led me to examine the role of another lambda gene, Q, and indicate the need for future experiments. Finally, I consider how viral replication may provide a mechanism for coinfecting viruses to ’vote’ on the decision outcome, a previously hypothesized feature of the lambda decision motivated by single-cell experiments [45]. 6

Chapter 2

Decision making in temperate phages

This chapter consists of the following book chapter:

Golding, I., Coleman, S., Nguyen, T.V.P., Yao, T. (2021). Decision making in tem- perate phages. In Bamford, D. & Zuckerman, M. (Eds.), Encyclopedia of Virology (4th ed.). Academic Press

My contributions: (1) Researched decision-making in temperate phages besides bacteriophage lambda, and provided references for the chapter. (2) Contributed ma- terial for Figure 2.2. (3) Co-wrote book chapter. 7

2.1 Introduction

The defining lifestyle of most bacteriophages, analogous to the behavior of higher viruses, is one in which infection is followed by rampant viral replication and the release of numerous mature progeny, often accompanied by death of the host cell (lysis). However, a subset of phages, denoted as temperate, are capable of an al- ternative lifestyle called lysogeny, where the violent cataclysm is replaced by viral dormancy: Following infection, the phage genome is maintained inside the bacterial host—either integrated (as a prophage) into the bacterial chromosome, or replicating extra-chromosomally—with all virulent functions shut off. The dormant phage, now an integral part of the bacterial cell, is inherited from generation to generation. How- ever, while dormant, the phage typically maintains the potential for lytic induction: a switch back to the virulent pathway in response to specific signals indicating stress to the host cells. Multiple aspects of the lysogenic lifestyle are the subject of current studies. These include the physiological costs and benefits, to both phage and host, of viral dormancy, as well as the ecological, medical, and evolutionary consequences of lysogeny. Here we focus on a single element: The decision between lysis and lysogeny, made by temperate phages upon infection of the host. In particular, we will devote most of our attention to the decision by bacteriophage lambda, which infects Escherichia coli (Fig. 2.1). Through more than half a century of genetic, biochemical, and biophysical studies, lambda has become arguably the best characterized biological system, albeit one that still presents many open questions. As we review the lambda lysis/lysogeny decision, we will also provide examples for some of the ways that other temperate phages make this choice. In light of the overwhelming diversity of phage lifestyles, it is certain that many more ways in which phages pursue their lysis/lysogeny decisions remain to be discovered. Owing to the relative compactness and tractability of bacteriophages, many of their functions have been studied not only for their intrinsic value, but also as sim- 8

Figure 2.1 : The post-infection decision of bacteriophage lambda. Following infection of an E. coli cell, a binary choice is made between lysis, defined by rampant viral replication and cell death, and lysogeny, in which the viral DNA is integrated into the host’s genome to become a dormant prophage. The lysogenic state is stably maintained, but a switch back to lysis can be induced by cellular damage. Adapted with permission from Golding, I., 2016. Single-cell studies of phage λ: Hidden trea- sures under Occam’s rug. Annual Review of Virology 3 (1), 453–472, permission conveyed through Copyright Clearance Center, Inc. 9 plified models for the behavior of higher biological systems. This is also the case for the lysis/lysogeny decision, which serves as a paradigm for the way genetic circuits, receiving external inputs, make binary cell-fate choices. In that role, for example, the decision by phage lambda has provided insights about the transition in and out of dormancy by HIV, and, beyond viruses, regarding the process of cellular differ- entiation during metazoan development. At the heart of lambda’s ability to inform us on higher organisms are common features of binary decision circuits across bio- logical systems, notably, the utilization of auto-regulating, fate-determining genes to provide high cell-state stability while simultaneously allowing efficient fate switching in response to external signals. As part of its paradigmatic role, lambda has also served as a testbed for the idea of creating a quantitative description, formulated in mathematical terms, for the func- tion of the genetic circuit, with the goal of predicting the decision outcome for given initial conditions. While efforts in this direction have yielded significant progress, they have also been limited by the phenomenon of cellular individuality, whereby geneti- cally identical cells, in a uniform environment, nevertheless end up pursuing different paths. To explain the indeterminacy of single-cell choice, researchers invoked the effect of stochastic biochemical fluctuations (“noise”) on the decision circuit. With this concept, too, lambda has paved the way for the elucidation of cellular stochastic- ity and its consequences across the fields of microbiology, development, ecology and medicine.

2.2 The post-infection decision of phage lambda

A decision between lysis and lysogeny first takes place following lambda infection, upon entry of the viral genome into the E. coli cell (Fig. 2.1). A genetic circuit encoded by the phage, integrating inputs from the host (and, indirectly, from the extracellular environment), converges on one of the two developmental pathways. If lysis is chosen, the decision is irreversible and, hence, the role of the decision 10 circuit is complete. If, however, the choice is lysogeny, that fate needs to be actively maintained over the long term, using a small subset of the decision gene network. In addition to repressing all virulent functions, this maintenance circuit (known as a genetic “switch”) perpetuates a continuous version of the lysis/lysogeny decision: at any given time, whether to maintain stable dormancy or, if cellular conditions change, undergo lytic induction, switching back to the lytic pathway by relieving the repression of virulent functions. The choice between lysis and lysogeny can be seen as one of timescale, namely, whether to reproduce (and kill the host) immediately, or at a later, yet to be de- termined, time. An optimal choice, one which maximizes—in the long term—the expected number of progeny, requires knowledge of the state of the infected cell, e.g., how likely it is to support successful viral reproduction, as well as its environment, e.g., what the chances are that newly-released phages will find additional targets to infect. Below we discuss how lambda and a few other phages measure these critical parameters. From a theoretical point of view, the optimization problem that underlies decision making by temperate phages continues to be an area of considerable interest. The genetic circuit governing the lambda decision is depicted in Fig. 2.2. The de- cision involves a dense network of regulatory interactions between phage genes, as well as multiple inputs from various host functions. The regulatory interactions involve diverse molecular mechanisms, modulating all stages of gene expression: transcription initiation (through transcription factors, alternative promoters, and interference be- tween neighboring promoters) and elongation (anti-terminators), post-transcription (antisense RNA), translation, and post-translation (protein lifetime). The role of some of these interactions in the decision process has been elucidated through ge- netics, biochemistry, and mathematical modeling of the circuit. The utility of other interactions, however, is less clear. It is possible that those come into play under in- fection conditions that are not emulated by standard laboratory conditions, serving to increase the robustness of the decision to environmental and biochemical fluctuations. 11

Figure 2.2 : Key lambda genes and host factors involved in the ly- sis/lysogeny decision. Regulatory interactions between the various nodes rely on diverse molecular mechanisms, at the level of phage DNA, mRNA and proteins.

To gain insight into the decision process, it is helpful to follow the cascade of transcriptional events taking place on the lambda genome following infection, and how these events diverge en route to each of the two possible outcomes (Fig. 2.3). Upon entry of the viral genome into the E. coli cell, transcription begins from the

left (PL) and right (PR) early promoters. Transcription is initially attenuated at

the tL1 and tR1 terminators, such that only a single protein is expressed from each

promoter: N from PL and Cro from PR. One of these proteins, N, is an anti-terminator

that allows readthrough at tL1 and tR1, as well at another terminator, tR2, further downstream of PR. This leads to the expression of additional lambda genes. These “delayed early” genes include those that allow progression of the lytic pathway: O and P, required for phage genome replication, and Q, which controls the expression of multiple lytic genes. However, cII and cIII, whose products drive the establishment of lysogeny, are also produced from those same left and right transcripts. If the Q protein accumulates to a sufficient level, it abrogates termination at the 12

Figure 2.3 : The cascade of transcriptional events during the lambda ly- sis/lysogeny decision. Regulatory elements and their interactions are depicted on the relevant region of the lambda genome. The expression pattern of early genes is qualitatively similar irrespective of the eventual fate. After a decision has been reached, different sets of genes are expressed to execute the lysis/lysogeny choice. 13

tR’ site, to enable expression from the late lytic promoter PR’ of genes responsible for the production of new viral particles and lysis of the host cell, and thus completion of the lytic cycle. For the alternative route of lysogeny to be taken, Q production needs to be reduced. This is achieved by CII, which activates the PaQ promoter to produce an antisense transcript to Q, resulting in inhibition of Q expression and thus pre- venting the expression of the late lytic genes. In addition, CII promotes the lysogenic pathway by activating the PI promoter, whose product Int catalyzes integration of the phage genome into the bacterial chromosome. Finally, CII activates the repression establishment promoter PRE, to produce CI. CI shuts off transcription from PL and

PR and regulates its own transcription, now from the repression maintenance pro- moter, PRM. In the lysogenic state, PRM expression of CI is the only transcriptional activity from the decision network of the dormant prophage. What makes the post-infection decision elusive, even after decades of meticulous interrogation, is that, qualitatively, the expression patterns of early genes are similar in lytic and lysogenic cells, and therefore appear unpredictive of the eventual outcome.

Consider cII, the key driver of lysogenic choice. Its expression from PR takes place in a limited time window of 15 min, regardless of the eventual choice. Whether this transient expression will result in lysogeny depends on the exact timing and on the maximal CII level obtained. CII kinetics are regulated at multiple levels—at the promoter (PR) and terminator (tR1), as well as the mRNA and protein lifetimes. This multi-layer regulation provides the means by which the lysis/lysogeny decision is modulated by host and phage parameters, as discussed later. Whereas the decision by lambda is by far the one best characterized, progress has also been made towards understanding the decision circuits of other phages. In multiple cases, the decision network is quite reminiscent of lambda’s, with the cI - cII -cIII module conserved, and an orchestrated gene expression cascade taking place. Even in phages that are otherwise very different from lambda, a dual repressor motif similar to the cI /cro pair (discussed in more detail below) is often present, and an 14

auto-regulatory viral “repressor”, which both establishes and maintains the lysogenic state, appears to be near universal. Among temperate phages whose decision process has been studied in some detail, a few offer slight variations on lambda, for example, P22, a “lambdoid” phage (i.e., one with a similar genome architecture to lambda), where the Cro-analog (ant) directly inactivates the CI-analog (c2 ) by binding to it and preventing it from effectively binding to DNA. Others exhibit dramatically different behavior. P4, for example, is a “satellite phage”, a genetic element that requires other viruses for its own propagation. Rather than having a global repressor, which establishes and maintains lysogeny by regulation of transcription initiation, the outcome and maintenance of the decision in P4 are driven by control of transcription termination, mediated by RNA-RNA interactions. The default pathway in P4 appears to be lysogeny, with the alternative state, in the absence of a co-infection by a helper phage, being that of a multi-copy plasmid. In that state, infection by a helper phage induces the transition into lysis.

2.3 Counting by infecting phages

The perceived role of the decision circuit, as stated above, is to choose between lysis and lysogeny based on the conditions of infection. A key question is, thus, how these conditions are sensed by the infecting phage and processed by the decision circuit to yield an optimal outcome. This question is far from settled. To begin with, which aspects of the infection event are pertinent to the outcome? The parameter best characterized in terms of its effect on the lambda lysis/lysogeny choice is the multiplicity of infection (MOI), namely, the number of phages co-infecting the cell. It was found long ago that, the higher the MOI, the higher the probability of lysogeny. Fig. 2.4 depicts the results of an experiment measuring the relation between the two observables. A known number of bacteria is mixed with varying concentrations of phages, and, once infection is allowed to proceed, the number of resulting lysogens is measured using selection for an antibiotic marker that was engineered into the viral 15 genome. To interpret the experimental results, the measured values are compared to a simple mathematical model, in which random phage-bacteria encounters (following mass-action probability and Poisson statistics) result in lysogeny if the number of phages co-infecting a single cell reaches some number m*. As seen in Fig. 2.4, the measured data is consistent with a value of m* = 2, i.e., a scenario where infection by a single phage leads to lysis, whereas simultaneous infection by two or more phages results in lysogeny.

Figure 2.4 : The dependence of lysogenization on the multiplicity of in- fection: bulk data. A known number of E. coli bacteria is infected with varying concentrations of lambda phage, and the number of resulting lysogens is measured using selection for an antibiotic marker that was engineered into the viral genome. The experimental trend is reproduced by a simple mathematical model, where infec- tion by a single phage leads to lysis, whereas simultaneous infection by two or more phages results in lysogeny.

The notion that infecting phages are able to count their numbers in the cell, and then decide on the mode of action based on that number, is intriguing both 16 mechanistically—how do viruses count?—and in terms of its utility—why do they do so? In terms of mechanism, it is commonly held that counting is mediated through the level of CII reached during infection. The MOI affects both CII production (through increased gene dosage) and degradation (through production of CIII, which protects CII from degradation by FtsH, see below). As described earlier, high CII level triggers lysogeny by driving the expression of the integrase and CI, and inhibiting late-lytic gene expression from PR. However, an alternative hypothesis, motivated by recent single-cell experiments and mathematical modeling, posits that phage counting simply reflects the number of cI gene copies, whereas CII levels respond only weakly to MOI, possibly due to the auto-repression of PR by Cro. If enough CI accumulates, it will then activate its own transcription from PRM and shut down expression of the lytic genes. Elucidating phage counting is complicated considerably by the fact that the lambda genome begins replicating shortly after infection, i.e., during, not after, the lysis/lysogeny decision. That early replication plays a role in the decision is evidenced by the fact that replication-deficient lambda mutants require a higher number of initial viruses to achieve lysogeny, compared to wild type phages. As for the biological utility of choosing lysogeny at higher MOI, the common in- terpretation is that the number of co-infecting phages is used by lambda as a proxy for the ratio of phage-to-bacteria abundance in the surrounding environment. That ratio, in turn, serves to assess the chances of successful infection by the next genera- tion of phages, should the lytic pathway be chosen. Specifically, high MOI indicates that phages outnumber bacteria and that, therefore, releasing more phages into the environment is futile, since they are unlikely to find new bacterial targets. Hence, high MOI advocates lysogeny. Conversely, low MOI indicates the availability in the envi- ronment of yet-uninfected bacteria, thus promoting the release of more viral particles through lysis. Consistent with the idea that lambda measures the multiplicity of infection in order to assess the abundance of uninfected bacteria in the environment, recent studies 17 found additional ways by which phages can infer this kind of information. During initial rounds of infection of Bacillus subtilis, phage phi3T expresses the genes aimR and aimP. AimR binds to the phage DNA and activates the transcription of aimX, which blocks the lysogenic pathway in a mechanism not yet elucidated. The lytic pathway is thus favored in the initial infections. Meanwhile, AimP, a short peptide, is secreted into the extracellular medium, where it accumulates and is taken up by uninfected cells. When some of these cells are later infected, the intracellular AimP, now at high concentration, binds to and inactivates AimR, resulting in repression of aimX expression. Consequently, lysogeny is favored in later rounds of infection. Using this intercellular communication system (termed “arbitrium”), infecting phi3T phages are able to record past infections of other cells and tune their lysis/lysogeny decision based on this knowledge. The ability to hijack the host’s quorum sensing system is also found in VP882, a temperate phage that infects Vibrio cholerae and other Vibrio species. The outcome of infection by VP882 depends on a repressor (Gp59) that inhibits the expression of the lytic regulator (Gp62), thus promoting the lysogenic pathway. During infection of

Vibrio cholerae, the phage also expresses VqmAPhage, a homolog of the host’s endoge- nous quorum sensing receptor, VqmA. When bound by Vibrio’s autoinducer, DPO,

VqmAPhage promotes the expression of an antirepressor (Qtip), which sequestrates the repressor Gp59, allowing genes involved in the lytic pathways to be expressed. Consequently, when the bacterial density is high, the increased DPO level in the medium promotes the lytic pathway of infecting VP882. Note that, in contrast to phi3T above, which relies on a phage-specific secreted molecule, here, VP882 assesses the bacterial density via an autoinducer encoded by the host. The cases described above, in lambda and other phages, as well as the ecologi- cal argument mentioned, all support the idea that increased phage-to-bacteria ratio promotes lysogeny. However, whether this rule applies universally is still unclear. The propensity for lysogenization by bacteriophage P1, for example, is reported to 18 be insensitive to the number of viruses infecting the host cell. In infections by phage Mu, the probability of lysogeny appears to decrease with the multiplicity of infection, although this trend may reflect the virus’ toxicity to the host at high MOI. There is also an ongoing debate whether, outside the artificial lab environment, bacterial density is correlated – positively or negatively – with the occurrence of lysogeny, and how to interpret the observed trends. Ecological studies of lytic in- fections support a “kill the winner” model, in which viral infection increases host diversity by preventing overabundance. However, the dynamics of temperate viruses are much harder to interpret, and the relationship between host density and outcome frequencies is unclear. Previous studies of prophage induction found that lysogeny is more prevalent at low host density, consistent with the picture above of increased lysogenization at high MOI. A more recent work, using metagenomics analysis, re- ported an inverse trend, but the interpretation of these newer findings is a subject of some controversy. Whereas the multiplicity of infection is the best-characterized driver of the lambda lysis/lysogeny decision, it is definitely not the only one. Recall that multiple host factors interact with the decision circuit (Fig. 2.2 above). One such factor is FtsH, a membrane-bound ATP-dependent protease. During lambda infection, FtsH degrades CII and thus impacts the choice between lysis and lysogeny. Part of FtsH’s influence comes about through the MOI, specifically, the dosage-dependent production of CIII, which is believed to protect CII by serving itself as a target for FtsH. However, beyond the response to MOI, the level and activity of FtsH are regulated by the physiological state of the cell, thus providing a means for the condition of the host cell to inform the lysis/lysogeny decision. For example, the increase in lysogenization at low temperature can be attributed to a decrease in FtsH levels, as well as an increase in the thermodynamic stability of CII, to which FtsH is highly sensitive. Temperature also impacts the decision circuit through its effect on another bacterial protease, Lon, which is expressed in a temperature-dependent manner and targets 19 the N anti-terminator. Two other cellular sensors, cyclic adenosine monophosphate (cAMP) and guanosine tetraphosphate (ppGpp), have also been reported to affect the lysis/lysogeny decision, possibly by inhibiting FtsH. Cellular state also influences the lambda decision through RNase III, whose levels are modulated by the E. coli growth rate. RNase III promotes degradation of cII transcripts, stimulates ribosome binding and translation initiation of cIII, and blocks auto-repression of N translation during early infection, thus affecting multiple nodes of the decision network. The ways in which the decision circuit assesses the state of the cell via these and other physiological sensors remain a promising direction for future interrogation.

2.4 The view from the single cell

Most of what we know about lambda’s post-infection decision comes from studies that used traditional genetic and biochemical assays, performed in bulk cultures, and thus involving the averaging of all measured observables over millions of cells. But these individual cells may, in fact, exhibit very different phenotypes. Over the last decade, traditional bulk assays have begun to be supplemented by microscopy-based studies, in which the infection process is followed in real time, at the level of individual cells and phages. Fig. 2.5(A) shows an example of such an experiment. Here, the lambda capsid was labeled using multiple copies of a fluorescent protein, such that each phage particle appears under the microscope as a diffraction-limited spot. The infected cells were simultaneously imaged using phase contrast microscopy. Time- lapse images show the infection and its outcome for two individual cells. The first cell, infected by a single phage, proceeds to produce more viral proteins (green) and lyse within two hours. The second cell, co-infected by three phages, survives to grow and divide. That cell has chosen the lysogenic pathway, as indicated by the production of a red fluorescent protein, here expressed from the lysogeny establishment promoter,

PRE. Following many infection events in this manner allows one to determine how the 20

Figure 2.5 : The lysis/lysogeny decision at the single-cell level. (A) Images from a live-cell movie following the fate of two E. coli cells, infected by fluorescently- labeled lambda phages. The upper cell, infected by a single phage, proceeds to produce new viral particles and undergo lysis. The lower cell, co-infected by three phages, enters lysogeny, as indicated by a fluorescent reporter for PRE activity, and proceeds to divide normally. (B) The fraction of cells undergoing lysogeny as a function of the multiplicity of infection, as measured from 41,000 infection events. In contrast to the original modeling of the bulk data (Fig. 2.4 above), the single- cell curve rises gradually, suggesting that the MOI dependence is probabilistic rather than deterministic. (C) Incorporating the effect of intracellular viral concentration (MOI divided by cell volume) captures the experimental data and yields a decision curve that is markedly more step-like. Adapted from Zeng, L., Skinner, S.O., Zong, C., et al., 2010. Decision making at a subcellular level determines the outcome of bacteriophage infection. Cell 141 (4), 682–691, Copyright 2010, with permission from Elsevier. 21 decision outcome—lysis or lysogeny—depends on the infection parameters, such as the MOI. Fig. 2.5(B) shows that the fraction of cells choosing lysogeny increases with MOI, a trend consistent with the observations in bulk. However, in contrast to our original interpretation of the bulk data (Fig. 2.4 above), the single-cell data suggests that the MOI dependence is probabilistic rather than deterministic: At MOI of 2, for example, an infected cell has about a 50% chance of going either lytic or lysogenic. In other words, when we observe a cell co-infected by two lambda phages, we have no way of telling what route will be chosen! We are thus confronted with the indeterminacy of single-cell behavior: genetically identical cells, all subject to the same environment, exhibiting different phenotypes from each other. This phenomenon is observed throughout biology, and its origins are a subject of intensive interrogation. According to the prevailing picture, cellular individuality reflects the inherent randomness of biochemical reactions in the cell. In this view, the unavoidable fluctuations in molecular copy number and in the timing of events render the lambda decision “noisy” and unpredictable, rather than precise and deterministic. The plausibility of this argument was first demonstrated using a computational simulation of the lambda decision circuit, showing that fluctuations in biochemical reactions can result in diverging cell fates among infected cells. The concept of noise-driven decisions then evolved and was used to explain cell-fate inde- terminacy in higher systems, including the transition in and out of HIV latency, as well as the differentiation and reprogramming of metazoan cells. But the fact that we can describe cell fate decisions probabilistically does not necessarily mean that we should settle for such a narrative and give up seeking a deterministic description of the decision process. While biochemical stochasticity is undisputedly present, automatically attributing all cellular indeterminacy to un- knowable “noise” may be taking the easy path. One must consider the alternative hypothesis, which is, that our inability to predict the decision outcome reflects a fail- ure to account for additional cellular variables that have a deterministic effect on the 22 decision. So long as these “hidden variables” remain unknown to us, the decision will appear more random that it truly is, and our understanding of it remain limited. And, in fact, a number of lambda studies suggest that incorporating additional variables can reveal a more precise decision at the single-cell level. It was first found that, for a given MOI, smaller cells are more likely to be lysogenized than larger ones. This should not surprise us, since decreasing the size of the infected cell is, to a first approximation, the same as infecting with a larger number of phages: both result in an increased concentration of viral gene products in the cell. Detailed analysis revealed that a unique arithmetic combination of the MOI and cell size yields a more step-like (and therefore, more deterministic) probability of lysogenization (Fig. 2.5(C)). The way in which the infection parameters combine to yield a sharp decision curve points to a nonlinear interaction between the co-infecting phages as they converge on the cell’s fate. Elucidating the nature of this interaction will require characterizing the spatiotemporal dynamics and genetic activity of individual phages within the infected cell. Fluorescent reporters for phage capsid, genome, RNA and protein products, needed for such an investigation, are now becoming available (Fig. 2.6).

2.5 The decision to remain dormant

If, following infection, the lysogenic route is chosen, control of cell fate is then handed over from the post-infection decision circuit to a smaller circuit, whose role is to maintain dormancy by repressing all virulent functions. The maintenance circuit must also be able to trigger a switch back to lysis (induction) when cellular conditions change. Thus, the dormant virus continuously reevaluates its lysis/lysogeny decision. Some elements of lysogenic maintenance in lambda and other phages are covered by a separate article in this encyclopedia (Shearwin). Here we focus on the decision- making aspect of the maintenance circuit. In lambda, lysogenic maintenance is handled by a subset of the post-infection decision network discussed above. The smaller maintenance circuit, known as the 23

Figure 2.6 : Detecting the transcriptional activity of individual lambda phages. (A) Each phage is detected through the binding of fluorescently-tagged ParB proteins to the parS sequence, engineered into the lambda genome. mRNA molecules transcribed by the phage are simultaneously detected using single-molecule fluorescence in situ hybridization (smFISH). (B) Four lambda genomes (cyan) inside a single infected cell, at 10 min after infection. Individual phages vary in their tran- scriptional activity, with one transcribing cro (green) and two others producing cI (red). 24

lambda “switch”, consists of two phage genes, cI and cro, transcribed respectively from two diverging promoters, PRM and PR (Fig. 2.7). The two gene products, CI

and Cro, compete for binding to six operator sites (OR1–3 and OL1–3) that regulate

PRM and PR transcription, resulting in mutual repression by the two proteins. In the prophage state, high cellular level of CI represses transcription from PR and PL, thus maintaining viral dormancy. The lysogenic state is further stabilized by the formation of a DNA loop between OR1–3 and OL1–3, secured by oligomerization of CI dimers bound at the two loci. Perturbations that reduce the level of CI (such as activation of the bacterial SOS response, discussed below), can lead to lytic induction.

In this process, the inhibition of PR and PL is relieved, leading to transcription of early lytic genes, including cro. Cro then represses PRM, leading to further reduction of CI level and allowing the lytic cascade to proceed. Despite decades of meticulous studies, recent experiments continue to reveal new features of the lambda maintenance circuit, such as the role of mechanical coupling between transcription, DNA supercoiling, and looping, and how this coupling may affect the stability of the lysogenic state. As in the case of infection, the phage (now, prophage)-encoded circuit requires input from the bacterial host in order to sense the state of the infected (now, lysoge- nized) cell and use that information to choose optimally between lysis and lysogeny. Specifically, during lysogenic maintenance, the role of host input is to alert the prophage when the bacterial cell is in danger, indicating that it is time to escape the host through the lytic pathway. Lambda receives this information via E. coli’s SOS system, which, in response to cellular DNA damage, halts progression of the cell cycle and triggers DNA repair and mutagenesis. Under normal growth, expression of the SOS response genes is repressed by LexA. However, in the presence of DNA dam- age (due to, e.g., UV radiation), regions of single-stranded DNA accumulate, leading to recruitment and activation of RecA. Activated RecA facilitates LexA self-cleavage, de-repressing SOS genes, including RecA itself. Phage lambda is coupled to the SOS response through the self-cleavage of CI by activated RecA. The consequent drop in 25

Figure 2.7 : The maintenance of lambda lysogeny. (A) The lysogenic state is maintained by a regulatory circuit consisting of CI and Cro, expressed from the PRM and PR promoters, respectively. CI and Cro compete for binding at six operator sites (OR1–3 and OL1–3), to determine which promoters (PRM, or PR and PL) are active, and thus decide whether lysogeny is maintained or, instead, lytic genes induction takes place. Two examples of binding configurations are shown. On the left, CI dimers bind to four operator sites, resulting in DNA looping that ensures repression of PR and PL during lysogeny. On the right, binding of Cro to OR3 represses transcription of CI from PRM and allows lytic genes to be expressed. (B) The regulatory interactions between CI and Cro form a bistable switch. The system can alter its state in response to a large perturbation, such as depletion of CI by RecA, leading to lytic induction, but is immune to small perturbations. Adapted with permission from Golding, I., 2016. Single-cell studies of phage λ: Hidden treasures under Occam’s rug. Annual Review of Virology 3 (1), 453–472, permission conveyed through Copyright Clearance Center, Inc. 26

cellular CI concentration leads to the relief of cro repression, activation of the lytic pathway, and prophage induction. Many other temperate bacteriophages are also induced following treatment with UV radiation or DNA-damaging agents. However, some, like P2, appear to be non-inducible, and immune to the bacterial SOS system. The cI /cro pair serves as a canonical example for a so-called “toggle switch”, a genetic module exhibiting two stable states, here corresponding to lysogeny and to the onset of lysis. Despite consisting of only two genes, the lysogeny maintenance circuit of lambda captures key features of cell-fate choice, as observed across the spectrum

of biological complexity. Through the use of feedback (PRM autoregulation by CI), the system achieves extremely high stability in the absence of external perturbations, with fewer than one spontaneous switching event per 106 cell doublings. At the same time, almost 100% of lysogenic cells switch to lysis in response to an inducing signal. These properties have made the lambda maintenance circuit an attractive starting point for understanding cellular differentiation and reprogramming in metazoans. In particular, lambda has served as a fertile test ground for the attempt to formulate a detailed biophysical description of cellular behavior, in the form of a mathematical model that uses the known molecular interactions to predict the resultant cellular phenotype. This effort has been, at least partly, successful. For example, a ther- modynamic model can be written, describing the different binding configurations of

CI at the operator sites that control transcription from PRM, and this model used

to predict the regulatory curve relating CI concentration in the cell to PRM activity. The theoretically predicted curve shows good agreement with experimental measure- ments of this regulatory relation (Fig. 2.8). The theoretical calculations can next be utilized to estimate the amount of CI protein present in a lysogenic cell, by requiring that CI production is exactly balanced by CI elimination (via dilution, due to cell growth and division) (Fig. 2.8). The eventual test for a theory of the lambda switch is to successfully predict the key phenotype, namely, whether a given cell will remain in the lysogenic state 27

Figure 2.8 : A theoretical biophysical model captures PRM regulation by CI. The regulatory curve relating CI concentration in the cell to PRM activity can be predicted from a thermodynamic description of the possible binding configurations of CI at the OR1–3 and OL1–3 operator sites. The theoretically predicted curve agrees with the experimental measurement of the regulatory relation. The average amount of CI protein present in each lysogenic cell can be estimated by requiring that CI production is exactly balanced by CI elimination (via dilution, due to cell growth and division). Adapted from Sep´ulveda, L.A., Xu, H., Zhang, J., Wang, M., Golding, I., 2016. Measurement of gene regulation in individual cells reveals rapid switching between promoter states. Science 351, 1218–1222, reprinted with permission from AAAS; and from Golding, I., 2011. Decision making in living cells: Lessons from a simple system. Annual Review of Biophysics 40 (1), 63–80 with permission conveyed through Copyright Clearance Center, Inc. 28 or, instead, switch to lysis. In attempting to answer this question, we are again confronted with the challenge of cellular individuality: In the absence of an external signal, only one out of a million lysogens in a growing culture will spontaneously switch. Can we predict which cell this will be? It is widely believed that the answer is negative, and that we may only aspire to predict the probability of induction, not the actual fate of an individual cell. This is because spontaneous induction is considered a stochastic process, driven by random fluctuations of CI levels in the cell.

Small drops in CI number will be corrected by the negative feedback in the PRM-CI circuit, raising CI level and thus reverting to the mean. However, rare, larger drops will overcome the feedback and lead to de-repression of PR, Cro production and onset of the lytic pathway. In this picture, spontaneous lytic induction is analogous to the way random thermal motion drives a physical system to transition from one stable state to another (Fig. 2.7 above). The challenge of theoretically predicting the behavior of the cI /cro switch dwarfs in comparison to the larger goal of predicting cell fate following infection, when the full decision network (Fig. 2.2 above) comes into play.

2.6 Conclusion

The prevailing narrative for the lambda lysis/lysogeny decision, both following infec- tion and during lysogenic maintenance, offers only a probabilistic prediction, rather a deterministic one, as to which path an individual cell will choose. This proba- bilistic point of view is similarly applied, further afield, to the choice of latency by mammalian viruses and to cellular differentiation and reprogramming. These deci- sions are all held to be indeterminate, noise-driven processes. Bacteriophage lambda, where this picture originally emerged, also bears the potential to challenge the proba- bilistic view by revealing previously-hidden variables that bias the decision outcome, or even determine it in full. Future studies, describing the lysis/lysogeny decision in individual phages and cells, in real time, will be key to delineating true randomness 29 from the hidden precision of cellular decision-making.

Acknowledgements

We are grateful to Ian Dodd and Keith Shearwin for commenting on an earlier draft of this article. Work in the Golding lab is supported by grants from the National In- stitutes of Health (R01 GM082837), the National Science Foundation (PHY 1147498, PHY 1430124 and PHY 1427654), the Welch Foundation (Q-1759) and the John S. Dunn Foundation (Collaborative Research Award). We gratefully acknowledge the computing resources provided by the CIBR Center of Baylor College of Medicine.

Further Reading

• Arkin, A., Ross, J., McAdams, H.H., 1998. Stochastic kinetic analysis of devel- opmental pathway bifurcation in phage lambda-infected Escherichia coli cells. Annual Review of Genetics 149 (4), 1633–1648.

• Casjens, S.R., Hendrix, R.W., 2015. Bacteriophage lambda: Early pioneer and still relevant. Virology 479–480, 310–330.

• Dodd, I.B., Shearwin, K.E., Perkins, A.J., et al., 2004. Cooperativity in long- range gene regulation by the λ CI repressor. Genes & Development 18 (3), 344–354.

• Erez, Z., Steinberger-Levy, I., Shamir, M., et al., 2017. Communication between viruses guides lysis-lysogeny decisions. Nature 541 (7638), 488–493.

• Golding, I., 2011. Decision making in living cells: Lessons from a simple system. Annual Review of Biophysics 40 (1), 63–80.

• Golding, I., 2016. Single-cell studies of phage λ: Hidden treasures under Oc- cam’s rug. Annual Review of Virology 3 (1), 453–472. 30

• Knowles, B., Silveira, C., Bailey, B., et al., 2016. Lytic to temperate switching of viral communities. Nature 531 (7595), 466–470.

• Oppenheim, A.B., Kobiler, O., Stavans, J., et al., 2005. Switches in bacterio- phage lambda development. Annual Review of Genetics 39 (1), 409–429.

• Ptashne, M., 2004. A Genetic Switch: Phage Lambda Revisited, third ed. Cold Spring Harbor, NY: Cold Spring Harbor Laboratory Press.

• Silpe, J.E., Bassler, B.L., 2019. A host-produced quorum-sensing autoinducer controls a phage lysis-lysogeny decision. Cell 176 (1–2), 268–280.

• St-Pierre, F., Endy, D., 2008. Determination of cell fate selection during phage lambda infection. Proceedings of the National Academy of Sciences of the United States of America 105 (52), 20705–20710.

• Tal, A., Arbel-Goren, R., Costantino, N., Stavans, J., 2014. Location of the unique integration site on an Escherichia coli chromosome by bacteriophage lambda DNA in vivo. Proceedings of the National Academy of Sciences of the United States of America 111 (20), 7308–7312.

• Weitz, J.S., Beckett, S.J., Brum, J.R., et al., 2017. Lysis, lysogeny and virus–microbe ratios. Nature 549 (7672), E1–E3.

• Zeng, L., Skinner, S.O., Zong, C., et al., 2010. Decision making at a subcellular level determines the outcome of bacteriophage infection. Cell 141 (4), 682–691. 31

Chapter 3

Bacteriophage self-counting in the presence of viral replication

This chapter consists of the following manuscript:

Coleman, S.*, Yao, T.*, Nguyen, T.V.P., Golding, I., Igoshin, O. (2021). Bac- teriophage self-counting in the presence of viral replication. bioRxiv. (submitted manuscript)

(*Co-first authors).

My contributions: (1) Developed the 3-gene mathematical model used throughout the manuscript (see Appendix A, Section A.2). (2) Fitted the 3-gene mathemat- ical model (see Appendix A, Section A.2.4. (3) Performed all analysis for the 3-gene mathematical model. (4) Co-created all main text figures, with help from T.V.P. Nguyen (particularily on Figure 3.1) and T. Yao (on Figure 3.6D). (5) Created Supplementary Figures A.6–A.10, A.12–A.13, and A.15–A.16. (6) Co-wrote the manuscript. 32

3.1 Introduction

Following genome entry into the host cell, temperate bacteriophages must choose between two developmental pathways [28]. In the default, lytic pathway, rapid viral replication typically culminates in the death of the host cell (lysis) and release of viral progeny. By contrast, in the lysogenic pathway, phages suppress their virulent functions and enter a dormant prophage state [28]. To decide on the infected cell’s fate, temperate phages assess the environmental abundance of potential hosts [27, 28]. If susceptible host cells are scarce, then producing hundreds of new phages via the lytic pathway would be futile, and, instead, lysogeny should be chosen. To evaluate the relative abundance of viruses and cells, phages use diverse methods. Some achieve this by measuring the number of simultaneously coinfecting phages (multiplicity of infection, MOI) and increasing the frequency of lysogeny at higher MOI [46, 47]. Other bacteriophages harness cell-cell communication to assess the frequency of virus-host encounters in their vicinity [48, 49]. Notwithstanding the mechanism by which the measurement is performed, a regulatory circuit encoded by the virus must interpret a biological signal reflecting the relative abundance of viruses and host cells and use it to bias a decision between the two possible outcomes of infection. Phage lambda, a temperate bacteriophage that infects Escherichia coli, has long served as the paradigm for viral self-counting [31, 11, 1, 32]. Direct measurements, both in bulk [1, 25] and in single cells [45], demonstrated that a higher number of coinfecting phages increases the probability of lysogeny. Decades of experimental interrogation have resulted in a comprehensive genetic understanding of the virus and the identification of key players involved in the lambda post-infection decision [19, 50, 51]. However, despite this detailed molecular knowledge of the underlying circuitry, our system-level understanding of how MOI drives the infection outcome is far from complete [33, 29, 45]. In contrast to the two-gene “switch” governing lysogenic maintenance [30], the network driving the lysis/lysogeny decision comprises 33

multiple genes, regulating each other through diverse molecular interactions [19]. The common theoretical view of the decision is that this genetic network is biased by MOI towards either of two attractors, one corresponding to lytic onset, another to lysogeny [31, 36, 34, 33, 29, 35, 2, 32]. However, the way this takes place varies between models. Further challenging our ability to decipher the circuit’s function, and seemingly inconsistent with the two-attractors picture, is the fact that, while the eventual gene expression patterns in lysis and lysogeny clearly differ, the initial gene expression cascade following infection appears indistinguishable for both pathways [12]. Complicating phages’ task of measuring MOI—–and our attempts to decipher how they do it—–is the fact that viral copy number is rapidly increasing inside the infected cell (Figure 3.1). Phage replication begins within minutes of genome entry [2] and coincides with the expression of early genes in the decision circuit [12](see Figure 3.2 below). In other words, the initial MOI, which the viral circuitry presumably attempts to measure [5, 52], is soon obfuscated by the presence of additional phage genomes in the cell. Elucidating how lambda succeeds in distinguishing the initial genome number from the instantaneous number present in the cell has remained a challenge partly due to experimental limitations. Within a population, single-cell MOI is broadly distributed [1, 45](Figure A.1), necessitating measurements at the level of the individually infected cell. However, simultaneous measurement of viral copy number and the expression of phage genes has previously not been possible at single-cell resolution.

3.2 Results

3.2.1 In the absence of viral replication, gene expression does not diverge into lytic and lysogenic trajectories

To characterize the behavior of the lambda decision circuit, we sought to measure gene expression kinetics during infection across a range of single-cell MOI values. To 34

Figure 3.1 : The lambda decision circuit measures the multiplicity of in- fection (MOI) even as viral copy number is changing. A higher multiplicity of infection (MOI) increases the propensity to lysogenize. Here, infection by a sin- gle lambda phage (top) results in lysis, whereas coinfection by two phages (bottom) leads to lysogeny. In choosing cell fate, the infecting phage must respond to the initial number of viral genomes in the cell but ignore the subsequent increase in number due to viral replication. 35

decouple the gene-regulatory aspects from the effects of time-varying dosage, we first followed the approach of [1] and [31] and examined infection by a replication-deficient mutant (Pam80, henceforth denoted P-)[1]. We focused on the expression of three lambda genes at the heart of the decision circuit—–cI, cro, and cII [36, 2, 32](Figure

3.2A). Cro, transcribed from PR, is a repressor that inhibits transcription of multiple lambda genes (including itself) from the two early promoters, PR and PL [50]. Cro is required for successful lysis, to prevent the accumulation of CI and the overpro- duction of lambda proteins deleterious to later development [50]. Transcription of cro can also be used as a proxy for the presence of Q, a critical lytic gene produced

from the same polycistronic transcript [12]. CI, too, inhibits transcription from PR

and PL, in addition to regulating its own expression from PRM, and is required for establishing and maintaining lysogeny [12]. CII is a short-lived protein that activates

early transcription of cI from PRE, a critical event for the establishment of lysogeny [50]. Both CI and Cro also suppress viral replication by inhibiting expression of the

lambda replication proteins O and P [12] and by repressing transcription from PR, required for early replication [19]. To measure the MOI dependence of expression dynamics in this three-gene sub- network, we combined single-molecule quantification of mRNA and phage genomes in individual cells [53](Figure 3.2B). Following infection by a replication-deficient phage (cI 857 Pam80 P1 parS; see Experimental Methods for strain construction and experimental protocols), samples were taken at different time points and chem- ically fixed. The lambda genomes present in each cell were detected and counted using the ParB-parS system [54, 55](Experimental Methods and Figure A.2). In the same cells, mRNA copy numbers for cI, cro, and cII were measured using single-molecule fluorescence in situ hybridization (smFISH)[56, 53]. The cells were then grouped based on the measured single-cell MOI, and the averaged mRNA level for each gene, time, and MOI, was calculated (Figure 3.2C). All three genes exhibited a transient pulse of expression, with mRNA numbers 36

Figure 3.2 : A simplified model of the decision network captures the ki- netics of mRNA and viral copy number following infection. (A) Top, the three-gene circuit at the heart of the lysis/lysogeny decision. Bottom, the corre- sponding segment of the lambda genome. Upon viral entry, PR expresses both cro and (following a leaky terminator) cII. CII then activates cI expression from PRE. CI and Cro repress PR and PL, as well as phage replication. In a lysogen, CI regulates its own expression from PRM. (B) Images of a single E. coli cell at 10 minutes following infection by λ cI 857 Pam80 P1parS. Phage genomes are labeled using ParB-parS, and the mRNA for cI, cro, and cII using smFISH. Yellow dashed line indicates the cell boundary. (C) The numbers of cI, cro, and cII mRNA per cell, at different times following infection by λ cI 857 Pam80 P1parS, at MOI = 1—5. Markers and error bars indicate experimental mean ± SEM per sample (see Table A.9 for detailed sample sizes). Solid lines indicate model fit. (D) Viral copy number, measured us- ing qPCR, following infection at MOI = 1 by P+ (λ cI 857 Sam7) and P- (λ cI 857 Pam80 P1parS) phages. Markers and error bars indicate experimental mean ± stan- dard deviation due to qPCR calibration uncertainty. Lines indicate model prediction. See Experimental Methods and Theoretical Methods for detailed experimental procedures, image and data analysis, and modeling. 37

first rising, then decaying (Figure 3.2C). The main difference between genes was in the timing of the expression peak, with cro and cII reaching their highest level approximately ∼10 minutes after the entry of viral genomes, and cI peaking later, at about ∼20 minutes. Biological replicates yielded consistent results (Figure A.3). The observed dynamics were consistent with our current understanding of the gene expression cascade following infection: upon viral entry, cro and cII are transcribed

from PR (see Figure 3.2A) and this promoter is later repressed by Cro [12]. cI tran- scription requires activation of the PRE promoter and is hence delayed until enough CII protein, driving this activation, has accumulated [12]. Previous models of the lysis/lysogeny decision, both in the presence [36, 34] and absence [33, 35, 32] of viral replication, predicted the existence of distinct patterns of gene expression, identifiable as the two possible infection outcomes. In light of this prevailing picture, we were surprised to observe no clear divergence of mRNA trajectories between low and high MOI, reflecting a transition from lysis to lysogeny. Instead, we found that for each of the genes, a simple scaling by a factor MOI (suggested previously by [33]), with  ≈ 1 for cI and  ≈ 0.5 for cro and cII, yielded a near collapse of the different MOI-gated trajectories to a single curve (Figure A.4). Numerically integrating the mRNA numbers to estimate the instantaneous concentrations of Cro and CI likewise did not reveal a divergence of trajectories between low and high MOI (Figure A.5), indicating that the lack of divergence is not merely an artifact of examining the short-lived mRNA species.

3.2.2 Modeling network dynamics reveals that viral replication is re- quired for a lysis-to-lysogeny transition

The absence of a clear MOI-driven lysis-to-lysogeny transition following infection by a non-replicating phage led us to hypothesize that viral replication is required for such a transition to take place. To explore this hypothesis, we constructed a de- terministic mathematical model that describes the regulatory interactions between 38

cI, cro, and cII, as well as the coupling between gene expression and viral replica- tion (Figure 3.2A and Theoretical Methods). Building on previous theoretical efforts [36, 2, 32], our model captures, phenomenologically, both direct interactions

between the three genes (e.g., the activation of cI transcription from PRE by CII

and the repression of cro transcription from PR by CI) and indirect ones, mediated by players that are not modeled explicitly (e.g., CIII-mediated suppression of CII degradation—–see [57]). As its output, the model produces the population-averaged temporal dynamics of the viral copy number and mRNA and protein concentrations, for a given initial MOI. To estimate the parameters governing gene regulation in the network, we fitted the model to the experimental mRNA kinetics during P- infection, by minimizing the least-squares error using particle swarm optimization [58](see The- oretical Methods). This procedure yielded a good agreement between the model and experiments (Figure 3.2C and Figure A.6). To relate the gene expression dynamics computed by our model with the infection outcome, we assumed that cell fate is determined by whether Cro or CI concentration in the cell reaches a threshold value [33]. The lytic threshold reflects Cro’s role in repressing cI transcription from PRM and (indirectly, by repressing cII transcription from PR) from PRE. The Cro threshold also reflects the requirement that Q (encoded by the same transcript as Cro) reaches sufficient level to enable readthrough of the late lytic genes transcribed from PR’ [29]. The lysogenic threshold, on the other hand, corresponds to CI levels sufficient to turn off the PL and PR promoters, thereby repressing the expression of lytic genes and phage replication [12]. Using our model to analyze the network dynamics following P- infection, we found that, consistent with what we inferred from the mRNA data above, the predicted shape of protein trajectories in the CI-Cro plane (Figure 3.3A) was largely un- changed with MOI, with both species merely increasing in concentration with the viral copy number. While a range of CI thresholds can be defined, which ensure lysogeny above some critical MOI, no Cro threshold can result in lytic decision below 39

Figure 3.3 : Phage replication is required for an MOI-driven lysis-to- lysogeny transition. (A) Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by P- (non-replicating) phage at varying MOI. Protein concentrations were normalized by the lytic and lyso- genic thresholds. (B) Same as panel A, for the case of infection by a replicating (P+) phage. See Theoretical Methods for detailed information.

that critical MOI value (Theoretical Methods). Instead, the inferred trajectories suggest that the lytic threshold is never reached during P- infection, and that, at low MOI, neither lysis nor lysogeny is selected. This interpretation is consistent with the known absence of lysis following infection by P- phages [59, 2], but suggests that this failure reflects the state of the decision circuit, rather than merely a failure to exe- cute the chosen lytic pathway, which was the implicit assumption in previous works [31, 33, 35]. We next sought to evaluate what effect viral replication would have on the sys- tem’s behavior. To calibrate the model parameters pertaining to lambda replication and its regulation, we used qPCR measurements of genome number kinetics from low-MOI infection by a replicating (P+) phage (Figure 3.2D), as well as published data for cI and cII expression following infection with P+ phages, cI-, cro-, and cro-P- mutants [2](Figures A.7, A.8). The resulting model allowed us to calculate the CI-Cro trajectories following infection by a replicating phage at various MOIs. 40

In contrast to what we observed for the nonreplicating phage, these trajectories show a clear divergence between MOI = 1 and higher MOIs, and, in particular, support a transition from lysis (Cro threshold crossing) to lysogeny (CI threshold crossing) with increasing MOI (Figure 3.3B). Moreover, a single choice of thresholds is simul- taneously consistent with the experimental phenotypes of both P- and P+ phages in terms of the MOI value at which the transition to lysogeny occurs (≈ 3—4 and 2 for P- and P+ respectively; see Theoretical Methods and Figure 3.6 below)[1].

3.2.3 CII activation of PRE defines a time window for the network’s re- sponse to MOI

Having successfully recapitulated the decision phenotype, we next sought to under- stand how lambda reliably responds to the initial MOI even in the presence of viral replication. Since establishing lysogeny requires reaching a critical CI concentration, we focused on the response of cI expression to MOI. Two lambda promoters drive cI transcription, PRE (activated by CII) and PRM(autoregulated by CI) [12]. PRM is solely responsible for CI production in a lysogen [12], but whether it plays a role during the initial decision has remained unresolved [60]. Our model indicates that the

MOI-driven increase in CI is caused by transcription from PRE, and that removing cI autoregulation does not eliminate that response (Figure A.9). This finding is con- sistent with reports that a wide range of mutations in PRM permit the establishment of lysogeny [61, 62], whereas mutating PRE or CII prevents it [12]. Thus, to elucidate CI’s response to MOI, we focused on characterizing the CII-activated expression of cI during infection.

Our model indicates that CII-activated PRE expression occurs in a single pulse, taking place within the first ∼30 minutes of infection (Figure 3.4A). The amplitude

(per phage) and duration of this PRE activation pulse depend only weakly on MOI (Figure 3.4A and Figure A.10). These predictions are supported by direct mea- surement of nascent cI mRNA level at individual phage genomes (Figure A.11). 41

Figure 3.4 : CII activation of PRE defines a time window for the network’s response to MOI. (A) Model-predicted PRE promoter activity (per phage) following infection by P+ phage at MOI = 1—5. Gray shading indicates the MOI-averaged CII activity window, defined as the period during which PRE activity is greater than 10% of its maximum. (B) Cellular CI concentration (normalized by the threshold concentration for lysogeny, dashed red line) following infection by P+ phage at MOI = 1—5. (C) The strength of cro repression by CI, calculated as the magnitude of the CI repression term in the PR transcription rate, following infection by P+ phage at MOI = 1—5. (D) The strength of repression of replication by CI, calculated as the magnitude of the CI repression term in the viral replication rate, following infection by P+ phage at MOI = 1—5. See Theoretical Methods for detailed information.

They are also consistent with our findings above that, in non-replicating phages, cel- lular cII numbers are dosage compensated whereas cI level scales linearly with MOI (Figure A.4). The MOI independence of cI expression from individual viral copies during the

PRE activation window provides a necessary element for the lysogeny decision, by guaranteeing that cellular CI concentration increases with MOI. Specifically, the max- imum CI concentration at MOI = 2 is approximately 2-fold higher than for MOI =

1, and this fold-change in CI concentration is first reached within the PRE window (Figure 3.4B). Higher MOI further increases CI concentration within the same time 42

window (Figure 3.4B). The end result is that, at MOI ≥ 2, CI concentration reaches the critical level sufficient to repress both cro expression (Figure 3.4C) and viral replication (Figure 3.4D), leading to lysogeny. This repression is established during

the PRE activation window and persists throughout infection (Figure 3.4C & D).

3.2.4 Changes in viral copy number outside the CII activity window do not alter the decision

The findings above reveal how CI levels–—and the propensity to lysogenize–—increase with MOI. However, a reliable MOI-based decision requires also that viral replication inside the cell will not obfuscate the initial response to MOI. We reasoned that, for this to hold, the system should become insensitive to changes in viral dosage once

the window for CII activation of PRE is closed. To test this hypothesis, we followed the approach of [34] and used the model to examine what happens in the case of delayed infection. Specifically, we modeled an infection by a single phage, followed

by a second single-phage infection at time τd later (Figure 3.5). We found that, indeed, if the second infection takes place after the end of the PRE activity window,

τPRE, the outcome is lysis, and the CI and Cro trajectories are indistinguishable from those at MOI = 1 (Figure 3.5A). If, in contrast, the delayed infection occurs early

enough within the PRE activity window, infection results in lysogeny, with CI and Cro trajectories similar to those for a synchronized infection at MOI = 2 (Figure 3.5A). Infecting with higher numbers of late-arriving viruses lengthens the time window

where the late phages can affect the decision, but not beyond τPRE (Figure A.12). Thus, the outcome is insensitive to changes in viral copy number that take place outside the time windows determined by CII activity. Why does delayed infection result in a diminished response? Our model indicates

reduced PRE expression from the second virus, since that virus is not present for the entire duration of the CII activity pulse (Figure 3.5B). This results in lower cellular accumulation of CI during that time window, as compared to simultaneous coinfection 43

Figure 3.5 : Changes in viral copy number outside the CII activity window do not alter the decision. (A) Model-predicted system trajectories, in the plane of Cro and CI concentration, during the first 60 minutes following infection by P+ phages, at 4 different scenarios: Infection by a single phage (solid line, light blue), infection by a single phage followed by a second phage at time τPRE/3 (with τPRE the end of the CII activity window; dashed line, dark blue) and τPRE (dotted line, dark blue), and simultaneous infection by two phages (solid line, dark blue). (B) PRE activity from the second arriving phages, for the cases modeled in panel A. The shaded grey region indicates the MOI-averaged CII activity window estimated for synchronized infections. (C) Viral copy number over time, for the cases modeled in panel A. See Theoretical Methods for detailed information. 44

(Figure 3.5A). Furthermore, the delay impacts not only the late-infecting virus itself but also all viruses produced subsequently through viral replication (Figure

3.5C). We note that, through the combination of Cro repression of PR and active CII degradation [12], the decision network constrains CII activity to a single time window. Subsequent changes in viral copy number after the initial infection cannot overcome this constraint. These features of the system’s response to delayed infection can also explain why rampant viral replication during infection at MOI = 1 does not cause a switch to lysogeny: While replication generates > 100 additional genome copies, viruses produced outside the CII activity window are unable to express cI

from PRE (Figure A.13). The opportunity to ‘flip’ the decision switch has already been lost.

3.2.5 Phage replication enables the lytic choice and lowers the MOI re- quired for lysogeny

We have thus seen how phage replication is tolerated, i.e., how a reliable response to the initial MOI is achieved despite the subsequent change in viral copy number. Recall, however, that comparing CI-Cro trajectories in replicating (Figure 3.3B) and nonreplicating (Figure 3.3B) phages indicated that viral replication is not only tolerated but, in fact, required for the existence of an MOI-dependent lysis-to-lysogeny transition. To understand why that is the case, we first addressed the absence of a lytic choice following P- infection at MOI = 1. Our model indicates that, in both P+ and

P-, cro transcription from PR (per phage) is repressed > 2-fold within ∼10 minutes of infection (Figure 3.6A; Figure A.14 depicts the corresponding experimental data). However, the presence of additional gene copies in the P+ case results in higher Cro concentration later in the infection, sufficient to cross the lytic threshold (Figure 3.6B). Viral replication thus serves to boost total Cro expression despite repression of its transcription at the individual phage level. As for the effect of replication on the lysogenic choice, we find that cI transcription 45

Figure 3.6 : Replication is required for the lytic outcome and lowers the MOI required for lysogeny. (A) Model-predicted PR activity per phage during P- (solid blue line) and P+ (dashed blue line) infection at MOI = 1. (B) Cellular Cro concentration, normalized by the lytic threshold, for the cases modeled in panel A. (C) Cellular CI concentration, normalized by the lysogenic threshold, during P- infection at MOI = 2 and 4 (solid light and dark blue lines, respectively), and during P+ infection at MOI = 2 (dashed line). (D) The fraction of cells undergoing lysogeny as a function of average MOI, during bulk infection with P- (circles), P+ (squares), and phages with prolonged CII lifetime (P+ phages infecting hflKC- hosts; triangles). The experimental data was fitted to a model (black lines) where virus-cell encounters follow Poisson statistics, and infection at MOI ≥ MOI* results in lysogeny [1]. The hflKC- strains are either ∆hflK or ∆hflC (see Table A.4). (E) Predicted infection outcome as a function of MOI and viral replication rate (normalized by the fitted replication rate for P+ phage). See Experimental Methods and Theoretical Methods for detailed experimental procedures, data analysis, and modeling. 46

from PRE during P- infection follows a similar pattern to P+, namely, a single pulse whose duration and amplitude per phage depend only weakly on MOI (Figure A.15, compare to Figures 3.4A and A.10). However, as in the case of Cro, the presence of added gene copies during P+ infection leads to considerably higher CI accumulation than for P- (Figure 3.6C). Consequently, while replicating phages reach the lysogenic CI threshold at MOI = 2, nonreplicating ones require a higher MOI to reach that threshold and establish lysogeny. The theoretical prediction that P- phage lysogenizes at higher MOI than P+ is born out in experiments ([1] and Figure 3.6D). To generalize the effect of viral replication on the lysis/lysogeny decision, we simulated infections at MOI in the range 1-–7, while varying the phage replication rate from zero to 1.5x that of P+ phage. Determining the infection outcome at each MOI and replication rate yielded the two-dimensional “fate diagram” shown in Figure 3.6E. Consistent with the discussion above, we find that both the existence of a lytic outcome and the minimal MOI at which the transition to lysogeny occurs depend on the viral replication rate. The ability to replicate is not by itself sufficient to enable the lytic pathway. Rather, there is a minimum required replication rate, below which, MOI = 1 infections fail to achieve either outcome. As for lysogeny, the MOI at which this fate is chosen decreases with the viral replication rate (Figure 3.6E). When replication is sufficiently rapid, the model predicts a lysogenic outcome even at MOI = 1. While we are unaware of an experimental test for this prediction, we note that the model predicts a similar behavior when the CII activity window is extended by inhibition of CII degradation, a result validated by experiments [29](Figure 3.6D and Figure A.16). Interestingly, there are two regions where the system’s trajectories cross both the lytic and lysogenic thresholds in the course of infection (Figure 3.6E). The first of those occurs at high replication rates, in the MOI range between lysis and lysogeny. We are uncertain how to interpret this feature, but it is intriguing to note that, at the single-cell level, it may correspond to the range of infection parameters where stochas- 47 tic effects become important and, consequently, individual cells exhibit different fates, as reported experimentally [25, 63, 45]. The analysis of cellular heterogeneity is out- side the premise of our current model, which captures the population-averaged be- havior only. A second region where mixed outcomes are predicted is found for MOI & 5, and may correspond to a scenario where the overexpression of viral proteins results in the halting of cell growth, rather than a lytic or lysogenic outcome, consistent with experimental data [45].

3.3 Discussion

The combination of single-molecule genome and mRNA measurement in individual cells with theoretical modeling provided us with new insights into the way lambda counts coinfecting phages to bias the lysis-lysogeny decision. Early theoretical studies of the post-infection decision sought insight to the phage’s binary choice in the toggle switch comprised of the mutually antagonistic CI and Cro, which governs lysogenic maintenance and lytic induction [36, 64, 65, 32]. This famous “genetic switch” ex- hibits bistability [66], with well-defined states characterized by high CI (lysogeny) and high Cro (lytic onset), respectively. These features, and the tremendous body of experimental and theoretical knowledge that has accrued about the pairwise CI/Cro interactions [50, 30], explain the focus on this element as the key to the decision. Moreover, theoretical work has shown that MOI could indeed drive a bifurcation of the CI/Cro switch’s steady state, consistent with a transition from lysis to lysogeny [36, 32]. Our analysis above, however, suggests that this is not how the lambda de- cision unfolds. Instead, each phage initially attempts to execute a preset pattern of gene expression, independent of the MOI. In this cascade of events, cI transcription takes place predominantly through the transient activation of PRE by CII, while cI au- toregulatory expression from PRM is unnecessary. The resulting cellular CI expression is approximately linear in MOI, indicating the absence of an ultrasensitive response to CII as previously suggested [34, 29, 39]. Given this fixed gene expression cascade 48

at the individual phage level, it is the introduction of time-varying gene dosage due to viral replication—rather than the standalone topology of the viral circuit—that enables the subsequent divergence of gene expression trajectories and cell-fate choices at low and high MOI. The time-varying viral copy number is found to be critical for both possible out- comes of infection. The choice of lysogeny depends on the number of lambda copies

present during the early decision window, set by CII activation of PRE (Figure 3.4A above). The finite response window immunizes the lambda decision to changes in viral copy number that take place outside it, thus allowing reliable detection of the initial MOI. However, the role of viral copy number does not end then, and is, in fact, crucial for reaching the protein threshold required for establishing either fate: Cro for lysis, or CI for lysogeny. Cro (and by proxy, the lytic activator Q) continues to accumulate through late infection, reaching the lytic decision threshold 50 minutes post-infection (Figure 3.6B). Cro’s continued accumulation is driven by late viral

replication, despite the repression of PR at the single phage level (Figure 3.6A). As for lysogeny, while this fate is achievable in the absence of viral replication, it is replication that ensures that coinfection by more than one phage is sufficient to drive lysogeny; in the absence of replication, CI accumulates insufficiently, and higher MOI is required to reach the lysogenic threshold (Figure 3.6C). The relation between gene dosage and the output of genetic networks has been explored in diverse biological contexts, both natural [67, 68, 69] and synthetic [70, 71, 72]. In some instances, changes in gene copy number were found to have a signif- icant effect on phenotype [73, 74, 75] whereas in other cases, mechanisms of dosage compensation buffer the network output from such changes [76, 77, 78, 79]. The lambda decision exhibits a richness of dosage response beyond what was previously documented, with the transcriptional output either linear in dosage, or partially com- pensating it, for different genes in the network, at different times during the infection. Consequently, viral replication is found to facilitate, rather than hinder, the imple- 49 mentation of a reliable decision. Since cellular decisions frequently take place even as gene copy number is changing [80, 81, 15], this inextricable coupling of gene dosage and network output cannot be ignored if one aims for a predictive description of cellular decision making.

Acknowledgements

We are grateful to the following people for their generous advice and for providing reagents: R. Arbel-Goren, S. Austin, G. Bal´aszi,M. Cortes, I. Dodd, M. Feiss, J. Harris, C. Hayter, K. Shearwin, J. Stavans, F. St-Pierre, L. Thomason, G. Vasen, J. Weitz, L. Weinberger, Z. Yu, L. Zeng, and all members of the Golding and Igoshin groups. This work was supported by the National Science Foundation Center for Theoretical Biological Physics (NSF PHY-2019745) and grant PHY-1522550. The work was supported in part by the Big-Data Private-Cloud Research Cyberinfras- tructure MRI-award funded by NSF under grant CNS-1338099 and by Rice Univer- sity. Work in the Golding lab is supported by the National Institutes of Health grant R01 GM082837 and the National Science Foundation grant PHY 1430124. Igoshin also acknowledges support by the Welch Foundation Grant C-1995 and the National Science Foundation grant MCB-1616755.

Supplementary Information

• Experimental Methods

• Theoretical Methods

• Supplementary Figures 1—16

• Supplementary Tables 1—6 50

Chapter 4

Future Directions

4.1 Extending the model to capture stochasticity in gene ex- pression

4.1.1 A stochastic reformulation of the model deviates from deterministic predictions

While the deterministic model in Chapter 3 was well-constrained and provided useful insights into the function of viral replication in the lambda decision, those predictions are restricted to population-averaged behavior. To extend our predic- tions to the single-cell level, we need a model that can simulate the impact of noise on network dynamics [41]. While there are many potential sources of noise [82], a common starting point in modeling single-cell behavior is incorporating the inherent stochasticity of biochemical reactions at low reactant copy number [83]. There are multiple ways to model stochastic chemical kinetics [84], however due to its ease of implementation, we chose to rewrite the model in terms of a continuous-time Markov jump process. In this formulation, the state of the system at time t is defined by the 7-dimensional vector of the copy numbers of chemical species in the system,

T n = [mcI , mcro, mcII , CI, Cro, CII, λ] , and stochastic chemical reactions model tran- sitions in the nonnegative integer state space [84]. In reformulating the model, we followed the approach of previous works [33, 85, 86, 87] which equated fluxes in the ODEs of the deterministic model to chemical reactions in the stochastic model (see Appendix B, Section B.1). This approach is both easy to implement and allows us to reuse all fitted parameters (Chapter 51

3), rather than refitting. An alternative approach would be to instead construct a new stochastic model based on the network interactions included in our deterministic model (Appendix A, Section A.2.2), and fit parameters by comparing the model- predicted mean trajectories (or the time-dependent mRNA distributions themselves) to the experimental data [88]. However, fitting a stochastic model presents signifi- cant technical challenges. For anything but very simple models, the chemical master equation describing the Markov jump process cannot be analytically solved [84], so solutions must either be numerically estimated (for example, by finite state projection (FSP)—[89]) or directly simulated [90]. In the former approach, for models of the size used in Chapter 3, standard methods for implementing the FSP algorithm fail, as these methods inherently rely on matrix computations using the estimated state space, whose size explodes with the number of chemical species modeled [91]. This leaves direct simulation (based on kinetic Monte Carlo methods—[92]), which even when parallelized can be computationally expensive when used to estimate probabil- ity distributions or their moments. This expense is compounded when fitting, where thousands of iterations may be required to reach a solution [91, 88]. While our pragmatic approach of reformulating the model by equating fluxes in the ODEs with stochastic reactions in the chemical master equation avoids these technical hurdles, strictly speaking this conversion process is only exact for linear models. Nonlinear terms (such as those in the regulatory functions in the determinis- tic model—see Appendix A, Section A.2.2) imply coupling between higher order moments of the species in the joint probability distribution described by the chem- ical master equation [84]. This coupling generates terms dependent on higher order moments in the differential equations for the first moments derived from the chemical master equation [84]. These higher order terms result in a mismatch between the differential equations derived from the chemical master equation and the differential equations comprising our deterministic model (which do not contain higher order mo- ments). Nevertheless, this pragmatic conversion approach can still sometimes yield 52

Figure 4.1 : The stochastic model recaptures the deterministic model’s trajectories during P- infection. The numbers of (A) cI, cro, and cII mRNA and (B) CI, Cro, and CII protein per cell, during the first 60 minutes following simulated infection by P- phages, at MOI = 1—5. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3).

good agreement between the average behavior of the stochastic model and the corre- sponding deterministic model [33, 85, 93]. To test our stochastic model, we first simulated P- infection, using the direct simulation method for the stochastic simulation algorithm [92]. The mean of the generated stochastic trajectories generally matches the output of the deterministic model (Figure 4.1). There are small deviations evident, particularly for early cII mRNA numbers (starting at ∼10 min), and later for CII protein and cI mRNA 53

Figure 4.2 : The stochastic model yields outcomes consistent with the deterministic model during P- infection. Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by P- (nonreplicating) phage at varying MOI. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3). Protein concentrations were normalized by the lytic and lysogenic thresholds. 54

Figure 4.3 : The stochastic model strongly deviates from the deterministic model’s trajectories during P+ infection. The numbers of (A) cI, cro, and cII mRNA, (B) CI, Cro, and CII protein, and (C) viral copy number per cell, during the first 60 minutes following simulated infection by P+ phages, at MOI = 1—5. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3).

levels (starting at ∼20 min; see Figure 4.1). However the stochastic simulations capture the overall dynamics and MOI-dependence predicted by the deterministic model. Examining the behavior in the CI-Cro plane, the mean trajectories of the stochastic model predict the same phenotypic outcomes as the deterministic model (Figure 4.2). We next tested whether the stochastic model could replicate the predicted network dynamics during P+ infection. Here the mean stochastic trajectories show signifi- cant deviations from the deterministic model (Figure 4.3), for the same species and time windows observed during P- infection (Figure 4.1). There is also considerable disagreement between the models in predicted viral copy number (Figure 4.3C). Examining mean stochastic trajectories in the CI-Cro plane (Figure 4.4), we find 55

Figure 4.4 : The stochastic model generates outcomes that are inconsis- tent with the deterministic model during P+ infection. Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by P+ phage at varying MOI. Solid lines indicate the mean of stochastic trajectories, while dashed lines indicate the output of the deterministic model (see Chapter 3). Protein concentrations were normalized by the lytic and lysogenic thresholds. 56

that the deviations from deterministic predictions result in a failure by the stochastic model to replicate decision outcomes at the population-averaged level. Specifically, at MOI = 1 the stochastic model crosses the lysogenic threshold, and fails to cross the lytic threshold (Figure 4.4).

4.1.2 Estimated single-cell lysogenization frequencies fail to capture ob- served trends

We next tested whether, in spite of the stochastic model’s deviations during P+ in- fection (Figures 4.3 & 4.4), the model-predicted lysogenization frequencies match experimental measurements under conditions similar to those used to calibrate the deterministic model [2]. When using the decision thresholds for the deterministic model used to produce the figures in Chapter 3, we find significant disagreement between experiments and model predictions (Figure 4.5A). Not only is the model unable to quantitatively match the measured lysogenization frequencies, but the MOI- dependent trends during P+ infection also disagree, with the model-predicted fre- quency decreasing abruptly at MOI = 4 (Figure 4.5A). Additionally, the model does not capture the observed convergence of P- and P+ lysogenization frequencies at MOI = 3 (Figure 4.5A). Given that the decision thresholds in Chapter 3 were derived based on population- averaged behavior, we next attempted to recover the observed lysogenization frequen- cies by choosing the decision thresholds which yielded the best fit to the experimental measurements (see Appendix B, Section B.4). The lysogenization frequencies ob- tained with fitted thresholds still show strong disagreement between model-predicted and experimentally-observed lysogenization frequencies (Figure 4.5B). Additionally, the model still fails to capture both the monotonic increase in frequency over the mea- sured range of measured MOI and the convergence of frequencies at MOI = 3 for P+ and P- infection. 57

Figure 4.5 : The stochastic model is unable to replicate experimentally observed lysogenization frequencies. The measured single-cell frequencies of lysogeny during infection with P- (red markers) and P+ (blue markers) phages, at MOI = 1–4 (data from [2]). Model-predicted frequencies (solid lines) are calculated using (A) decision thresholds used in Chapter 3, and (B) fitted thresholds (see Appendix B, Section B.4).

4.1.3 Possible explanations for deviations in predicted single-cell behav- ior

There are several possible explanations for why the stochastic model is unable to reca- pitulate the observed lysogenization frequencies (Figure 4.5). First, as demonstrated in Figure 4.3, the stochastic model is unable to reproduce the deterministic model’s trajectories during P+ infection. This disagreement likely stems from our inexact re- formulation of the model—while this approach can sometimes yield consistent results [33, 85, 93], it has also been reported to result in significant deviations in predictions between deterministic and stochastic models [94, 95, 96], particularly for strongly nonlinear interactions. A second possibility is that the stochastic model, beyond fail- ing to capture the population-averaged behavior, is also not truly capturing the noise at the single-cell level. The stochastic model reuses parameters from the deterministic model, which fitted to the first moment only of the underlying mRNA distributions (see Chapter 3). The stochastic model is thus ignorant of the underlying single-cell distributions, beyond their time-dependent means. Lastly, it’s possible that, even with a correctly fitted stochastic model which matches the (averaged) trajectories of 58 the deterministic model used in Chapter 3, our CI-Cro formulation of the decision may be unable to correctly predict lysogenization frequencies, particularly at MOI = 3 and 4. The increase in maximum CI and Cro concentrations with MOI during P+ infection predicted by the deterministic model for MOI > 2 may invariably lead to a decrease in lysogenization frequencies in this MOI range, in contrast with the reported steady increase in lysogenization frequency at MOI = 3–4 [2]. We may thus need to explore alternative formulations of the decision (as in Section 4.2.1, where we find that a CI-Q decision yields alternative predictions at high MOI). Based on the observed deviations in the stochastic model’s behavior from both the deterministic model’s trajectories and measured lysogenization frequencies, it appears the best path forward for obtaining a useful single-cell formulation is to refit the stochastic model, at least partially. While this is a significant technical task (as outlined in Section 4.1.1), it is the most direct way to both incorporate more information from measured mRNA distributions and ensure coupled mean and single- cell predictions. If fitting is implemented using a massively parallelized optimization procedure [97], this approach may be feasible, even for a model of this size.

4.2 A Q-based lytic decision yields alternative predictions of infection outcome at high MOI

4.2.1 The role of Q during infection

While the lytic outcome in our deterministic model (Chapter 3) was defined based on Cro concentration, in line with the common hypothesis that the decision is a competition between CI and Cro [31, 32, 36], more recent models of the lysis/lysogeny decision have focused on the importance of Q [33, 34]. Q forms an antiterminator complex with host proteins that allows readthrough of PR0 , facilitating expression of late lytic genes (including capsid proteins and holons—see [12]). This readthrough is critical for the lytic pathway, and mutations that knockout Q eliminate the lytic fate 59

[29]. Additionally, as PR0 lies outside the regulatory domain of either CI or Cro, Q- mediated transcription of the late lytic genes is not directly countered by accumulation of either transcription factor once Q has reached sufficient concentration [12, 98]. Based on the apparent importance of Q, it’s worth considering whether our simple model would reach different predictions if lytic outcomes were based on Q concentra- tion, rather than Cro. Q mRNA, like cro and cII, is transcribed from PR, and is hence similarily repressed by both Cro and CI. However, Q is under additional regulation from CII [12]. CII activates transcription of the promoter PaQ, whose transcript is antisense to Q. It remains unclear whether transcription from PaQ directly interferes with Q transcription, or whether the antisense mRNA transcript binds to Q mRNA and either enhances degradation or prevents translation [50]. Regardless of the exact mechanism, multiple works have claimed CII inhibition of Q is critical to the deci- sion [29, 33, 34]. Specifically, a recent theoretical work [34] hypothesized that this interaction, combined with CIII protection of CII from degradation [12], causes the lambda network to function like an incoherent feedforward loop, with MOI as the source node. According to this view, at MOI = 1 active degradation prevents CII from reaching the threshold necessary to inhibit Q accumulation; only at MOI > 1 is CII sufficiently stable to accumulate to the required threshold concentration. An- other recent work focused on the implications of the CII-Q interaction on the relative timing of regulatory events, hypothesizing that early CII inhibition of Q expression helps create a window during which the circuit can respond to CI [29]. Regardless of the biological function of this interaction, it has been observed that removal of PaQ significantly decreases the frequency of lysogeny [50].

4.2.2 A toy module of Q regulation predicts only lysogeny at high MOI

To date, Q transcription has not been measured at the single-cell level as a function of MOI. This, combined with our lack of knowledge of the molecular details of the CII-Q interaction, make Q dynamics difficult to properly constrain and model. Nevertheless, 60

we made a toy module of Q regulation which phenomenologically incorporates the CII-Q interaction at the translation level (see Appendix C). We parameterized the

interactions in this toy module using the fitted parameters for CII activation of PRE, motivated by the identical binding sequence of both promoters [12], and the previously hypothesized synchronization of CII regulatory functions [12, 50]. As we assume CII inhibition of Q expression occurs only at the translation level, we use cII mRNA as a proxy for q mRNA, as both are transcribed from PR and both require readthrough of terminators [12]. Examining the predicted trajectories of the Q-inclusive model in the CI-Q phase plane (Figure 4.6A), we find that the dynamics are qualitatively different from those in CI-Cro phase space (see Chapter 3, Figure 3.3). Specifically, in both P- and P+ infection, maximum Q concentration monotonically decreases with MOI. Consequently, in contrast to the replication-dependent qualitative difference between MOI = 1 & 2 trajectories predicted in CI-Cro space (Chapter 3, Figure 3.3), trajectories in CI-Q space exhibit similar dynamics at MOI = 1 & 2 regardless of whether infection is by P- or P+ phages (Figure 4.6A). In spite of this lack of qualitative difference between trajectories during P- and P+ infection, replication still produces a quantitative difference—at MOI = 1, maximum Q concentration during P+ infection is 1.5-fold higher than during P- infection (Figure 4.6A). This amplification of Q accumulation by viral replication echoes our findings in Chapter 3 on the importance of replication to the lytic pathway. To test the robustness of the predicted inverse MOI-dependence of Q concentration in this toy module, we analyzed model trajectories when KQ, the threshold for CII inhibition of Q, is two-fold higher and lower than the default value used (KQ = KPRE, where KPRE is the concentration threshold for CII activation of PRE; Figure 4.6 B & C). In both extremes, the resulting CI-Q trajectories show the same qualitative trends—maximum Q concentration decreases with MOI (both during P- and P+ infection), and at MOI = 1 is higher for P+ infection. 61

Figure 4.6 : CI-Q trajectories support a lysis-to-lysogeny transition. Model- predicted trajectories, in the CI-Q plane, during infection by P- (left) and P+ phages, for three values of the CII-Q interaction threshold (KQ): (A) KQ = KPRE (the default value), (B) KQ = 0.5KPRE, and (C) KQ = 2KPRE. Lytic and lysogenic threshold ranges are indicated by the green and red bands, respectively. 62

Having demonstrated that this inverse MOI scaling is a robust feature of this module, we next tested how the difference in MOI-dependence of CI-Q trajectories versus CI-Cro trajectories affects the decision outcome. Two notable features in the genotype- and MOI-dependencies of the simulated CI-Q trajectories ensure these trajectories are consistent with all criteria previously used to derive the decision thresholds used in Chapter 3 (see Appendix A, Section A.2.5). First, while the shape of the CI-Q trajectories at MOI = 1 is qualitatively similar during P+ and P- infections, the maximum Q concentration reached during P+ is higher, enabling us to define a range of lytic thresholds that are not reached during P- infection (green bands, Figure 4.6). Second, there is still a qualitative shift in behavior during P+ infection with MOI—at MOI = 1 trajectories result in a high Q, low CI state, while trajectories at MOI > 1 reach lower Q and higher CI concentrations. It is therefore possible to define a range of lysogenic thresholds which predict a transition from lysis at MOI = 1 to lysogeny at MOI = 2 during P+ infection (red bands, Figure 4.6). The principle difference in predicted phenotype between curves in CI-Cro and CI- Q space occurs at high MOI. As shown in Chapter 3 (Figures 3 & 6), both lytic and lysogenic thresholds are crossed during high MOI infection in a CI-Cro formula- tion of the decision. However, as a result of the inverse MOI-dependence displayed by Q, this behavior is absent in CI-Q space. As maximum Q concentration mono- tonically decreases with MOI, while maximum CI concentration conversely increases, trajectories support a clear transition from lysis at MOI = 1 to lysogeny at MOI > 1. To demonstrate this explicitly, we applied the procedure used in Chapter 3 for constraining lytic and lysogenic thresholds (see Appendix A) to obtain new thresh- olds for a CI-Q decision, and then resimulated the fate diagram (Figure 4.7). As suggested by the topology of CI-Q trajectories (Figure 4.6A), high MOI infections in this formulation of the decision exclusively result in lysogeny. Given the model-predicted difference in Q- and Cro-dependence on MOI for this module, despite similar transcriptional regulation [12], we next sought to examine 63

Figure 4.7 : A CI-Q formulation of the decision exhibits only lysogenic outcomes at high MOI. Predicted infection outcome as a function of MOI and viral replication rate (normalized by the fitted replication rate for P+ phage). 64

Figure 4.8 : MOI-dependent CII inhibition of Q translation generates the observed inverse MOI-dependence of Q during P+ infection. (A) Strength of CII inhibition of Q translation during P+ infection, calculated as the magnitude of the CII-dependent term in the regulatory function fQ (see Appendix C). Gray shading indicates the MOI-averaged CII activity window, defined as the period during which PRE activity is greater than 10% of its maximum. (B) Cellular Q concentration during P+ infection at MOI = 1–5. (C) Strength of CII inhibition of Q translation when this interaction occurs during a fixed time window at each MOI (green curve), defined by the MOI-averaged turn-on and turn-off times for CII activation of PRE (see Appendix A, Figure A.10). (D) Same as panel B, for the case where CII inhibition of Q occurs during the fixed time window (green shaded region) shown in panel C. 65

Figure 4.9 : MOI-dependent CII inhibition of Q translation generates the observed inverse MOI-dependence of Q during P- infection. (A) Strength of CII inhibition of Q translation during P- infection, calculated as the magnitude of the CII-dependent term in the regulatory function fQ (see Appendix C). Gray shading indicates the MOI-averaged CII activity window, defined as the period during which PRE activity is greater than 10% of its maximum. (B) Cellular Q concentration during P- infection at MOI = 1–5. (C) Strength of CII inhibition of Q translation when this interaction occurs during a fixed time window at each MOI (green curve), defined by the MOI-averaged turn-on and turn-off times for CII activation of PRE (see Appendix A, Figure A.15). (D) Same as panel B, for the case where CII inhibition of Q occurs during the fixed time window (green shaded region) shown in panel C. 66

how CII inhibition results in the predicted inverse MOI-dependence of Q concentra-

tion. Transcription of Q from PR, as with cro and cII (which serves as our proxy for Q mRNA) is repressed early during infection regardless of MOI (Chapter 3, Figure 6A). As CII inhibition of Q translation is dependent on the concentration of CII (which is actively degraded—see [99]), this interaction occurs during the same early time window as CII activation of PRE (Figure 4.8B). Thus the early time

window when PR is highest overlaps with the window during which Q translation is inhibited by CII, and the duration of that inhibition increases slightly with MOI (Figure 4.8A). To test whether this MOI-dependence is necessary to produce the observed inverse MOI-dependence in maximum Q concentration (Figure 4.8B), we also simulated infection with this module when CII inhibition occurs in a fixed, MOI- independent time window (Figure 4.8C) defined by the MOI-averaged turn-on and turn-off times of CII activation of PRE (Appendix A, Figure A.10). When CII inhi- bition of Q accumulation is MOI-independent, the observed inverse MOI-dependence is broken—infections at high MOI result in higher concentrations of Q than at MOI = 1 (Figure 4.8D). The effect on Q accumulation is largest at low MOI, where the increase in the duration of inhibition of Q translation as a result of this fixed window perturbation results in a substantial decrease in maximum Q concentration reached during the first 30 minutes of infection (∼5-fold at MOI = 1, and ∼2-fold at MOI = 2). This suggests that the weak MOI-dependence of the CII-Q interaction, specifically the small increase in duration with MOI (Figure 4.8A), may be necessary to achieve the inverse MOI-dependence in maximum Q concentration predicted by this module. Performing this test during P- infection yielded similar results (Figure 4.9).

4.2.3 Experimental measurements of Q mRNA dynamics are needed to extend this analysis

While these results are promising, it must be noted that our treatment is only a toy module of the CII-Q interaction, and is not constrained by any direct observations. 67

There are additionally multiple hypothesized mechanisms for this interaction [50]. To accurately model CII inhibition of Q therefore requires, at the very least, direct measurements of Q mRNA expression as a function of MOI. While these experiments are possible, using the techniques described in Chapter 3, they have yet to be performed.

4.3 Viral replication may enable subcellular decision-making

4.3.1 The subcellular decision-making hypothesis

While we’ve considered the decision at the population-averaged level in Chapter 3, and attempted to extend our analysis to the single-cell level by the addition of noise (Chapter 4, Section 1), we have so far only used ’mean field’ approaches in describing network dynamics. Specifically, we have ignored spatial effects in gene expression, and in our stochastic model we have maintained the phenomenological description of regulation used in the deterministic model. This description effectively coarse-grains over processes like transcription factor binding/unbinding which might provide intrinsic degrees of freedom to viruses (in the form of stochastic states). While these assumptions follow common practice [41], they nevertheless prevent us from addressing one of the more interesting hypotheses [45] about the lambda decision: that individual viruses ’vote’ for lysis or lysogeny. This hypothesis stems from the observation that, when viewed as a function of initial viral concentration (MOI/L, where L is cell length), the measured lysogeniza- tion frequency at each MOI during P+ infection exhibits a unique scaling structure with respect to MOI [45]. By rescaling the lysogenization frequency at each MOI

1/MOI as Plys(MOI/L) → Plys(MOI/L) , the frequencies for all MOI collapse onto 1/MOI 1/MOI a single curve, given by Plys(MOI/L) = Plys(MOI = 1/L) . An inter- pretation of this collapse is that it implies the decision at the cellular level is the integration of individual ’coin flips’ (dependent on viral concentration) made by each of the coinfecting phages (Figure 4.10). In this hypothesized ’voting’ picture, the 68

Figure 4.10 : The subcellular voting hypothesis. Coinfecting phages make independent decisions to lyse the cell (green) or transition to dormancy (red). Only if all phages vote for dormancy does the infection result in lysogeny. 69 decision at the cellular level exhibits an AND-gate logic: if each virus infecting a cell of length L votes for lysogeny (with probability Plys(MOI = 1/L)), the outcome is viral dormancy. However if even a single virus votes for lysis, the cell lyses. As interesting as this hypothesis is, the authors were unable to supply a specific mechanism that could explain it. Recent works have demonstrated deviations from ’mean field’ behavior in E. coli, such as 1) nonhomogeneous spatial patterns of DNA and proteins [100] and 2) formation of RNAP condensates which indicate spatial organization in transcription [101, 102]. Yet in the decade since the publication of the subcellular decision-making hypothesis [45], no theoretical attempt has yet been made to provide a mechanistic origin for phage voting.

4.3.2 Control of replication by initially infecting viruses is a candidate voting mechanism

One candidate mechanism motivated by our work is control of viral replication at the single phage level. As shown in Chapter 3, viral replication is essential for a lysis/lysogeny decision, and drives both pathways. Early rounds of replication push the infection towards lysogeny at lower MOI than in P- infection, while late viral replication at MOI = 1 enables the lytic predictor (Cro in our work) to reach the decision threshold in spite of transcriptional repression at the single phage level (see Chapter 3, Figure 3.6). Additionally, our work, consistent with previous observa- tions [19], found that CI repression of replication is important for differentiating lytic and lysogenic trajectories, with CI turning off replication at MOI > 1 (Chapter 3, Figure 3.4). These observations motivate us to ask whether independent control of replication can provide initially coinfecting viruses with a voting mechanism. Specifically, is the regulatory control of replication in initially coinfecting viruses sufficiently local for an individual virus to override ’mean field’ CI repression and continue replicating, thereby pushing the cell towards lysis? Investigating this question would require 70 single-cell resolution measurements of viral replication, gated by MOI, which to date have not been possible. Nevertheless, given the experimental advances showcased in Chapter 3, we believe such an experiment may be possible in the near future. 71

Appendices 72

Appendix A

Supplementary Information for Chapter 3

A.1 Experimental methods

A.1.1 Growth media and conditions

Media

Unless otherwise noted, the growth medium for all E. coli strains was LB (Lennox recipe [103]): One liter of medium was prepared with 10 g tryptone (BD Biosciences), 5 g yeast extract (BD Biosciences), and 5 g NaCl (Fisher Scientific), pH adjusted using 1 µM NaOH (Fisher Scientific). When applicable, the LB medium was supplemented with 10 mM MgSO4 (Fisher Scientific) and 0.2% maltose (Fisher Scientific), hereafter denoted as LBMM, or with 10 mM MgSO4 and 0.2% glucose (Fisher Scientific), hereafter denoted as LBGM. Phage plaque assays (see below) were performed using NZYM agar. One liter of NZYM medium was prepared with 22 g NZYM media (Teknova) and adjusted using 10 µM NaOH. LB and NZYM agar plates were prepared using the media above, with 1.5% weight/volume agar (BD Biosciences). All media– —with and without agar—–were autoclaved using a liquid cycle (121◦C, at least 25 minutes) for sterilization.

Growth conditions

Cells were streaked and grown overnight (14—16 hours) on LB agar plates supple- mented with antibiotics when applicable: 100 µg/mL ampicillin (Fisher Scientific), 50 µg/mL kanamycin (Fisher Scientific). Plates were incubated at the appropriate temperature: 30◦C for temperature sensitive lysogens, 37◦C otherwise. From plates, 73

fresh colonies were inoculated in 2 mL of LB or LBMM in 14 mL round-bottom test tubes (Falcon), supplemented with antibiotics when applicable. The overnight cul- tures were grown for 14–16 hours at the appropriate temperature with aeration (220 rpm) in a MaxQ 4000 Benchtop Orbital Shaker (Thermo Scientific). The growth con- ditions of overday cultures are described later for each specific experiment (Sections 3, 4 and 9). When applicable, the overday cultures were also supplemented with isopropyl-β-thiogalactoside (IPTG, Sigma-Aldrich) at the required concentration to induce ParB production. The overday cultures were grown in a MaxQ 7000 Water Bath Orbital Shaker (Thermo Scientific).

A.1.2 Bacterial strains and plasmids

All bacterial and phage strains, plasmids, primers and DNA oligos are listed in Tables

S1–S5. We used E. coli strains LE392 and MG1655. The phage used, λTY11 (λ cI 857 Pam80 stf::P1parS-kanR, described in Section 3 below), carries an amber mutation in the P gene [104], which allows replication of the phage genome in the amber suppressor strain LE392 but not in the wild type strain MG1655 [105]. We therefore utilized LE392 during construction and propagation of the phage (Section 3), and MG1655 for infection experiments where viral copy numbers were to be held constant (Sections 4 and 9). Intracellular phage genomes were labelled using the ParB-parS system [106, 54]. The parS sequence of phage P1, engineered into the lambda genome, was bound by a fluorescent fusion version of the P1 ParB protein (CFP-ParB) expressed from the plasmid pALA3047 (gift of Stuart Austin).

A.1.3 Phage construction

To fluorescently label the phage genome in the cell, we inserted the parS sequence and a kanamycin resistance cassette to the stf region of λ Pam80 (gift of Lynn Thomason). Next, we converted the wild-type cI in the phage’s genome to a temperature-sensitive allele (cI 857), to allow prophage induction via temperature shift [105] and for con- 74 sistency with previous studies [54, 1, 45, 2]. The details of phage construction are provided in the following subsections.

Construction of the pTY001 plasmid containing the parS sequence

We first constructed the plasmid pTY001, which is used to insert the parS sequence into the phage genomes, as follows. The plasmid carries a cassette with parS and kanR sequences, flanked by homologies to lambda stf region (considered nonessential for phages derived from lambda PaPa [107]). Our template was a phage strain in which the kanR-parS cassette was placed within the stf region (gift of Joel Stavans) [54]. We first amplified the kanR-parS cassette (with stf homologies) using the primers p1-parS-FP and p1-parS-RP (Table S4). Then, we cut both the PCR product and a pBS-SK vector (Stratagene) using KpnI and SacII (New England Biolabs). The insert and the vector backbone were then purified by electrophoresis using 1% agarose gel (Bio-Rad), followed by DNA extraction from the gel using Wizard SV Gel and PCR Clean-Up System (Promega). Finally, the insert and the vector were ligated using T4 DNA ligase (New England Biolabs). The pTY001 plasmid was then transformed into strain LE392 using the Bio-Rad MicroPulser Electroporator in accordance with the instrument’s instruction manual.

Construction of the λTY8 phage containing the parS sequence

We next constructed phage λTY8 (λ Pam80 stf::P1parS-kanR) by performing a phage- by-plasmid cross [61] between the parental phage λ Pam80 (carrying a C-to-T mu- tation at the 39,759 position in the lambda genome [104], an amber mutation in the P gene) and the plasmid pTY001 (described above), as follows. An overday culture of the host LE392 carrying plasmid pTY001 was prepared by diluting the overnight culture 1:1000 into 110 mL LBMM. The overday culture was grown in a 1 L baf- fled Erlenmeyer flask at 37◦C with 220 rpm aeration. Upon reaching OD600 ≈ 0.4 (measured using the Bio-Rad SmartSpec Plus Spectrophotometer), 20 mL of culture 75

(containing approximately 2×109 cells) were centrifuged at 4,000×g for 10 minutes at 4◦C (using the Thermo Scientific Sorvall Legend XTR Centrifuge), and the su- pernatant was removed. The cells were then resuspended in 200 µL of fresh LBMM medium. For infection, 200 µL of phage λ Pam80 was mixed 1:1 (equal volume) with the concentrated cells in a 1.5 mL Eppendorf tube, to reach MOI ≈ 0.5. The infection mixture was incubated at 37◦C for 15 minutes, then diluted into 15 mL of LBGM (prewarmed to 37◦C) in a 125 mL baffled Erlenmeyer flask and shaken at 37◦C for 120 minutes to complete a lytic cycle. Chloroform (final concentration of 5%) was then added to the media to lyse all remaining cells. The solution was centrifuged at 2,000×g for 10 minutes at 4◦C to remove the cell debris, and the phage stock (con- taining both the unmodified λ Pam80 and the modified λ Pam80 stf::P1parS-kanR, named λTY8) was harvested.

To screen for the recombinant phage λTY8, we first performed lysogenization and selected for lysogens harboring kanamycin resistance, as follows. An overday culture of LE392 was prepared by diluting the overnight culture 1:1000 into 25 mL of LBMM, and the culture was grown in a 250 mL baffled Erlenmeyer flask at 37◦C with 220 rpm aeration. Upon reaching OD600 ≈ 0.4, cells were harvested by centrifugation at 4,000×g for 10 minutes at 4◦C. The supernatant was removed, and cells were resuspended in 2.5 mL of fresh LBMM (at approximately 1×109 cells/mL). The phage

stock above, containing λTY8 as well as non-recombinant phages, was diluted in SM buffer (Teknova) to reach a concentration of approximately 1×109 plaque-forming units (PFU)/mL, then mixed with the concentrated cells to reach MOI ≈ 1. The infection mixture was incubated at room temperature for 20 minutes, then diluted into 1 mL of LBGM (prewarmed to 30◦C) in 14 mL round-bottom test tubes (Falcon) and shaken at 30◦C for 1.5 hours. 100 µL of the culture was plated on LB agar plates supplemented with 50 µg/mL kanamycin. The plates were incubated overnight at 30◦C. Only lysogens harboring recombinant prophages, which carry the kanamycin resistance cassette, survived and formed colonies under this selection. 76

Poly-lysogenic cells, carrying multiple tandem prophages, may harbor a mix- ture of nonrecombinant and recombined prophages [108], with the latter conferring kanamycin resistance to the cell. To screen for monolysogens, we picked a number of colonies from the lysogens above and performed PCR using primers e.coli attB, lambda attB and lambda int (Table S4) to detect for the presence (or absence) of the junction between multiple prophages within a polylysogen [109]. Next, lysogens

carrying a single recombinant prophage (denoted λTY8) underwent prophage induc- tion using mitomycin C (Fisher Scientific) following the protocol of [105], and the

phage lysate containing λTY8 was harvested.

Construction of the temperature-sensitive phage λTY11

We next constructed λTY11 (λ cI 857 Pam80 stf::P1-parS-kanR) by converting the wt cI allele in λTY8 (described above) to the temperature-sensitive allele cI 857 (con- sisting of a C-to-T mutation at the 37,742 position in the lambda genome [105]). The phage genotype was modified using a recombineering protocol for modifying an

infecting phage (here, λTY8) [110]. As host we used strain LE392 carrying the plas- mid pKM208 (Addgene). This plasmid contains the lambda gam, beta and exo genes [111], required for lambda recombination [19]. We designed the single-strand oligos cItsoligo-R and cIts-oligo-F (Table S4), which contained the target single-nucleotide mutation flanked by around 40 nt homologous sequences at both sides. We then fol- lowed the protocol of [110] and harvested the phage lysate, which contained both the

unmodified λTY8 and the recombinant λTY11. To screen for the recombinant phage ◦ λTY11, we took advantage of the fact that, at 37 C, the phage would form clear plaques

due to its temperature-sensitive cI 857 allele, whereas the unmodified λTY8 would form turbid plaques [105, 99]. We first diluted the phage lysate into SM buffer (Teknova), then mixed 10 µL of the diluted phage lysate (contains approximately 5×103 phages) with 100 µL of LE392 cells (at approximately 1×109 cells/mL) in a 1.5 mL Eppendorf tube. The mixture was incubated at 37◦C for 15 minutes, then added to 3 mL of 77

molten 0.7% NZYM agar (maintained at 47◦C) in a 14 mL round-bottom test tube (Falcon). The mixture was then poured onto a 1.5% NZYM agar plate and left to solidify at room temperature for approximately 15 minutes. The plate was incubated overnight at 37◦C to allow for plaque formation. The next day, 0.1% of plaques were clear, indicating successful conversion to the temperature-sensitive allele. To collect phages from the clear plaques, we first picked a whole plaque by penetrating the top agar around the clear plaque using a 100 µL pipet tip with a widecut end. Then, the agar piece with the plaque was soaked in 100 µL SM buffer in a 1.5 mL Eppendorf tube, yielding a lysate of the recombinant phages. Using a lysogenization protocol similar to Section 3.2 above,

we generated the lysogenic strain LE392 λTY11, for storage. The genotypes of cI 857 and Pam80 in the recombinant prophage were confirmed by sequencing (Lone Star Labs Genetic Sequencing) using the primers cI-seq-F, cI-seq-R, P-seq-F and P-seq-

◦ R (Table S4). The LE392 λTY11 lysogen underwent prophage induction at 42 C

following the protocol of [105], to obtain a phage lysate of λTY11.

A.1.4 Phage infection

Bulk lysogenization assay

We measured the probability of lysogenization as a function of MOI, using a protocol adapted from [45, 112]. An overnight culture of the host was diluted 1:1000 into 25 mL LBMM and grown in a 125 mL baffled Erlenmeyer flask at 37◦C with 220 rpm aeration. The phage lysate was diluted in SM buffer (Teknova) to yield a 4-fold dilution series between 107 and 1011 PFU/mL. Upon reaching OD600 ≈ 0.4, cells were centrifuged at 1,000×g for 10 minutes at 4◦C, and the supernatant was removed by pipetting. The cell pellet was resuspended in 1 mL of fresh, ice-cold LBMM. Then, 20 µL of the concentrated bacteria was combined with an equal volume of the diluted phage solution (at different concentrations, measured by standard plaque assay [113]), resulting in MOI in a range of ≈ 0.01-100. The infection mixture was 78

incubated on ice for 30 minutes, followed by an additional 5-minute incubation in a 35◦C water bath to trigger injection of the phage DNA [45]. Next, we diluted 10 µL of each infection mixture into 1 mL LBGM (prewarmed at 30◦C) in 14 mL round-bottom test tubes (Falcon), and incubated the diluted mixtures at 30◦C for 45 minutes with 220 rpm aeration. The cells were then diluted in ice-cold 1×PBS buffer to create a 10-fold dilution series between 103 and 107 cells/mL, and 100 µL of the diluted cells were plated on LB agar plates supplemented with 50 µg/mL kanamycin. The uninfected cells were also diluted and plated in a similar manner on LB agar plates. All plates were incubated overnight at 32◦C. The lysogenization frequency was determined by dividing the number of lysogen colonies (on kanamycin-selective plates) and uninfected colonies (on non-selective plates) on the next day.

Infection followed by smFISH

An overnight culture of MG1655 carrying plasmid pALA3047 was diluted 1:1000 into 75 mL of LBMM supplemented with 10 µM IPTG and grown in a 500 mL baffled Erlenmeyer flask at 37◦C with 220 rpm aeration. Upon reaching OD600 ≈ 0.4, cells were transferred to 50 mL centrifuge tubes (Corning) and centrifuged at 1000×g for 10 minutes at 4◦C. The supernatant was carefully decanted, and the cells were resuspended in fresh, ice-cold LBMM supplemented with 10 µM IPTG at 100× the original concentration (approximately 7×109 cells/mL after resuspension). We left some host cells uninfected as the negative control, treating them according to the smFISH procedure described in Section 5 below. Next, 500 µL of the concentrated host cells were mixed with 70 µL of λTY11 phage lysate (prepared as described in Section 3.3) to reach MOI ≈ 2. The infection mixture was incubated on ice for 30 minutes, followed by an additional 5 minute incubation in 35◦C water bath to trigger injection of the phage DNA [45]. Next, the infection mixture was diluted 1:1000 into 400 mL of LBGM (prewarmed to 30◦C) supplemented with 10 µM IPTG, split into two 2 L baffled Erlenmeyer flasks, and shaken at 30◦C. At various time points, 30 mL 79

of the culture was collected and treated according to the smFISH procedure described in Section 5 below.

Infection followed by DNA extraction and qPCR

The crude lysates of phage λ cI 857 Sam7 (for the λ P+ data series) and of phage λ Pam80 (for the λ P- data series) were produced by heat induction as in [105] from lysogenic cells (gift of Mike Feiss and Lynn Thomason, respectively). For λ P+, the Sam7 genotype was selected because the amber mutation in the phage S gene prevents cell lysis in the non-suppressor MG1655 strain [105]. This allowed us to measure the number of phages in the cells throughout the lytic cycle. An overnight culture of MG1655 was diluted 1:1000 into 100 mL of LBMM in a 1 L baffled Erlenmeyer flask and grown at 37◦C with 220 rpm aeration. Upon reaching OD600 ≈ 0.4, cells were centrifuged at 2,000×g for 10 minutes at 4◦C. The supernatant was removed, and the cells were resuspended in fresh, ice-cold LBMM at 100× the original concentration (approximately 7×109 cells/mL). The phage lysate was diluted in ice-cold SM buffer (Teknova) to achieve a titer of approximately 4×109 PFU/mL. Next, 300 µL of the cold phage lysate was added to 300 µL of the cold concentrated bacteria in a 1.5 mL Eppendorf tube to reach MOI ≈ 0.5. The infection mixture was gently mixed by pipetting. A negative control was also prepared by mixing cells with DEPC-H2O (Invitrogen), using the same 1:1 volume ratio. The infection mixture and the negative control were incubated on ice for 30 minutes to allow phage adsorption. Genome injection was triggered by shifting the samples to a 35◦C water bath for 5 minutes [45]. Then, 50 µL aliquots of the infection mixture (or the negative control) were each diluted 1:500 into 25 mL of LBGM (prewarmed to 30◦C) in multiple 250 mL baffled Erlenmeyer flasks, then grown at 30◦C and 220 rpm aeration. At each time point, the entire diluted infection mixture from single flasks was treated according to the DNA extraction procedure described in Section 9 below. 80

A.1.5 Single-molecule fluorescence in situ hybridization (smFISH)

The smFISH protocol was described in detail previously [56]. Briefly, sets of antisense DNA oligo probes were designed against lambda cI, cro and cII mRNA, synthesized with a 3’ amine modification (LGC Biosearch Technologies), and all oligos against a given gene were pooled together. We covalently linked the cI, cro and cII probe sets to TAMRA, Alexa 594 and Alexa 647 (Invitrogen) respectively, and purified them using ethanol precipitation. The probe concentration and dye labeling efficiency were measured using a NanoDrop 2000 spectrophotometer (Thermo Scientific). Probe se- quences are listed in Table S5. Following the cell growth and infection procedure described in Section 4.2 above, cells were fixed and permeabilized, then incubated with the fluorescently labeled probe sets, washed, and finally imaged as described in Section 6 below. We made the following modifications relative to the original proto- col from [56]. At each time point, 30 mL cell culture was directly mixed with 7.5 mL of 18.5% formaldehyde solution in 5×PBS (to a final concentration of 3.7% formalde- hyde in 1×PBS) and nutated for 30 minutes at room temperature. In addition, when permeabilizing the cells, we resuspended the cell pellets in 750 µL of DEPC-H2O (In- vitrogen), mixed thoroughly with 250 µL of 100% ethanol by pipetting, then mixed gently for 1 hour at room temperature using a nutator. We found that a final ethanol concentration of 25%, instead of 70% as in the original protocol, better preserved the fluorescence signal of CFP-ParB without harming the permeability of cells to smFISH probes (Data not shown).

A.1.6 Microscopy

We used an inverted epifluorescence microscope (Eclipse Ti, Nikon), equipped with motorized stage control (ProScan III, Prior Scientific), a universal specimen holder, a mercury lamp (Intensilight C-HGFIE, Nikon), and a CMOS camera (Prime 95B, Photometrics). A ×100, NA 1.40, oil-immersion phase-contrast objective (Plan Apo, Nikon) was used, as well as a ×2.5 magnification lens (Nikon) in front of the camera. 81

The fluorescent filter sets used in the study were as follows: CFP (Nikon, 96341), Narrow Cy3 (Chroma, SP102v1), Narrow Cy5 (Chroma, 49307), and a customized set for imaging Alexa 594 (Omega, excitation filter: 590±10 nm; dichroic beam splitter: 610 nm; emission filter: 630±30 nm). After the fixation, hybridization and washing steps (described in Section 5), we mounted the cells between two coverslips as described in [56]. The sample was then placed onto the microscope’s slide holder and the cells were visually located using the phase-contrast channel. Images were acquired in the following order: phase-contrast (100 ms, to detect the cell outline), CFP (200 ms, CFP-ParB), Narrow Cy3 (500 ms, cI-TAMRA), 594 cube (500 ms, cro-Alexa 594), and Narrow Cy5 (500 ms, cII-Alexa 647). Snapshots were taken at 5 z-positions (focal planes) with steps of 300 nm. A set of images with multiple z-positions is denoted as an “image stack” and the image of each z position as a “z-slice”. Images were acquired at multiple positions on the slide, to image a total of 400–2000 cells per sample (typically 9-20 positions).

A.1.7 Cell segmentation and spot recognition

Cell segmentation

Our procedure for identifying cells from the phase-contrast channel follows [56, 53]. In every image stack, we first identified the “in-focus” z-slice, defined to be the one with the largest variance among pixels. Then we used Schnitzcells [114] to generate cell segmentation masks, i.e., matrices with the same dimension as the phase-contrast image, in which pixels of cells have integer values corresponding to the identification number of the cell, while non-cell pixels have a value of zero. Finally, we visually inspected the segmentation results; poorly segmented cells were either manually cor- rected or discarded using the graphical user interface within the software. 82

Spot recognition

Following [56, 53], we used Sp¨atzcells[56] to identify and measure the properties of fluorescent foci (spots) from the different fluorescent channels. Briefly, the Sp¨atzcells software first identified spots by finding two-dimensional local maxima in the fluo- rescence intensity above a user-defined “detection threshold”. Next, the fluorescence intensity profile around each spot was fitted to a 2D elliptical Gaussian. The proper- ties of each spot were obtained from the fitting results, including the position, spot area, peak height of the fitted Gaussian, and spot intensity (integration of the volume underneath the fitted Gaussian). In cases where other spots were close to the spot being fitted, the software performed a 2D multi-Gaussian fit over all detected spots instead.

Discarding false-positive spots

Using a low detection threshold during spot recognition (Section 7.2) ensured that all genuine spots were detected. However, because of the low threshold, the number of false positives (from background noise, or nonspecific binding by smFISH probes and CFP-Par) also increased. To discard the false positives, we performed a gating procedure following [56, 53]. Briefly, we compared the 2D scatter plots of peak height versus spot area for all detected spots in the experimental samples (infected cells) to that from the negative control (uninfected host cells). A polygon was manually chosen in the plane of peak height and spot area, such that most spots from the negative sample were located outside of it. This polygon served as the gating criteria for all samples, with spots located outside of discarded. The choice of gating was confirmed by manual inspection of spots in a subset of images. 83

A.1.8 Data analysis following cell segmentation and spot recognition mRNA quantification

Defining the fluorescence intensity of a single mRNA molecule was performed as de- scribed in [56, 53]. Briefly, after discarding the false positive spots, we examined the early infection samples, which exhibit low mRNA levels, and in which individ- ual mRNA molecules were spatially separated. The histograms of spot intensities were fitted to a sum of three Gaussians corresponding to one, two, and three mRNA molecules per spot. The fluorescence intensity of a single mRNA molecule was es- timated to be the center of the first Gaussian. We then divided the measured spot intensity of each mRNA spot by the single-mRNA intensity to obtain the number of mRNA molecule number in that spot. The total number of mRNA molecules in a given cell was calculated by summing the mRNA molecule numbers represented by all spots within the cell.

Identification of single-cell MOI

To verify that CFP-ParB spots correspond to individual phage genomes, we con- firmed that, in lysogenic cells, the mean number of CFP-ParB spots per cell was consistent with the expected prophage copy-number under the specific growth condi- tions [115, 116] (Figure S2). Therefore, our estimated number of infecting phages in each individual cell (single-cell MOI) was the CFP-ParB spot number in the cell, as measured by Sp¨atzcells(Section 7.2). Infected cells were grouped based on single- cell MOI and the corresponding levels of cI, cro and cII mRNA calculated for each group were used in constraining the mathematical model, as described in Theoreti- cal Methods. 84

A.1.9 DNA extraction and quantitative PCR (qPCR)

Calibration curves

We used two pairs of primers: One targeting a 154-bp region in the cI gene of phage lambda, and another targeting a 150-bp region in lacZ of E. coli [117]. The primer sequences are provided in Table S4. The amplification efficiencies of the primer sets were first determined as follows. Two 25 mL cultures of non-lysogens

(MG1655, containing lacZ ) and lysogens (MG1655 λIG2903, containing both cI and ◦ lacZ ) were grown at 30 C in LB supplemented with 10 mM MgSO4 with 220 rpm aeration. Upon reaching OD600 ≈ 0.4, 2 mL of cells (containing approximately 2×108 cells) were centrifuged at 21,130×g (max speed) for 1 minute using a Eppendorf 5420 centrifuge, and the supernatant was removed. Genomic DNA was extracted from the cell pellet using the PureLink Genomic DNA Mini Kit (Invitrogen), according to the kit’s instructions for Gramnegative bacteria. The concentration and quality (A260/A280 and A260/A230) of the DNA extracts were measured using a NanoDrop 2000 spectrophotometer (Thermo Scientific). A serial dilution series of the lysogen DNA was prepared, spanning five orders of magnitude (5 pg to 50 ng of DNA per 20 µL reaction). The DNA extracted from non-lysogens served as the negative control for the cI target. Each DNA sample was then subjected to both pairs of primers (for cI and lacZ targets). qPCR was performed using SsoAdvanced Universal SYBR Green Supermix (Bio-Rad), in either a MiniOpticon Real-Time PCR System (Bio-Rad) or a CFX Connect Real-Time PCR Detection System (Bio-Rad). We prepared 20 µL of each qPCR reaction comprising 1× supermix, 500 nM of the forward and reverse primers (each), DNA template (5 pg to 50 ng, diluted as above), and DEPC-H2O. No-template controls (NTC) were also prepared by replacing DNA with DEPC-H2O. Each sample was prepared in technical duplicates or triplicates. The thermal cycle was chosen based the instruction manual, specifically, 3 minutes at 98◦C, followed by 40 cycles of 15 seconds at 98◦C and 30 seconds at 60◦C, then finally a melt 85 curve analysis from 65◦C to 95◦C. The amplification efficiencies of the cI and lacZ targets were determined from the calibration curves following standard procedure [25, 26]XXX. Calculated from 10 and 8 biological replicates, respectively, the efficiencies for the cI and lacZ targets were 1.998 (or 99.8%) and 1.956 (or 95.6%), respectively.

Quantification of phage copy numbers following infection

Infection of MG1655 by λ cI 857 Sam7 or λ Pam80 was performed as described in Section 4 above. At each time point, 25 mL of the diluted infection mixture (con- taining approximately 2×108 cells) was poured into a pre-chilled 50 mL centrifuge tube (Corning) and incubated in an iced water bath to stop cell activity. The nega- tive control (cells mixed with DEPC-H2O, no phage) was collected at the end of the experiment. All samples were centrifuged at 4,000×g for 10 minutes at 4◦C, and DNA was extracted from the cell pellets, followed by concentration and quality measure- ments as above. The qPCR reactions were set up and performed as above, with 50 pg of template DNA for each sample. Using the target-specific amplification efficiency (measured above), we performed a relative quantification of cI, with lacZ serving as a reference gene in accordance with the efficiency-correction quantification method [118, 119], yielding the ratio of cI to lacZ templates in each sample. We then used these values and the estimated copy number of the lacZ locus per cell (approximately 2.6 under the relevant growth conditions [115]) to estimate the average copy number of the cI locus—a proxy for the number of lambda genomes per cell in each sample.

A.2 Theoretical Methods

A.2.1 Overview of the Governing Differential Equations

As discussed in the main text, our model focuses on three genes at the center of the lambda cell-fate decision: cI, cro, and cII. Based on the known interactions in the lambda network (discussed in the main text and in more detail below in Section 86

A.2.2), we constructed a deterministic model describing the dynamics of cI, cro, and cII mRNA and protein concentrations, as well as viral concentration. The interac- tions within this 3-gene network are shown schematically in Figure 2A (main text).

Denoting the mRNA and protein concentrations of gene x as [mx] and [X] respec- tively, and viral concentration as [λ], the model consists of the following differential equations:

d[m ]   cI = [λ] k1,RMf ([CI], [Cro]) + k1,REf ([CII]) − k1 [m ] (A.1) dt tx cI,RM tx cI,RE m cI

− kd[mcI ], (A.2) d[m ] cro = [λ]k2 f ([CI], [Cro]) − k2 [m ] − k [m ], (A.3) dt tx cro m cro d cro d[m ] cII = [λ]k3 f ([CI], [Cro]) − k3 [m ] − k [m ], (A.4) dt tx cII m cII d cII d[CI] = k1 [m ] − k1 [CI] − k [CI], (A.5) dt tr cI P d d[Cro] = k2 [m ] − k2 [Cro] − k [Cro], (A.6) dt tr cro P d d[CII] = k3 [m ] − k3 d ([CII])[CII] − k [CII], (A.7) dt tr cII P CII d d[λ] = [λ]k r ([CI], [Cro], t) − k [λ]. (A.8) dt λ λ d

α α α α Here, ktx, ktr, km, kP denote the rate constants for transcription, translation, mRNA degradation, and protein degradation respectively (with superscript α denoting the affiliated gene: 1 = cI, 2 = cro, and 3 = cII ), kλ is the rate of viral replication, and kd is the rate of dilution due to cell growth. As cI is transcribed from two promoters (as discussed in the main text and Section A.2.2 below), we use additional RM and RE superscripts to differentiate transcription rates from each promoter. Values of these and other parameters used to generate the figures in the main text are estimated as described in Section A.2.4 and summarized in Theoretical Methods Table 3 87

(Section A.2.7). Equations A.1–A.8 are based on several simplifying assumptions commonly used for modeling lambda and other bacterial systems. Following previous works in lambda [32, 33], we assumed that transcription rates are proportional to viral concentration, i.e. that transcription fluxes from each additional viral copy are the same. Follow- ing standard practice [41], we modeled translation as being proportional to mRNA concentration in Eqs. A.5–A.7, and modeled mRNA degradation (Eqs. A.1–A.4) as a first-order reaction. Assuming exponential growth of cell volume (following pre- vious models of lambda [34, 120] and of other bacterial systems [121, 122, 123]) leads to the first-order effective degradation term in Eqs. A.1–A.8 with rate kd. To model transcriptional regulation in Eqs. A.1–A.4, we introduced transcription reg- ulatory functions (fcro, fcII , fcI,RM , and fcI,RE). These functions are dependent on transcription factor concentrations and used as multiplicative factors to change the transcription rate of individual genes [124]. To describe protein degradation, we mod- eled non-specific decay of the stable proteins CI (Eq. A.5) and Cro (Eq. A.6) as a first-order reaction with a constant rate (again in line with common practice [41]). In contrast, CII is actively degraded, and its degradation is regulated [12]. Therefore, the degradation term in Eq. A.7 is multiplied by a degradation regulatory function, dCII. Viral replication in Eq. A.8 is modeled as exponential growth of the number of viruses with the rate proportional to a replication regulatory function (rλ). The func- tional forms and biological basis for all regulatory functions are described in Section A.2.2. All codes and parameter/datasets can be found in the following GitHub repository: https://github.com/sethtcoleman/Replication-Manuscript.git

A.2.2 Formulation of Regulatory Functions

Although phage lambda is a comparatively well-characterized system, the molecular details of many regulatory interactions are still uncertain [19]. We therefore designed 88

our model to capture known regulatory interactions phenomenologically. Below we

describe how transcription (fcro, fcII , fcI,RM, and fcI,RE), CII degradation (dCII), and

viral replication (rλ) regulatory functions were formulated.

Transcription

Motivated by previous work that characterized transcription regulation phenomeno- logically [36, 37], we modeled the effect of transcription factors on promoter activity at the single copy level using functions composed of standard Hill terms for activa- tion ([X]n/ (Kn + [X]n)), repression (Kn/ (Kn + [X]n)), and their generalizations to combinatorial gene regulation by multiple factors [124]. We applied this approach to

cI transcription from PRM and PRE and cro and cII transcription from PR.

cI Transcription cI is transcribed from two promoters: PRE (CII-activated) and

PRM (regulated by CI and Cro). For modeling cI transcription from PRE, we fol- lowed common practice and ignored basal transcription [32, 33, 34]. The single phage

transcription regulatory function for cI expression from PRE is a Hill function

n  [CII]  RE K f ([CII]) = RE . (A.9) cI,RE  nRE 1 + [CII] KRE

This function is used to define PRE activity in the main text (Figure 4A).

CI activates its own expression from PRM at low concentration, but represses PRM

at high concentration [30]. We assume competitive binding for Cro and CI at PRM, in line with previous models [65, 32, 34]. We extend this competitive binding logic to CI-activated and CI-repressed states, modeling each with distinct Hill activation

or repression terms. Defining αRM as the fold-change for transcription from PRM in the CI-activated state, the single phage transcription regulatory function for cI

expression from PRM is 89

na  [CI]  RM,CI 1 + αRM a KRM,CI fcI,RM([CI], [Cro]) = na nr n ,  [CI]  RM,CI  [CI]  RM,CI  [Cro]  RM,Cro 1 + a + r + KRM,CI KRM,CI KRM,Cro (A.10)

where αRM is the fold-change for transcription from PRM in the CI-activated state. The parameters for CI-repressed and CI-activated states are denoted with r and a superscripts for repression and activation respectively.

cro Transcription cro is transcribed from PR, and its transcription is repressed by itself and by CI. Employing the competitive binding assumption outlined in Section A.2.2, the single phage transcription regulatory function is

1 f ([CI], [Cro]) = . (A.11) cro  ncro,CI  ncro,Cro 1 + [CI] + [Cro] Kcro,CI Kcro,Cro

This function defines PR activity in the main text (Figure 6A). To describe the strength of CI repression of cro transcription (main text, Figure 4C), we used the normalized weight of the CI-dependent term in Eq. A.11, which we heuristically define as the probability of CI repressing cro:

n  [CI]  cro,CI K Probability of CI repressing cro = cro,CI . (A.12)  ncro,CI  ncro,Cro 1 + [CI] + [Cro] Kcro,CI Kcro,Cro 90

cII Transcription cII is also transcribed from PR, however its transcription re-

quires readthrough of the terminator tR1 [12]. This readthrough is facilitated by the action of another lambda protein, N, that forms an antiterminator complex with host factors [12]. N is short-lived [12], and its transcription from promoter PL is repressed by CI and Cro at concentrations similar to those that PR is represssed at [30]. Based on this, the qualitative similarity in the cro and cII mRNA kinetics (main text,

Figure 2C), and the observation that the termination efficiency at tR1 is relatively low (66% per [43]), we phenomenologically account for antitermination in our model by allowing the transcription regulatory function governing cII transcription to have parameters different from those describing cro transcription (Eq. A.11). The single phage transcription regulatory function describing cII transcription is then

1 f ([CI], [Cro]) = . (A.13) cII  ncII,CI  ncII,Cro 1 + [CI] + [Cro] KcII,CI KcII,Cro

CII Degradation

CII is actively degraded by the host protease FtsH, and has a half-life on the order of minutes [12]. This degradation rate is reduced by the presence of another lambda protein, CIII, through an undetermined mechanism [57]. Transcription of cIII (from

promoter PL) is repressed by both Cro and CI at concentrations similar to those that

PR is represssed at [30]. Like CII, CIII is also actively degraded by FtsH, and also has a half-life on the order of minutes [19]. Given the similarilty in transcriptional regulation and degradation of CII and CIII, we use CII concentration as a proxy for that of CIII. As a result the degradation regulatory function takes the following form:

1 d ([CII]) = . (A.14) CII  nCII 1 + [CII] KCII 91

This functional form is motivated by a previous approach [36].

Viral Replication

Viral copy number changes during infection as a result of viral replication [125, 2] which occurs through two primary modes: theta (also called circle-to-circle) and sigma (also called rolling circle) [19]. Lambda replication proteins O and P, which are

transcribed from PR, are important for both modes of replication, and the dominant mode of replication switches from theta to sigma during infection [19]. Theta mode

replication requires active transcription from PR [12]. Repression of PR by CI and Cro therefore prevents both theta mode of replication and transcription of O and P [12]. The molecular details of the different modes of replication and the switch from theta to sigma are still under investigation [19]. We follow the approach of [34, 2] and coarse-grain replication into a single mode: exponential replication with a CI- and Cro-dependent rate given by the regulatory function

H(t − τ ) r ([CI], [Cro], t) = λ . (A.15) λ  nλ,CI  nλ,Cro 1 + [CI] + [Cro] Kλ,CI Kλ,Cro

Notably, to account for the previously reported [2] delay in the onset of replication following infection (assumed to occur at t = 0), we have introduced the factor H(t −

τλ) with the Heaviside function (H(t) = 1 if t ≥ 0, 0 otherwise). Parameter τλ characterizes the duration of this delay. The functional form of Eq. A.15 captures the repression of PR by both CI and Cro. However, we allow the parameters used in this equation to differ from those in Eq. A.11 to reflect the uncertainty in the molecular details of replication suppression. To characterize the role of CI in repression of replication (main text, Figure 4D), we heuristically define the probability of CI repressing replication as the normalized 92

weight of the CI-dependent term in the denominator of Eq. A.15:

n  [CI]  λ,CI K Probability of CI repressing replication = λ,CI . (A.16)  nλ,CI  nλ,Cro 1 + [CI] + [Cro] Kλ,CI Kλ,Cro

A.2.3 Infection Simulation Methods

We numerically integrate the model ODEs (Eqs. A.1–A.8 using MATLAB’s built-in ode15s solver, to account for possible stiffness due to nonlinear regulation terms and differences in time scales. For infection with P- phages, the replication rate in Eq.

A.8 is 0, and the ODE can be solved analytically. Defining initial cell volume as V0

(so that V (t) = V0exp(kdt)) and initial viral concentration as

[λ]0 = C · MOI/V0, (A.17)

(where C is the conversion constant from volume to molar concentration), we express the viral concentration for P- infection as

−kdt [λP -](t) = [λ]0e . (A.18)

Model results in the main text are from simulations run for 60 minutes, which marks the end of the time series of P- mRNA measurements (Figure 2C, main text). All initial mRNA and protein concentrations are set to 0, and initial viral concentration (Eq. A.17) is determined by the MOI and the initial cell volume. All concentrations are in units of nM. To simulate the delayed infection scenario where infection by a second phage

occurs τd minutes after initial infection (main text, Figure 4), the model ODEs in

Section A.2.1 were integrated until τd minutes after infection. Then we adjusted 93 phage concentration by computing the corresponding change in viral concentration due to the addition of the second phage at the current volume, V (τd). ODEs were then integrated from τd to t = 60 min using the mRNA and protein concentrations at τd and the updated viral concentration as the initial conditions.

A.2.4 Parameter Fitting

Three data sets are used to fit the model: 1) cI, cro, and cII mRNA numbers from smFISH measurements during P- infection (described in the main text; see Exper- imental Methods for details), 2) measurements of viral copy number using qPCR during MOI = 1 P+ infection (also described in the main text; see Experimental Methods for details), and 3) previously published cI and cII mRNA numbers from smFISH measurements during infection with wild-type (cI +cro+P+), cro-, cI-, and cro-P- mutants (collectively denoted by subscript ’Z’ in this text; see [2]). All model parameters are fitted simultaneously, by minimizing the error between model output from simulations run for 60 minutes and experimental data, as described in the next section. As we fit to population-averaged mRNA numbers, data points with only a single sample were ignored when fitting (see Table S6).

Objective Function

Denoting model parameters by the vector θ, and using hat notation for the model output (i.e.,y ˆ), we fit the model to the data by minimizing the objective function 94

5 3  2 X X X yP -,α(MOI = i, t) − yˆP -,α(MOI = i, t, θ) J(θ) = max [yP -,α(MOI = i, t)] i=1 α=1 tTP - ˆ !2 X λP +(MOI = 1, t) − λP +(MOI = 1, t, θ) + γP + max [λP +(MOI = 1, t)] tTP + (A.19)  2 X X X µg,α(MOI = 1, t) − µˆg,α(MOI = 1, t, θ) + max [µg,α(MOI = 1, t)] gGZ α{1,3} tTg,α 13 X + Pi (θ) . i=1

In Eq. A.19, genotype is indicated in the subscript of fitted terms (e.g. P+ in λP +), and the subscript α  {1, 2, 3} denotes mRNA species (1 = cI, 2 = cro, and 3 = cII). The first term in Eq. A.19 describes the error in fitting the P- data, the second term describes the error in fitting the P+ data, and the third term describes the error in fitting the previously published smFISH data. The last term describes penalties derived from qualitative relationships between parameters and observables described in the next section. To avoid problems with differences in scale in the residuals across data sets, we normalize each residual in the first three terms by the maximum of the relevant data subset. While there may be differences between our smFISH experiments and those in [2] affecting total mRNA counts, we assumed that relative scaling in mRNA copy number between genotypes is preserved across experiments. To fit the previously published smFISH data, we thus compared relative differences between rescaled simulated and measured mRNA numbers, using maximum mRNA values in selected reference geno- types as normalization factors. These normalized terms are denoted µ in Eq. A.19, and for a given genotype g and mRNA species α take the form

yg,α (MOI = 1, t) µg,α (MOI = 1, t) = , (A.20) max[yG(α),α (MOI = 1, t)] 95

where G(α) denotes the reference genotype for mRNA species α. For cII mRNA measurements, we used the P- mutant as the reference genotype. For cI mRNA measurements, we used the cro-P- mutant, as cI mRNA during P- infection was not measured in [2]. The number of data points in our P+ replication data is significantly smaller (7 data points) than that in either the P- smFISH measurements (150 data points) or the previously published smFISH data (50 data points). Therefore, to ensure simultaneous fits to all data captured trends in the replication data, we assigned an additional weight (γP + = 8) to the P+ error term. The value of this weight approximately corresponds to the ratio in the number of points between the previously published smFISH and P+ data sets.

Penalties

While most parameters in the model have never been directly measured, qualitative relationships between some parameters can be gleaned from previous experimental measurements, resulting in inequality constraints. Such constraints can also be de- duced for the kinetics of some observables, such as viral copy number. Because particle swarm optimization (described in Section A.2.4) does not require a differ- entiable objective function, we directly penalized violations of inequality constraints using a combination of simple Heaviside functions and quadratic terms (motivated by exact penalization in direct search methods—see [126, 127]). To avoid scaling issues, we normalized the quadratic penalties by the larger value in the violated inequality. We also included ’count’ penalties [128] to ensure penalty terms have a non-negligible weight in the objective function even when the violation of the inequality is small. For a given argument pair, xj(θ) and xk(θ), a penalty term based on the strict inequality constraint xk(θ) < xj(θ) has the form 96

2 Pi(θ) = γP,1H(xk(θ) − xj(θ)) ((xk(θ) − xj(θ)) /xj(θ)) + γP,2H(xk(θ) − xj(θ)). (A.21)

Here the first term corresponds to the quadratic penalty for the constraint, the second term is the count penalty, and H(x) is the Heaviside function.

The penalty weights (γP,1 and γP,2) were both fixed at 0.1, a value empirically found to result in fits that satisfy constraints without compromising fit quality. A full list of parameter constraints is given in Theoretical Methods Table 1 (Section A.2.7).

Particle Swarm Optimization

We minimized the objective function in Eq. A.19 using MATLAB’s built-in algorithm for particle swarm optimization (PSO). PSO is a global optimization algorithm [58] that performs well for a wide range of optimization problems [129, 130, 131]. Opti- mization runs were carried out using randomized initial positions in parameter space, and the fitting code was parallelized to run on the NOTS high performance comput- ing cluster at Rice University. Internal parameters of the PSO algorithm that were modified from their default values in MATLAB are listed in Theoretical Methods Table 2 (Section A.2.7). We use the top 25% of fits (as measured by the value of the objective function) obtained from end-points of multiple separate PSO runs for our ensemble of param- eter sets. 97

A.2.5 Decision Thresholds

Ranges for CI and Cro thresholds for the lysis-lysogeny decision were obtained using previously published observations on the frequency of each fate. For P+ infection, lysis is the dominant outcome during at MOI = 1 [45]. How- ever, lysis has not been observed with P- phages, either in bulk [59] or in single-cell resolution experiments over a range of MOI [2]. Based on these observations, we used our model simulations to constrain possible Cro thresholds for lysis:

• Maximum lytic threshold: the maximum Cro concentration reached in simula- tions of P+ infection at MOI = 1,

• Minimum lytic threshold: the maximum Cro concentration reaching in simula- tions of P- infection at MOI = 5.

To constrain the lysogenic threshold, we first defined the MOI at which the decision switches from lysis to lysogeny during P+ infection. As discussed in the main text, we interpret the observed separation of trajectories at MOI = 1 and MOI = 2 (main text, Figure 3B) as an indication of a switch in decision outcome. Therefore, we assume that the lysogenic threshold for CI is reached at MOI= 2 and not reached at MOI= 1 Based on this interpretation, we obtain the following constraints for the lysogenic threshold:

• Maximum lysogenic threshold: the maximum CI concentration reached in sim- ulations of P+ infection at MOI = 2.

• Minimum lysogenic threshold: the maximum CI concentration reached in sim- ulations of P+ infection at MOI = 1.

Notably, this identification of a transition in infection outcome from MOI = 1 to MOI = 2 is consistent with a previous simple model of Kourilsky et al. [1], who also found that the dominant outcome switches to lysogeny at MOI = 2 during P+ infection. 98

More recent lambda models have also assumed the lysis-to-lysogeny transition occurs at MOI = 2 based on this previous simple model [32, 132, 36]. To prevent cases where neither threshold is reached during P+ infection for 1 < MOI < 2 when scanning MOI as a continuous variable for the fate diagram (main text, Figure 6E), we scanned over MOI in this range, decreasing the maximum values of lytic or lysogenic thresholds further until a threshold is always crossed. In principle, any decision threshold within the defined bounds can lead to the prediction of cell fate that will satisfy the above-mentioned constraints. For simplic- ity and to achieve the maximally robust predictions, we set the lytic and lysogenic thresholds to the midpoints of their respective threshold ranges for all figures shown in the main text. We note that while changes in the values of the thresholds may affect the size and exact boundaries of fate regions in Figure 6E (main text), the topology of the fate diagram is conserved. For example, changes in the value of the lysogenic threshold affect the value of MOI at which the infection switches from failed infection to lysogeny during P- infection from 3 to 4. On the other hand, changes in the lytic threshold affect the MOI at which the mixed outcome regime occurs (grey region in Figure 6E, main text).

A.2.6 Modeling Bulk Lysogenization

The fraction of cells undergoing lysogeny as a function of the population-averaged MOI was measured as described in the Experimental Methods (Section 4.1). To estimate MOI*, the single-cell MOI at which the transition to lysogeny occurs, we followed the approach of [1, 45, 133]. First, we assume that phage-cell encounters in the infection mixture follow Poisson statistics. Therefore, the probability that a cell is infected by n phages, given an average MOI of M, is:

M ne−M P (n, M) = . (A.22) n! 99

We next assume that coinfection by MOI* phages or more results in lysogeny, while coinfection by fewer than MOI* does not. In other words, the probability Q that a cell is lysogenized at single-cell MOI = n follows:

 0, (n < MOI∗) Q(n) = (A.23) 1, (n ≥ MOI∗) .

ˆ The model-predicted lysogenization frequency Plys, given an average MOI of M, is then found by summation over all possible values of n:

∞ ∞ MOI∗−1 X X M ne−M X M ne−M Pˆ (M) = Q(n)P (n, M) = = 1 − . (A.24) lys n! n! n=0 n=MOI∗ n=0

To evaluate MOI*, we fitted Eq. A.24 to the experimentally measured lysogenization frequencies Plys(M) by minimizing the objective function:

2 MOI∗−1 n !! X X (a · M) e−a·M J(a, b, MOI∗) = log (P (M)/b) − log 1 − . lys n! M n=0 (A.25) Here, a and b are fitted normalization factors accounting for experiment-to-experiment errors in measuring the absolute numbers of phages and bacteria, and MOI* is al- lowed to take the values 1, 2, and 3. Fitting was performed in logarithmic space, since the serial dilutions used to scan bulk MOI result in data spanning multiple orders of magnitude. Figure 6D depicts the best fit for each dataset, with the experimental data rescaled by the fitting parameters, (i.e., 1/b · Plys(M) vs. a · M).

A.2.7 Theoretical Methods Tables 100

Table A.1 : Description of constraints used in fitting

Constraint Description Reference 1,RM 1,RE ktx αRM < ktx PRE has a higher maximum Herskowitz and Hagen,

transcription rate than PRM Annu. Rev. Genet., 1980 1,RM 2 ktx αRM < ktx PR has a higher maximum Dodd et al., Genes and

transcription rate than PRM Dev., 2004 1,RE 2 ktx < ktx PR has a higher maximum Palmer et al., Mol. Cell,

transcription rate than PRE 2009 1 2 ktr < ktr CI is translated at a slower Liu et al., PNAS, 2013 rate than Cro

1 3 ktr < ktr CI is translated at a slower Liu et al., PNAS, 2013 rate than CII

a Kcro,CI < KRM,CI CI represses cro transcrip- Ptashne, The Genetic tion before activating its Switch, 1986 own transcription

a r KRM,CI < KRM,CI CI activates its own tran- Ptashne, The Genetic scription at lower concentra- Switch, 1986 tion than that at which it re- presses its own transcription

a KcII,CI < KRM,CI CI represses cII transcrip- Ptashne, The Genetic tion before activating its Switch, 1986 own transcription

KRM,Cro < Kcro,Cro Cro represses PRM before re- Ptashne, The Genetic pressing its own transcrip- Switch, 1986 tion

KRM,Cro < KcII,Cro Cro represses PRM before re- Ptashne, The Genetic pressing cII transcription Switch, 1986 101

Kcro,CI ≤ Kλ,CI CI suppression of replica- Oppenheim, Annu. Rev. tion occurs at or above the Genet., 2005 concentration threshold for

repression of PR

Kcro,Cro ≤ Kλ,Cro Cro suppression of replica- Oppenheim, Annu. Rev. tion occurs at or above the Genet., 2005 concentration threshold for

repression of PR λP +(MOI=1, t=40 min) > 5 At least 5-fold more viruses Shao et al., iScience, 2018 λcro-(MOI=1, t=40 min) are produced by 40 min dur- ing P+ infection vs. cro- in- fection

Table A.2 : Internal particle swarm optimization parameters changed from MAT- LAB’s built-in defaults.

Swarm Size Min. Neighbors Max. Stall Inertia Range Fraction Iterations 100 0.1 200 [0.3, 1] 102

3 Table A.3 : Fitted parameters used for main text figures (with V0 = 1µm ).

Parameter Description Value (main text)

1,RM −1 ktx Basal cI transcription rate 0.205 min

from PRM

αRM Fold-change in PRM activity 5.00 1,RE −1 ktx cI transcription rate from 1.04 min

PRE 1 −1 ktr cI mRNA translation rate 0.887 min 2 −1 ktx cro transcription rate 2.54 min 2 −1 ktr cro mRNA translation rate 0.919 min 3 −1 ktx cII transcription rate 5.29 min 3 −1 ktr cII mRNA translation rate 1.10 min −1 kλ Viral replication rate 0.131 min −1 kd Dilution rate 0.250 min 1 −1 km cI mRNA degradation rate 0.100 min 1 −9 −1 kP CI degradation rate 3.22 × 10 min 2 −1 km cro mRNA degradation rate 0.158 min 2 −9 −1 kP Cro degradation rate 1.33 × 10 min 3 −1 km cII mRNA degradation rate 0.100 min 3 −1 kP CII degradation rate 0.369 min a nRM,CI Hill function sensitivity pa- 2.00 rameter for CI activation of

PRM r nRM,CI Hill function sensitivity pa- 6.00 rameter for CI repression of

PRM 103

nRM,Cro Hill function sensitivity pa- 3.00 rameter for Cro repression

of PRM nRE Hill function sensitivity pa- 5.00 rameter for CII activation of

PRE ncro,CI Hill function sensitivity pa- 2.55 rameter for CI repression of cro transcription ncro,Cro Hill function sensitivity pa- 2.22 rameter for Cro repression of cro transcription ncII,CI Hill function sensitivity pa- 3.95 rameter for CI repression of cII transcription ncII,Cro Hill function sensitivity pa- 3.00 rameter for Cro repression of cII transcription nλ,CI Hill function sensitivity pa- 6.00 rameter for CI repression of viral replication nλ,Cro Hill function sensitivity pa- 6.00 rameter for Cro repression of viral replication nCII Hill function sensitivity pa- 1.00 rameter for suppression of CII degradation 104

a KRM,CI Hill function threshold pa- 108 nM rameter for CI activation of

PRM r KRM,CI Hill function threshold pa- 528 nM rameter for CI repression of

PRM

KRM,Cro Hill function threshold pa- 20.4 nM rameter for Cro-repression

of PRM

KRE Hill function threshold pa- 78.1 nM rameter for CII activation of

PRE

Kcro,CI Hill function threshold pa- 64.1 nM rameter for CI repression of cro transcription

Kcro,Cro Hill function threshold pa- 94.2 nM rameter for Cro repression of cro transcription

KcII,CI Hill function threshold pa- 64.1 nM rameter for CI repression of cII transcription

KcII,Cro Hill function threshold pa- 94.2 nM rameter for Cro repression of cII transcription

Kλ,CI Hill function threshold pa- 241 nM rameter for CI repression of viral replication 105

Kλ,Cro Hill function threshold pa- 500 nM rameter for Cro repression of viral replication

KCII Hill function threshold pa- 241 nM rameter for CII degradation suppression

−1 τλ Time offset for onset of 7.50 min replication 106

A.3 Supplementary Figures

Figure A.1 : The distribution of single-cell MOI within a population. The distribution of single-cell MOI following infection by λ cI 857 Pam80 P1parS. The infection procedure is described in Experimental Methods Section 4.2, and the identification of single-cell MOI in Experimental Methods Section 8.2. Markers and error bars indicate mean ± SEM from samples taken at 1, 2 and 5 minutes fol- lowing infection (see Table A.9 for detailed sample sizes). The mean MOI calculated from these samples was 1.9 ± 0.5 (mean ± SEM). Red line: Poisson distribution of the same mean, reflecting the assumption of random encounters between phages and bacteria (Kourilsky, P., Mol Gen Genet, 1973). 107

Figure A.2 : The estimated number of lambda prophage copies in lysogenic cells. (A) Lysogenic strain MG1655 carrying prophage λ cI 857 ind− P1parS and plasmid pALA3047 (expressing CFP-ParB). Cells were grown at 30◦C in LBMM supplemented with 10 µM IPTG. Individual prophages are labeled using CFP-ParB. The imaging procedure is described in Experimental Methods Section 6. (B) Newborn lysogenic cells (“short cells”, defined as the 5–20 percentiles of cell lengths, N = 350) contain two lambda genome copies as expected (Bremer, H. and Churchward, G., J Theor Biol, 1977), while cells about to divide (“long cells”, the 80–95 percentiles of cell length, N = 349) contain four copies. Error bars indicate SEM. 108

Figure A.3 : The transcription kinetics of cI, cro and cII across biological replicates. The numbers of cI, cro, and cII mRNA per cell (mean ± SEM), at different times following infection by λ cI 857 Pam80 P1parS. The infection procedure is described in Experimental Methods Section 4.2, and the mRNA quantification in Experimental Methods Section 8.1. (A) Results from dataset 1 (see Table A.9 for detailed sample sizes). The mean MOI was 2.1 ± 0.3 (SEM). (B) Results from dataset 2 (see Table A.9 for detailed sample sizes). The mean MOI was 2.1 ± 0.2 (SEM). Solid lines are splines, used to guide the eye. 109

Figure A.4 : MOI scaling of cI, cro and cII trajectories. (A) The numbers of cI, cro, and cII mRNA per cell (mean ± SEM), at different times following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1—5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. Solid lines indicate splines, used to guide the eye. (B) The values in panel A, scaled by MOI, with  equal 1.16, 0.475, 0.441 for cI, cro and cII, respectively. The optimal value of  for each gene was obtained by minimizing the sum of squared deviation between mRNA values at different MOI at all time points. Solid lines indicate a single spline over all scaled mRNA numbers. 110

Figure A.5 : Estimated CI-Cro trajectories following infection. The es- timated trajectories in the plane of Cro and CI concentration, during the first 60 minutes following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1—5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. We first calculated the concentration of cI and cro mRNA, [mRNA], by dividing the number of molecules in each cell by the cell volume (approximated as a spherocylinder, with dimensions obtained from the au- tomated segmentation, see textbfExperimental Methods Section 7.1). [mRNA] was then used to estimate the protein concentration of the corresponding species using the relation: d[protein]/dt = translation rate × [mRNA] – decay rate × [protein], with the rates of translation and decay taken from the literature (Zong, C., et al., Mol Syst Biol, 2010; Reinitz, J. and Vaisnys J.R., J Theor Biol, 1990). 111

Figure A.6 : Model trajectories from ensembles of fits capture mRNA kinetics in biological replicates of P- infection. The numbers of cI, cro, and cII mRNA per cell, at different times following infection at MOI = 1—5 by P- phage (λ cI 857 Pam80 P1parS; see also Figure 2C, main text). (A) Results from dataset 1. (B) Results from dataset 2. Markers and error bars indicate experimental mean ± SEM of each sample. Solid lines indicate ensembles of model fits that yield consistent predictions, obtained from minimizing the objective function described in Theoretical Methods Section 4. See Table A.9 for samples sizes. 112

Figure A.7 : Model trajectories from an ensemble of fits capture the dy- namics of viral copy number during P+ infection. Viral copy number, mea- sured using qPCR, following infection at MOI = 1 by P+ phage (λ cI 857 Sam7 ; see also Figure 2D, main text). Markers and error bars indicate experimental mean ± standard deviation due to qPCR calibration uncertainty. Solid lines indicate an ensemble of model fits obtained from minimizing the objective function described in Theoretical Methods Section 4, using the P- mRNA measurements from dataset 2 (Figure A.6). 113

Figure A.8 : Model trajectories from an ensemble of fits capture cI and cII mRNA kinetics during infections with various lambda genotypes. The num- bers of cII mRNA (normalized by maximum cII mRNA count during P- infection) and cI mRNA (normalized by maximum cI mRNA count during cro-P- infection) per cell at different times following infection at MOI = 1 with various lambda geno- types (data from Shao, Q. et al., iScience, 2018). (A) cII mRNA following infection with P-, cI-, and cro- phages. (B) cII (top) and cI (bottom) mRNA following infec- tion with WT (cI +cro+P+) and cro-P- phages. Solid lines indicate an ensemble of model fits obtained from minimizing the objective function described in Theoretical Methods Section 4, using the P- mRNA measurements from dataset 2 (Figure A.6). 114

Figure A.9 : CII-activated cI expression from PRE is required for a lysis-to- lysogeny transition, while cI autoactivation is not. (A) The model-predicted fraction of total cI mRNA expressed from PRE (purple) and PRM (red) during the first 60 minutes of infection with P+ phages over a range of MOI. The majority of cI expression comes from PRE for all MOI simulated. (B) Model-predicted trajectories, in the plane of Cro and CI concentrations, during the first 60 minutes following infection by phages in which cI autoactivation of PRM has been removed. A lysis- to-lysogeny transition is achieved even in the absence of CI-activated transcription from PRM. (C) Same as panel B, for the case of infection by a phage in which CII activation of PRE has been removed. In the absence of cI transcription from PRE, a transition to lysogeny is not observed. Protein concentrations were normalized by the lytic and lysogenic thresholds. 115

Figure A.10 : Single phage PRE activity during P+ infection depends only weakly on MOI. Model-predicted CII-activated cI transcription from PRE, at the single phage level, shows weak MOI-dependence of its (A) amplitude, (B) turn-on time, (C), turn-off time (τ PRE), and (D) turn-on duration. The turn-on (turn-off) time is defined as the first (last) time that PRE activity is greater than or equal to 10% of its maximum value, while turn-on duration is the difference between these times. 116

Figure A.11 : Measured kinetics of nascent cI mRNA. (A) The number of nascent cI mRNA per phage genome following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1—5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. Nascent mRNA was quantified based on colocalization of the smFISH and ParB signals, following the method of Wang, M., et al., Nat Microbiol, 2019. The turn-on and turn-off of cI transcription were both fitted to a Hill function with coefficient h = 10 (solid lines). (B) The maximum number of nascent cI mRNA per phage genome as a function of MOI. (C-D) The time of turn-on and of turn-off of cI transcription, estimated using the midpoint of the fitted Hill curves, as a function of MOI. (E) The duration of cI transcription pulse, estimated using the time interval between activation and repression time, as a function of MOI. 117

Figure A.12 : Coinfection delays of τ PRE still result in lysis even when the number of coinfecting viruses is greater than 2. (A) Following infection with a single phage, ∆MOI additional phages infect at t = τd. (B) Model-predicted infection outcome as a function of coinfection delay τd and ∆MOI following MOI = 1 infection by P+ phage. Even when ∆MOI is 3-fold larger than MOI*=2, the critical MOI at which the system transitions to lysogeny during simultaneous coinfection (Figures 3.3B and 3.6D, main text), only coinfection delays below τ PRE (the time when PRE activity falls below 10% of its maximum possible value; see main text) result in lysogeny. 118

Figure A.13 : Delayed infection does not result in a second pulse of PRE activity. Model-predicted CII concentration during infection by P+ phages for 4 scenarios: Infection by a single phage (light blue solid line), simultaneous infection by two phages (dark blue solid line), infection by a single phage followed by a second phage at time τ PRE/3 (dark blue dashed line), and infection by a single phage followed by a second phage at time τ PRE (dark blue dotted line). The addition of a second phage after a delay (dark blue dashed and dotted lines) does not result in a second pulse of CII. CII concentration is normalized by the PRE activation threshold, and the MOI-averaged CII activity window (defined as the time span during which PRE activity is at least 10% of its maximum possible value) is indicated by the gray shading. 119

Figure A.14 : Measured kinetics of nascent cro mRNA. The number of nascent cro mRNA per phage genome following infection by λ cI 857 Pam80 P1parS, at single-cell MOI of 1–5. The infection procedure is described in Experimental Methods Section 4.2, the mRNA quantification and single-cell MOI identification in Experimental Methods Sections 8.1 and 8.2, respectively. Nascent mRNA was quantified based on colocalization of the smFISH and ParB signals, following the methods of Wang, M., et al., Nat Microbiol, 2019. Solid lines are splines, used to guide the eye. 120

Figure A.15 : CII activation of PRE during infection by P- phage is only weakly MOI-dependent. (A) Model-predicted activity of the PRE promoter follow- ing infection by P- phage at MOI = 1-–5. Similar to PRE activity following infection by P+ phage (Figure A.10; also see Figure 3.4A, main text), PRE activity during P- infection does not show strong MOI-dependence in (B) amplitude, (C) turn-on time, (D), turn-off time (τ PRE), or (E) turn-on duration. The turn-on (turn-off) time is defined as the first (last) time that PRE activity is greater than or equal to 10% of its maximum value, while turn-on duration is the difference between these times. 121

Figure A.16 : Modulation of CII’s degradation rate can generate lysogenic outcomes at MOI = 1. Model-predicted infection outcome as a function of MOI and CII degradation rate (normalized by the fitted wild-type (hflKC +) degradation rate). Perturbations which sufficiently decrease the CII degradation rate (hflKC- mutants, black dashed line) result in lysogeny even at MOI = 1. 122

A.4 Supplementary Tables

Table A.4 : Bacterial strains used in this study

Strain name Relevant genotype or Source description MG1655 Wild-type Lab stock LE392 glnV (supE44), tryT Lab stock (supF58) TY132 MG1655 ∆hflK Lab stock TY134 MG1655 ∆hflC Lab stock 123

Table A.5 : Phage strains used in this study

Strain name Relevant genotype or Reference or source description λ cI 857 ind- stf::P1parS- Tal et al., 2014 (Gift from kanR Joel Stavans) λ Pam80 Lynn Thomason λ cI 857 Sam7 Mike Feiss

λTY8 λ Pam80 stf::P1parS-kanR This work

λTY11 λ cI 857 Pam80 stf::P1parS- This work kanR

λIG2903 λ cI 857 bor::kanR Lab stock 124

Table A.6 : Plasmids used in this study

Plasmid name Description Reference or source pALA3047 Plac-cfp-P1-∆30parB-ampR Stuart Austin pTY001 Carrying P1parS-kanR This work flanked by lambda stf homology pKM208 Plac-gam-beta-exo-ampR Murphy and Campellone, 2003 125

Table A.7 : Primers used in this study

Primer name Sequence (5’–3’) Source p1-parS-FP GAACGGTACCTGAAT This work GAACTGGCCGCAGCG p1-parS-RP GAACCCGCGGACCGC This work AGAACGTTATTTCAT e.coli attB GAGGTACCAGCGCGG Powell et al., 1994 TTTGATC lambda attB TTTAATATATTGATA Powell et al., 1994 TTTATATCATTTTA CGTTTCTCGTTC lambda int ACTCGTCGCGAAC Powell et al., 1994 CGCTTTC cIts-oligo-R GAAGGGCTAAATTC This work TTCAACGCTAACTT TGAGAATTTTTG TAAGCAATGCGG CGT- TATAAGCAT TTAATG- CATTG ATGCCAT cIts-oligo-F ATGGCATCAATG This work CATTAAATGCTTA TAACGCCGCATT GCTTACAAAAATTCT CAAAGTTAGCGTT GAAGAATTTAGC CCTTC 126

cI-seq-F GATGATTATCAG This work CCAGCAGA cI-seq-R TCAGGGTTATGC This work GTTGTTCC P-seq-F GTGTGTGCTGTT This work CCGCTGGG P-seq-R TTCGCCAGACCTT AC- This work CTTCG lacZ-FP CGTGAGCGGTC So et al., 2011 GTAATCAGC lacZ-RP ACGACATTGGCGT So et al., 2011 AAGTGAAGCG cI-FP CAACAGCCTGC So et al., 2011 TCAGGGTCAAC cI-RP GGTGATGCGGAG So et al., 2011 AGATGGGTAAGC 127

Table A.8 : DNA oligos used for smFISH

Transcript Probe description (5’– Source 3’) 128

cI GGTTTCTTTTTT So et al., 2014 GTGCTCATCTCA AGCTGCTCTTGTG TTAAATTGCTT TAAG- GCGAC GTGGGGATAA GCCAAGTTCATT TATCTTGTCT GCGACAGATT CAATAAAGCA CCAACGCCTG AGCATTTAAT GCATTGATGCC TG- CAAGCAATGC GGCGT- TATCT TCAACGC- TAAC TTTGAGACTG GCGATTGAA GGGC- TAAACGCT TCATA- CATCT CGTAGATAA GTGACGGC TGCAT- ACT AACAGGGTA CTCATACTCA CTC- CCTGCC TGAA- CATGA GAAATTCTA AGCTCAGG TGA- GAACTC CGCATCACC TTTGGTAAAT TTG- GTTGTGC TTACC- CATC AGAATGCAGA ATCACTGGC TCG- GTCATGGA ATTAC- CTTCA AGCTTGGCTT GGAGCCTG TTA- GAATT AACATTCC GTCAGGAACA GCCT- GCTCAGGG TCAAC- TATGCA GAAATCAC- CTG GCAACTCATCA CCCCCAAGT CTCCT- GATCA GTTTCTTGAA GGGTAAAAA CACCT- GACC GCTATTGGGT ACTGTGGG TTTAGT- CAAC TCTCATTGCA TGGGATAG CGATAACT TTCCCCA CAAAAACG TCTCTTC AGGC- CACTA TTGTTATCAG CTATGCGCC GGGAGT- GAAA ATTCCCC- TAA CGGTAAGTCG CATAAAAACCGA GCGCTTATCT TTC- CCTTTGC CAGCA- GAGAA TTAAGGAAC AACCTGCAG GTGAT- GATTA CTGAACCAGA CTCTTGTCAT CAACT- GAAGCT TTAGAGC- GAG CGAGGCTGTT CTTAATATC GGAATC- CCA ATGATTCGT CAGTGTCGC CTTCAACA AACAAC- CGA AAACAGTTC TGGCAAAA ATCT- GTCAGA TCGGATGTG CCACTGCTT AATGA- CATT CCATCAGTGG CTCTATCTG AACAA- CATCG TCTTTGGTG GTTCTCGGC CGAT- GAAAT GCATATTAGC TTGGCTTCT ACCTTCA 129

cro ATACAACCTCC TTAG- This work TACAT TTTCAGGGTTA TGCGTTGTTGC CCAAAGCGCAT TGCATAAGAG ATCTTTAGCTGT CTTGGTTGTT GATCGCGCTT TGATAAAAATCTT TCGGCCTGCA TGTTC- CATCAGCG TTTATGT- TATTT ACCTCTTCCG CATAAACTGTTT TTTTGTTACTC GGGACTGGAAT GTGTAAGAGC GGGAATTTGA TGCC- CTTTTT CAGATGCATA CACCATAGGT GTG 130

cII CGTTTGTTTG CAC- This work GAACCAT TCTCGATTC GTAGAGCCTC GGC- GATTTT GTTAAG- CAACG CTGTCTTCTC AGTTCCAAG CAGCT- GATC TGCGACTTAT CAAAGAACTT TG- GAATCCA GTCCC- CCAT TCAAGAACA GCAAGCAAT CGAGC- CATG TCGTCGT- CAA TCGCAGCAAC TTGTCGCGCC GGGCGTTTT TTATTG- GTGC TGGATTTGT TCAGAACGCT 131

Table A.9 : Sample sizes for single-cell infection experiments

Dataset 1 Time af- Total Number of cells with ter infec- num- tion ber of cells (in- cluding unin- fected) MOI = 1 MOI = 2 MOI = 3 MOI = 4 MOI = 5 0.5 min 452 103 117 76 31 7 1 min 737 154 159 145 79 42 2 min 355 100 85 56 33 12 3 min 462 104 105 72 57 36 5 min 351 103 95 36 24 10 7.5 min 219 45 40 43 25 10 10 min 56 11 8 18 6 1 20 min 76 28 14 11 1 1 30 min 94 24 10 12 5 4 60 min 822 205 186 86 61 43 Dataset 2 132

Time af- Total Number of cells with ter infec- num- tion ber of cells (in- cluding unin- fected) MOI = 1 MOI = 2 MOI = 3 MOI = 4 MOI = 5 0.5 min 804 267 159 50 17 3 1 min 488 99 109 95 43 26 2 min 299 93 78 42 19 8 3 min 364 72 66 69 41 31 5 min 664 194 151 89 37 17 7.5 min 832 221 195 122 55 18 10 min 467 123 108 73 35 21 20 min 875 214 188 120 91 45 30 min 1387 338 223 174 142 94 60 min 820 202 167 120 65 32 133

Appendix B

Detailed description of the stochastic model

In this appendix we detail how the deterministic model in Chapter 3 (see also Appendix A, Theoretical Methods) is converted to a stochastic formulation, and how the resulting stochastic model is simulated.

B.1 Conversion of the deterministic model

We modeled stochastic network dynamics as a continuous time Markov jump process, in which the time-dependent joint probability distribution of the state of the system evolves according to the chemical master equation

∂ X P (n, t|n , t ) = [a (n − ν , t)P (n − ν , t|n , t ) − a (n, t)P (n, t|n , t )] , ∂t 0 0 j j j 0 0 j 0 0 j (B.1) where n is the 7-dimensional state vector describing the system, aj is the propensity of the j th reaction (describing a transition in state space), and νj is the stoichiometric vector describing the change in state space as a result of reaction j [84]. In this formulation, mRNA, protein, and viral copy numbers are discrete random variables, and reactions (which occur within the small time window (t, t + dt) with probabilities defined by the reaction propensities aj) move the system through the 7-dimensional nonnegative integer state space (Z≥)7. To convert the deterministic model to this formulation, we followed [33, 85, 86, 87] by converting fluxes in the ordinary deterministic equations to reaction propensities. As all fluxes in the deterministic model are written as first order reactions (possibly some modified by a nondimensional regulatory function—see Appendix A, Section 134

A.2.2) the only change in the composition of rates when converting to the stochastic model is that stochastic propensities are dependent on reactant copy number, whereas the ODEs are written in terms of reactant concentrations. To illustrate the conversion procedure, let [X] be the concentration of protein X, described by the ODE:

d[X] = −k [X]. (B.2) dt 1

The corresponding stochastic propensity for the first-order flux term k1[X] in Eq. B.2 is

a1 = k1X. (B.3)

This propensity describes the probability per unit time that the transition X → X−1 will occur in the Markov jump process formulation.

B.2 Reactions

Based on the conversion scheme outlined in B.1, and using the notation defined in Appendix A, we obtained the following reactions for the stochastic model:

1,RM ktx fcI,RM([CI], [Cro]) λ −−−−−−−−−−−−−→ λ + mcI {1}

1,RE ktx fcI,RE([CII) λ −−−−−−−−−−→ λ + mcI {2}

2 ktxfcro([CI], [Cro]) λ −−−−−−−−−−→ λ + mcro {3}

3 ktxfcII ([CI], [Cro]) λ −−−−−−−−−−→ λ + mcII {4} 135

1 km mcI −−→∅ {5}

2 km mcro −−→∅ {6}

3 km mcII −−→∅ {7}

1 ktr mcI −−→ mcI + CI {8}

2 ktr mcro −−→ mcro + Cro {9}

3 ktr mcII −−→ mcII + CII {10}

k1 CI −−→∅P {11}

k2 Cro −−→∅P {12}

k3 CII −−→∅P {13}

k f ([CI], [Cro], t) λ −−−−−−−−−−→λ λ 2λ {14}

B.3 Infection simulation methods

As outlined in Chapter 4, Section 4.1.1, we directly simulated trajectories of the stochastic process described by this reaction scheme (Reactions 1–14), with the goal 136

ˆ of generating estimates P (n, t|n0, t0) of the underlying time-dependent probability distribution. To do this, we employed the stochastic simulation algorithm (also called the Gillespie algorithim—see [92]). For all simulated trajectories, the initial state was

T (n0 = [0, MOI] , t0 = 0). Simulations were carried out on Rice’s NOTS high performance computing clus- ter. The individual simulation runs were parallelized to maximize computational efficiency. To estimate the underlying time-dependent probability distributions of the Markov jump process, we conducted 1000 simulations for each initial condition (MOI & genotype). To model dilution due to cell growth (which impacts the reaction propensities through the concentration-dependent regulatory functions inherited from the deter- ministic model—see Appendix A, Section A.2.2), we followed the approach of [134] and included a dummy reaction (with rate constant γ) which increments cell volume. Given the volume of the system at time t is V (t), and assuming exponential cell growth (see Appendix A, Section A.2.2), the volume increment after a waiting time of τ is:

∆V = V (t) expkdτ − 1
, (B.4)

where kd is the dilution rate. For sufficiently large values of γ, this approach yields an approximately determin- istic description of volume growth [134], however the increase in reaction events that results from a high rate constant also increases the runtime of simulations. To reduce simulation runtime, while still minimizing large stochastic jumps in volume which would add additional noise, we also add the volume increment when other reactions occur (Reactions 1–14). This approach yielded good agreement between the mean volume trajectory from stochastic simulations and simple exponential growth, as used in the deterministic model (see Appendix A, Section A.2.2). We set γ = 10kd for all simulations shown in Chapter 4, Section 4.1; increasing γ by a further factor 137

of 10 did not result in significant changes to simulation results.

B.4 Decision Thresholds

We defined new lytic and lysogenic decision thresholds (Klyt and Klys, respectively) for the stochastic model by fitting to the previously published experimental measure-

ments of lysogenization frequency during P- and P+ infection [2]. Specifically, Klyt

and Klys were defined as the values which minimized the objective function

4 X X ˆ 2 J(Klyt,Klys) = (Plys,g(MOI = i) − Plys,g(MOI = i, Klyt,Klys)) i=1 g∈{P-,P+} (B.5) 4 X ˆ + Plyt,P-(MOI = i, Klyt,Klys), i=1

where Plys,g(MOI) and Plyt,g(MOI) are the probabilities of lysogeny and lysis, re- spectively, during infection by lambda genotype g at a given MOI, and hat notation is used to denote model output. The second term in Eq. B.5 is a regularization penalty which penalizes nonzero lytic frequencies during P- infection, to reflect that lysis is not observed during infection over this range of MOI with this genotype [2]. Fitting was performed using the built-in simulated annealing algorithm in MAT- LAB. 138

Appendix C

Detailed description of the Q module

In this appendix, we describe the toy Q module used in Chapter 4, Section 4.2.

C.1 Formulation of CII inhibition of Q

As discussed in Chapter 4, Section 4.2, the exact mechanism by which CII inhibits Q accumulation is currently unknown [50]. We chose to implement this interaction at the level of mRNA translation (one of the hypothesized modes of this interaction— [50]), owing to the ease of implementation. To incorporate Q into our deterministic model (Chapter 3 and Appendix A, Section A.2.2), we added an ODE for Q protein concentration:

d[Q] = k4 [m ]f ([CII]) − k [Q], (C.1) dt tr Q Q d where the nomenclature follows the conventions defined in Appendix A, Section

A.2.1 and fQ is a regulatory function describing the CII inhibition of Q mRNA translation. For simplicity, in Eq. C.1 we have ignored Q degradation, due to the protein’s stability [19]. Owing to the lack of mechanistic understanding of CII inhibtion of Q, we for- mulated the regulatory function fQ in Eq. C.1 as a simple Hill function in CII concentration:

1 fQ([CII]) = nQ . (C.2) 1 + ([CII]/KQ) We parameterized this interaction with the fitted parameter values describing CII 139

activation of PRE. This was motivated by the observation that CII inhibition of Q is tied to CII activating transcription of PaQ, and it has been previously reported that CII activates transcription at both promoters at similar concentrations and coopera- tivities [19]. For simplicity, we take cII mRNA as a proxy for Q mRNA in Eq. C.1, as both are part of the same polycistronic transcript produced from PR [12], transcription of each requires readthrough of terminators (ibid), and we have assumed in this module that CII inhibition of Q occurs only at the level of translation.

C.2 Decision Thresholds

To define lytic and lysogenic thresholds for a CI-Q formulation of the decision, we followed the same procedure previously used to define decision thresholds in the CI- Cro formulation (Appendix A, Section A.2.5). 140

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