Cell Size Homeostasis and Optimal Viral Strategies

CELL SIZE HOMEOSTASIS AND OPTIMAL VIRAL STRATEGIES

FOR HOST EXPLOITATION

Cesar Augusto Vargas-Garcia

A dissertation submitted to the Faculty of the University of Delaware in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering

Fall 2017

I want to thank my advisor Abhyudai Singh. He discovered my professional potential and helped me to achieve this important goal in my life. I am grateful to the Dean, the Faculty, and the Staff of the Department of Electrical and Computer Engineering for providing their assistance and support through the years of my Ph.D. program. I want to thank also my co-advisor and friend, Dr. Ryan Zurakowski. He gave me the opportunity to start and enjoy this field. Also he encouraged me to be resilient in pursuing my degree in the hard days. I want to thank to professor and close friend Henry Arguello for all his support and advice through this years. I also want to thank my wife and daughter, Neyla Johanna and Victoria for their support and encouragement during my studies. They provided me the home to rest after every hard day. My students and alma-mater group HDSP gave me the motivation and encouragement to make the best effort in my research. They have been my friends and part of my family during this time. I appreciate their collaboration and company in this part of my life. Special thanks to my friends and colleagues Mohammad Soltani and Khem Ghusinga for their support, friendship and collaborations in uncountable and exciting projects which are nowadays the core of my research.

iii TABLE OF CONTENTS

ABSTRACT ...... vi

Chapter

1 PART 1: CELL SIZE CONTROL AND GENE EXPRESSION HOMEOSTASIS IN SINGLE-CELLS ...... 1

1.1 Cell-size regulation: going beyond phenomenological models ..... 1 1.2 Why do organisms control size? ...... 4 1.3 Living with size variations: gene expression homeostasis ...... 5

2 CONDITIONS FOR CELL SIZE HOMEOSTASIS: A STOCHASTIC HYBRID SYSTEMS APPROACH ...... 13

2.1 Timer-dependent growth and division ...... 14 2.2 Size-dependent growth rate ...... 16 2.3 Size-dependent division rate ...... 19

3 A MECHANISTIC STOCHASTIC FRAMEWORK FOR REGULATING BACTERIAL CELL DIVISION ...... 25

3.1 Distribution of the cell-division time given newborn cell size ..... 29 3.2 Distribution of the volume added between divisions ...... 31 3.3 Higher order moments of added volume ...... 34

4 PART 2: OPTIMALITY IN HOST-VIRUS SYSTEMS ...... 37

5 OPTIMAL ADSORPTION RATE: IMPLICATIONS OF THE SHIELDING EFFECT ...... 40

5.1 Traditional virus dynamics model ...... 41 5.2 Modeling the shielding eﬀect ...... 43 5.3 Single virus dynamics ...... 44 5.4 Competition dynamics ...... 45

iv 6 THE EFFECT OF MULTIPLICITY OF INFECTION ON THE TEMPERATENESS OF A BACTERIOPHAGE: IMPLICATIONS FOR VIRAL FITNESS ...... 51

6.1 Model ...... 53 6.2 Why do Bacteriophages display Temperateness? ...... 54 6.3 Probability of survival of a lysogen ...... 57 6.4 The Eﬀect of Multiplicity of infection (MOI) ...... 58

7 CONDITIONS FOR INVASION OF SYNAPSE-FORMING HIV VARIANTS...... 62

7.1 HIV Model ...... 64 7.2 Modeling Synaptic Virus ...... 66 7.3 Competition Model ...... 70

8 OPTIMAL MULTI-DRUG APPROACHES FOR REDUCTION OF THE LATENT POOL IN HIV ...... 75

8.1 HIV Model ...... 77 8.2 Optimal Control and Simulations ...... 83

9 DISCUSSION ...... 87

BIBLIOGRAPHY ...... 90

v ABSTRACT

The first part of this thesis address a question formulated more than 80 years ago (and still remains elusive): how does a cell control its size? Growth of a cell and its subsequent division into daughters is a fundamental aspect of all cellular living systems. During these processes, how do individual cells correct size aberrations so that they do not grow abnormally large or small? How do cells ensure that the concentration of essential gene products are maintained at desired levels, in spite of dynamic/stochastic changes in cell size during growth and division? In chapter 1, we introduce the reader to the field of cell size/content homeostasis. We review how advances in singe-cell technologies and measurements are providing unique insights into these questions across organisms from prokaryotes to human cells. More specifically, how diverse strategies based on timing of cell-cycle events, regulating growth, and number of daughters are employed to maintain cell size homeostasis. We further discuss how size-dependent expression or gene-replication timing can buffer concentration of a gene product from cell-to-cell size variations within a population. In chapter 2, we propose the use of stochastic hybrid systems as a framework for studying cell size homeostasis. We assume that cell grows exponentially in size (volume) over time and probabilistic division events are triggered at discrete time intervals. We first consider a scenario, where a timer (i.e., cell-cycle clock) that measures the time since the last division event regulates both the cellular growth and division rates. We also study size-dependent growth / division rate regulation mechanisms. We provide bounds on different statistical indicators (mean, variance, skewness, etc). Additionally, we assess the effect of different physiological parameters (growth rate, partition errors, etc) on cell size distribution.

vi Chapter 3 introduces a mechanistic model that might explain the recently uncovered added principle, i.e., selected species add a fixed size (volume) from birth to division, irrespective of their size at birth. To explain this principle, we consider a timekeeper protein that begins to get stochastically expressed after cell birth at a rate proportional to the volume. Cell-division time is formulated as the first-passage time for protein copy numbers to hit a fixed threshold. Consistent with data, the model predicts that the noise in division timing increases with size at birth. We show that the distribution of the volume added between successive cell-division events is independent of the newborn cell size. This fact is corroborated through experimental data available. The model also suggest that the distribution of the added volume when scaled by its mean become invariant of the growth rate, a fact also verified through available experimental data. In part 2 of this thesis, we study which strategies are implemented by a viral species, ranging from bacteriophages to human immunodeficiency virus (HIV), in order to exploit host resources. In chapter 4, we review the classical theory of viral-host dynamics and describe the key knobs that viruses tweak to exploit a cell population. This theory suggest that viruses might evolved to have infinite infectivity and virulence. In the case of infectivity, chapter 5 gives an alternative to infinite infectivity: virus will evolve to moderate infectivity because of local interactions. As an example, we study a phage attacking a bacterial population. We include the effect of local interactions by assuming that the phage needs to scape from bacterial death remains (debris). Infinite virulence is also challenged as evolutionary alternative for viral propa- gation. In chapter 5 we study environments where availability of susceptible bacteria fluctuates across time. Under such scenarios bacteria behaves contrary to classical ecology theory: phages evolve to a moderate virulence (lysis time). We present this insights through the use of the stochastic hybrid system framework. In chapter 7, we present a mathematical model of HIV transmission including cell-free and cell-cell transmission pathways. A variation of this model is considered including two populations of virus. The first infects cells only by the cell-free virus

vii pathway, and the second infects cells by either the cell-free or the cell-cell pathway (synapse-forming virus). Synapse-forming HIV is shown to provide an evolutionary advantage relative to non synapse-forming virus when the average number of virus transmitted across a synapse is a sufficiently small fraction of the burst size. HIV disease is well-controlled by the use of combination antiviral therapy (cART), but lifelong adherence to the prescribed drug regimens is necessary to prevent viral rebound and treatment failure. Populations of quiescently infected cells form a “latent pool” which causes rapid recurrence of viremia whenever antiviral treatment is interrupted. A “cure” for HIV will require a method by which this latent pool may be eradicated. Current efforts are focused on the development of drugs that force the quiescent cells to become active. Previous research has shown that cell-fate decisions leading to latency are heavily influenced by the concentration of the viral protein Tat. While Tat does not cause quiescent cells to become active, in high concentrations it prevents a newly infected cell from becoming quiescent. In chapter 8, we introduce a model of the effects of two drugs on the latent pool in a patient on background suppressive therapy. The first drug is a quiescent pool stimulator, which acts by causing quiescent cells to become active. The second is a Tat analog, which acts by preventing the creation of new quiescently infected cells. We apply optimal control techniques to explore which combination therapies are optimal for different parameter values of the model.

viii Chapter 1

PART 1: CELL SIZE CONTROL AND GENE EXPRESSION HOMEOSTASIS IN SINGLE-CELLS

1.1 Cell-size regulation: going beyond phenomenological models Size plays an important role in cellular processes and functions of a cell [1], and therefore should be actively maintained. Indeed, cell size distribution of proliferating cells is known to be stable through generations, suggesting regulation of growth and division to correct deviations from a desired cell size. Earlier attempts towards understanding cell size control was based on population-averaged data on model organisms, bacteria and yeast, and led to proposition of phenomenological models of cell size control. In particular, three models were hypothesized: Timer – a constant time between successive divisions, Sizer – cell division upon attainment of a critical size, and Adder – a constant size addition between consecutive generations. However, validation of these in various organisms remained inconclusive. With recent advances in single-cell technologies, high throughput measurements of cell size over several cell-cycles of individual cells can be made, generating correlation data between diﬀerent parameters such as cell size at birth, growth rate, cell size at division, division time, etc. This data has stimulated reexamination of phenomenological models of cell size homeostasis. For a broad range of microbes, an individual cell grows exponentially over time with a constant growth rate per size [2,3]. A Timer based mechanism to control division can thus be precluded since it would not be homeostatic [4,5]. Consistent with it, analysis of data for several bacterial species reveals a negative correlation between division time and cell size at birth, implying presence of a size control during cell-cycle. Further investigation reveals the phenomenological strategy: size added from birth to division is uncorrelated with cell size at birth, which is inconsistent with a Sizer model

1 and validates an Adder model (Fig. 1.1a) [6–12]. Existence of such phenomenological models, however, only provides limited perspective since it does not specify how other landmark cell-cycle events (e.g., initiation of DNA replication, assembly of division apparatus) are coordinated with division, and whether the size control is applied from birth to division or between two other cell-cycle events. Studies on E. coli have proposed several formulations that couple initiation of DNA replication to division while being consistent with an Adder between birth and division. One model postulates that size control is primarily exerted over the timing of initiation of DNA replication such that a constant volume per origin of replication is added between two consecutive initiation events. The corresponding division is assumed to occur a fixed time (C+D period; C–time to replicate DNA, D–time between end of replication to division) after initiation [13–15]. Another proposition, which suggests that initiation of DNA replication occurs at a constant size per origin and C+D period depends upon the growth rate, shows that the Adder model is valid only for fast growth rates and the size control behaves as a Sizer for slow growth conditions [16]. A third model argues that for slow growing cells, size control is exerted at two sub-periods (the time from birth to initiation, and the D period) whereas the C period resembles a Timer [17]. So far none of these models have been conclusively validated or falsified, and it would be worthwhile to carry out experiments to this end. Similar couplings between important cell-cycle events and division have been explored in other organisms as well. For C. crescentus, pre-constriction and post- constriction periods have been examined for size control, showing that it obeys a mixer model wherein the time until constriction acts as a Timer followed by the post- constriction period regulated via an Adder [18]. Likewise, the cell-cycle of budding yeast has been investigated by dividing it in two distinct periods: time until G1/S transition, and time from the G1/S transition to division. Despite having an overall Adder between birth to division [19, 20], independent control of both these periods is proposed. In particular, the first period is shown to be dependent upon the size of

2 mother cell, and the second period is controlled by the size of the daughter [20]. Col- lectively, dissecting the cell-cycle in biologically relevant periods for various organisms provides key insights in to how division might be coordinated with other events. An important step moving forward is to understand molecular mechanisms that implement size control over timing of cell-cycle events. To this end, two generic themes have been proposed. One approach is to accumulate a protein in size dependent manner up to a threshold. Some notable examples of this include FtsZ to control Z-ring formation, [21, 22], DnaA to control timing of initiation in E. coli [13], and Cdc25 to regulate timing of mitotic entry in S. pombe [23]. Another way to implement a size control over timing is to dilute a protein until a critical level as cell grows in size. A prominent example of this strategy is Whi5 for control of G1 duration in S. cerevisiae [19, 20, 24, 25]. Interestingly, an alternative model shows that an Adder-like behavior can also arise from a very different mechanism of maintaining a constant surface area to volume ratio [26]. Apparently, the nutrient intake imposes constraints on this ratio by affecting the synthesis of surface material. The candidate molecules that carry out such function have not been identified yet. It is plausible that molecular players underlying important cell-cycle events interact with each other, and therefore an overarching framework may emerge with further research. How is size control implemented in multicellular organisms? Arguably, these organisms operate in a more complex environment than bacteria and budding yeast; thus, size control strategies adopted by their cells are expected to be affected by physical constraints and thereby be relatively more complicated. Recent data indeed suggests that mammalian cells have different size control strategy in the G1 duration than budding yeast. This strategy phenomenologically resembles a Sizer for small cells, but Adder for larger cells [27]. Examining the data further reveals that for mammalian cells, not only the time spent in G1, but also the growth rate are negatively correlated with size at birth [27] (Fig. 1.1b). This observation has been strengthened by recent work showing size-dependent regulation of growth rate [30,31]. The molecular underpinnings of growth rate control are not well understood, although access to nutrients and physical

3 constraints are expected to play an important role. In a stark contrast to size control on timing of cell-cycle events and growth rate, the green alga C. reinhardtii controls the number of daughter cells it produces upon division in a size controlled manner. It has a prolonged G1 phase in which the size of a newborn cell increases by several folds. The cell then undergoes multiple divisions producing 2n daughter cells. The number of divisions n, on average, is large for large mother cells, so that the daughter cells are close to a target size (Fig. 1.1c) [28, 29]. The molecular mechanism for regulating the number of divisions in C. reinhardtii is suggested to rely on concentration dilution. Here, a cyclin dependent kinase CDKG1 is produced towards the end of G1 phase and its production depends on the size of the cell. With each division, this protein is degraded and further divisions stop once its concentration goes below a threshold [28].

1.2 Why do organisms control size? Recent experiments in proliferating animal cells unravel an interesting connec- tion between cell size homeostasis and cellular fitness. By sorting a population of growing cells based on their sizes, it was shown that cellular proliferation (i.e., fitness) is low for small and large cells, but high at intermediate sizes [32, 33] (Fig. 1.2). The decrease in the fitness for large cells could not be attributed to higher cell-death since the apoptosis rates here were similar for large and average–sized cells. Notably, a similar optimality also exists for mitochondrial activity (metabolism) of cells [33]. Since the metabolism is linked to the growth, it is not too surprising that existence of an optimal size can be theoretically shown to arise from joint regulation of timing and growth [34].

Do unicellular organisms also have an optimal size for a given growth condition? Intuitively, a bigger bacterial cell will divide faster and thus its proliferation would be higher than a smaller cell. However, the fact that the average bacterial cell size grows exponentially with growth rate (per size) imposed by the nutrients suggests that

4 there may be a speciﬁc target cell size for the given environment [36], which may be determined by several factors such as the surface to volume ratio [26], availability of fatty acids [37], etc. Observing an optimal size in bacteria via experiment such as [32] is perhaps hard because of a narrow range of cell sizes as compared to mammalian cells. It would be interesting to see results of competition between mutants with artiﬁcially large sizes, such as those obtained by expression of useless proteins in [21].

1.3 Living with size variations: gene expression homeostasis Going beyond cell size homeostasis, single-cell approaches are also elucidating mechanisms for gene-product concentration homeostasis. In particular, how is the concentration of given RNA/protein buffered to random or cell-cycle dependent fluctuations in cell size. This problem is especially acute for mammalian cells, where individual synchronized cells from the same population, exposed to identical nutrients, exhibit a six-fold variation in size [38]. Since rates of intercellular biochemical processes typically depend on the concentration of molecular species, cells must maintain concentration levels despite dramatic single-cell size deviations from the population average. Adding to size variations, unsynchronized cells also differ in the number of gene copies, and it is unclear to what degree expression is compensated for gene dosage across organisms.

In principle, one can envision diﬀerent strategies for maintaining a desired gene- product concentration: size-dependent regulation of synthesis and/or decay rates with perfect gene dosage compensation, or scaling of gene dosage with cell size accompa- nied by size-independent expression per gene copy (Fig. 1.3). Prior investigations on inferring these mechanisms have relied on bulk-assay expression measurement utilizing cell size mutants, or altering size through small-molecules drugs [39, 40]. Instead of artiﬁcially changing the average cell size, exploiting natural intracellular size variations within a population provides a more physiologically perturbation-free setting to study concentration homeostasis. Building up on this theme, recent works have used

5 mRNA fluorescence in situ hybridization (RNA FISH) to count mRNAs inside individual cell along with precise cell size measurement within a population of mammalian cells [38,41]. For most genes, mRNA copy numbers scale linearly with size, implying a gene-specific mRNA concentration set point (Fig. 1.4a). Intriguingly, the data further points to gene dosage compensation - two similarly sized cells in G1 and G2 phase will have same number of mRNAs, on average [38, 42]. While this mRNA count vs. size scaling observed in diverse mammalian cell types seems intuitive and also previously reported in bulk-assay experiments [40], its mechanistic underpinnings and molecular implementations remained elusive. Measurements of promoter activity of individual gene copies in single cells revealed for the first time that concentration homeostasis is orchestrated through modulation of the transcriptional burst size - with increasing cell size more RNA polymerases initiate transcription when the gene randomly switches to an active state (Fig. 1.4b). In contrast, gene dosage compensation occurs through the burst frequency (i.e., how often a gene become active) which is approximately halved upon gene duplication during the S phase. It is interesting to note that similar size-dependent transcription rates have recently been found in A. thaliana [43], suggesting the homeostasis principles uncovered through single-cell measurements are broadly applicable to animal and plant cells. Studies in budding yeast show similarities and striking differences with their mammalian counterparts. For example, as in mammalian cells, gene-dosage compensation occurs during the S-phase of S. cerevisiae, wherein incorporations of acety- lated histones into newly replicated regions leads to suppressed transcription per gene copy [44]. However, unlike mammalian cells, nascent transcription rates for RNA polymerase II genes have been reported to be size-independent in budding yeast [45]. This implies that a newborn yeast cell growing in G1 will have reduced mRNA synthesis with increasing size, in the sense of mRNAs added per volume per unit time. Intrigu- ingly, this decreased synthesis rate is compensated by a corresponding change in RNA stability to maintain mRNA concentration [45]. Previously, mRNA degradation rate modulation as a function of growth rate/temperature (which should indirectly affect the

6 cell-size) has also been observed in yeast cells [46–49]. In this case, the transcription- degradation cross talk for concentration homeostasis is hypothesized to occur through molecular factors that continuously shuttle between the nucleus/cytoplasm to coordinate and participate in both transcription and decay processes [50]. While transcription of RNA polymerase II genes is size-independent, transcription of ribosomal RNAs via RNA polymerase I is size-dependent [45], and this latter mechanism maybe critical in maintaining a ﬁxed concentration of ribosomes.

In contrast to eukaryotic cells, the bacterial cells do not seem to compensate for DNA dosage [51]. In this case, gene-product concentrations decrease with increase in size for fixed ploidy, suggesting that a strong coupling between gene-dosage and cell size can lead to near-constant gene product concentrations. In agreement with this, [52] has shown that cyanobacteria cells are able to maintain concentration of proteins by scaling their gene dosage with cell size. (Fig. 1.3c). A related issue is how change in gene dosage affects concentration of its products through the cell-cycle. Intuitively, the gene product concentration is expected to decrease until associated gene duplicates (based on its location on the chromosome) and increase back thereafter [53]. Since genes located at different places on the chromosome are duplicated at different times, such imbalance in gene dosage is utilized by B. subtilis for coordination of sporulation [54]. It is also worth noting that the bacterial cell size distribution is quite narrow, hinting that the scaling DNA dosage with cell size may suffice to maintain near constant gene product concentrations for some genes strategically placed on the chromosome. Recent experiments on E. coli indeed suggest that the fluctuations in protein level are not very large (about 4% ) for several genes and a near constant concentration is maintained during the cell-cycle [55]. These fluctuations may be further suppressed via autoregulation which is a prevalent feedback mechanism in prokaryotes. In the remaining chapters (Chapter 8 and 9), we discuss the conditions for achieving cell size homeostasis, and describe how a specific example explains size regulation mechanisms in E.coli bacterium.

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CONDITIONS FOR CELL SIZE HOMEOSTASIS: A STOCHASTIC HYBRID SYSTEMS APPROACH

Stochastic hybrid systems (SHS) constitute an important mathematical modeling framework that combines continuous dynamics with discrete stochastic events. Here we use SHS to model a universal feature of all living cells: growth in cell size (volume) over time and division into two viable progenies (daughters). A key question is how cells regulate their growth and timing of division to ensure that they do not get abnormally large (or small). This problem has ben referred to literature as size homeostasis and is a vigorous area of current experimental research in diverse organisms [3,7,8,11,18,59–69]. We investigate if phenomenological models of cell size dynamics based on SHS can provide insights into the control mechanisms needed for size homeostasis. The proposed model consists of two non-negative state variables: v(t), the size of an individual cell at time t, and a timer τ that measures the time elapsed from when the cell was born (i.e., last cell division event). This timer can be biologically interpreted as an internal clock that regulates cell-cycle processes. Time evolution of these variables is governed by the following ordinary diﬀerential equations

v˙ = α(v, τ )v, τ˙ = 1, (2.1) where the growth rate α(v, τ ) ≥ 0 can depend on both state variables and is such that (2.1) has a unique and well-defined solution ∀t ≥ 0 (i.e., cell size does not blow up in finite time). A constant α implies exponential growth over time. As the cell grows in size, the probability of cell division occurring in the next infinitesimal time interval (t, t + dt] is given by f(v, τ )dt, where f(v, τ ) can be interpreted as the division rate. Whenever a division event is triggered, the timer is reset to

12 zero and the size is reduced to βv, where random variable β ∈ (0, 1) is drawn from a beta distribution. Assuming symmetric division, β is on average half, and its coeﬃcient of variation (CVβ) quantiﬁes the error in partitioning of volume between daughters. To be biologically meaningful, α(v, τ ) is a non-increasing function, while f(v, τ ) is a non-decreasing function of its arguments. The SHS model is illustrated in Fig. 2.1 and incorporates two key noise sources: randomness in partitioning and timing of division. Next, we explore conditions for size homeostasis, in the sense that, the mean cell size does not converge to zero, and all statistical moments of v remain bounded.

Figure 2.1: SHS model for capturing time evolution of cell size. The size of an individual cell v(t) grows exponentially with growth rate α(v, τ ), where τ represents a timer that measures the time since the last division event. The arrow represents cell division events that occur with rate f(v, τ ), which resets τ to zero and divide the size by approximately half. A sample trajectory of v(t) is shown with cycles of growth and division.

2.1 Timer-dependent growth and division We begin by considering a scenario, where both the growth and division rates are functions of τ , but do not depend on v. The SHS can be compactly written as

v˙ = α(τ )v, τ˙ = 1, (2.2) with reset maps v 7→ βv, τ 7→ 0 (2.3) that are activated at the time of division. The timer-controlled division rate f(τ ) can be interpreted as a “hazard function” [70]. Let T1, T2, ... denote independent

13 and identically distributed (i.i.d.) random variables that represent the time interval between two successive division events. Then, based on the above formulation, the probability density function (pdf) for Ti is given by

− R x f(y)dy Ti ∼ f(x)e y=0 , ∀ x ≥ 0 (2.4)

[70]. Note that a constant division rate in (2.4) would lead to an exponentially distributed Ti. For this class of models, the steady-state statistics of v is given by the following theorem.

Theorem 1: Consider the SHS (2.2)-(2.3) with timer-dependent growth and division rates. Then

 T D R i α(y)dyE 0 e y=0 < 2 lim hv(t)i = (2.5) T t→∞ D R i α(y)dyE ∞ e y=0 > 2, where the symbol h i is used to denote the expected value of a random variable. More- over,

0 < lim hv(t)i < ∞, lim hv2(t)i = ∞ (2.6) t→∞ t→∞

T D R i α(y)dyE when e y=0 = 2.

th Proof of Theorem 1: Let vi−1 denote the cell size just at the start of the i cell cycle. Using (2.2), the size at the time of division in the ith cell cycle is given by

T R i α(y)dy vi−1e y=0 . (2.7)

Thus, the size of the newborn cell in the next cycle is

T R i α(y)dy vi = vi−1xi, xi := βie y=0 , (2.8) where βi ∈ (0, 1) are i.i.d random variables following a beta distribution and xi are i.i.d. random variables that are a function of βi and Ti. From (2.8), the mean cell size at the start of ith cell cycle is given by

i hvii = v0hxii (2.9)

14 and will grow unboundedly over time if hxii > 1, or go to zero if hxii < 1. Using the fact that hβii = 0.5 (symmetric division of a mother cell into daughter cells), βi and Ti are independent, (2.5) is a straightforward consequence of (2.9). It also follows from (2.8) that hv2i = v2hx2ii = v2hx i2i(1 + CV 2 )i (2.10) i 0 i 0 i xi where CV 2 represents the coeﬃcient of variation squared of x . When hx i = 1 then xi i i hvii = v0 and hv2i = v2(1 + CV 2 )i. (2.11) i 0 xi

Note that when the system is completely deterministic, i.e., pdfs for Ti and βi are given by delta functions, CV 2 = 0. However, the slightest noise in these variables will lead xi to CV 2 > 0, in which case (2.11) implies (2.6). xi T D R i α(y)dyE In summary, unless functions α(τ ) and f(τ ) are chosen such that e y=0 = 2, the mean cell size would either grow unboundedly or go extinct. Moreover, even if the mean cell size converges to a non-zero value, the statistical ﬂuctuations in size would grow unboundedly. Hence, size-based regulation of growth/division rates is a necessary condition for size homeostasis.

2.2 Size-dependent growth rate Recent work measuring sizes of single mammalian cells over time has reported lowering of growth rates as cells become bigger [71–73]. To explore the eﬀects of such regulation, we consider a growth rate α(v, τ ) that now depends on size. As in the previous section, timer-controlled division events occur with rate f(τ ) resulting in inter-division times Ti given by (2.4). The following result shows that size homeostasis is possible if growth rate is appropriately bounded from below and above. Theorem 2: Let the growth rate be bounded by

α(v, τ )v ≤ k(τ )vp, p ∈ [0, 1), ∀v ≥ 0 (2.12)

15 for some non-increasing function k(τ ). Moreover, the growth rate of a small cell is large enough such that

D R Ti E α0(y)dy e y=0 > 2, α0(τ ) := lim α(v, τ ). (2.13) v→0

Then 1 lhk(τ )ihT i 1−p 0 < lim hvl(t)i < i (2.14) t→∞ h1 − βli where l ∈ {1, 2,... }, hTii is the mean cell-cycle duration, and β ∈ (0, 1) is a random variable quantifying the error in partitioning of volume between daughters.

Proof of Theorem 2: Consider a newborn cell with a suﬃciently small size born at time t = 0. Then, the mean cell size will grow in successive generation iﬀ the second inequality in (2.5) is true for α0(τ ), which results in (2.13). Based on the Dynkin’s formula for the SHS (2.1) and (2.3), the time evolution of moments is given by

dhvli = f(τ )vl hβli − 1 + l α(v, τ )vl , (2.15) dt for l ∈ {1, 2,... } [74]. Using (2.12),

dhvli ≤ f(τ )vl hβli − 1 + l k(τ )vl−1+p . (2.16) dt Note that f(τ )vl = f(τ )hvl|τ i (2.17) where hvl|τ i is the expected value of vl conditioned on τ . Based on the time evolution of cell size in (2.1), hvl|τ i is an increasing function of τ (cells further along in the cell cycle, have on average, larger sizes). Since hvl|τ i and f(τ ) are monotone non- decreasing function of τ f(τ )vl ≥ hf(τ )ihvli. (2.18)

Similarly, since k(τ ) is a non-increasing function,

k(τ )vl−1+p ≤ hk(τ )ihvl−1+pi. (2.19)

16 Finally, using the fact that l − 1 + p ≤ l as p ∈ [0, 1)

l−1+p l−1+p hvl−1+pi = vl l ≤ vl l (2.20)

Using (2.18)-(2.20), (2.16) reduces to the following inequality

dhvli l−1+p ≤hf(τ )ihvli hβli − 1 + lhk(τ )i vl l . (2.21) dt Since at steady state 1 hf(τ )i = , (2.22) hTii [75], (2.21) implies (2.14).

An extreme example of size-dependent growth is k α(v, τ ) = , k > 0 (2.23) v which corresponds to cells growing linearly in size, as experimentally reported for some organisms [76]. For this case, the result below provides exact closed-form expressions for the ﬁrst and second-order statistical moments of v.

Theorem 3: Consider the growth rate (2.23) that results in the following SHS continuous dynamics v˙ = k, τ˙ = 1. (2.24)

Then, the steady-state mean and coefficient of variation squared of cell size is given by 2 khTii 3 + CV lim hv(t)i = Ti , (2.25) t→∞ 2 3 hTi i 2 4 4 9 3 − 9 − 6CVT − 7CVT 2 1 hTii i i CVv = + 27 27 3 + CV 2 2 Ti 16CV 2 + β , (2.26) 3(3 − CV 2)(3 + CV 2 ) β Ti where CV 2 and CV 2 denote randomness in the inter-division times (T ) and parti- Ti β i tioning errors (β), respectively, as quantified by their coefficient of variation squared.

17 The proof of Theorem 3 can be found in the Appendix. Interestingly, the mean cell size in (2.25) not only depends on the mean inter-division times hTii, but also on its second-order moment CV 2 . Thus, making the cell division times more random (i.e., Ti increasing CV 2 ) will also lead to larger cells on average. Similar eﬀects of CV 2 on Ti Ti mean gene expression levels have recently been reported in literature [77, 78]. More-

2 over, (2.26) shows that the magnitude of ﬂuctuations in cell size (CVv ) depend on 2 Ti through its moments up to order three. Note that if CVβ = 0 (no partitioning errors) and Ti = hTii with probability one (deterministic inter-division times), then 2 2 CVv = 1/27. This non-zero value for CVv in the limit of vanishing noise sources represent variability in size from cells being in diﬀerent stages of the deterministic cell cycle.

2 Theorem 3 decomposes CVv into terms representing contributions from diﬀerent noise 2 sources. The terms from left to right in (2.26) represent contributions to CVv from i) Deterministic cell-cycle and ii) Random timing of division events and iii) Partitioning errors at the time of division. Assuming lognormally distributed Ti,

hT 3i/hT i3 = 1 + CV 2 3 . (2.27) i i Ti

Substituting (2.27) in (2.26) and plotting CV 2 as a function of CV 2 and CV 2 , re- v β Ti veals that stochastic variations in cell size are more sensitive to partitioning errors as compared to noise in the inter-division times. In summary, our result show that appropriate regulation of growth rate by size (as seen in mammalian cells) can be an eﬀective mechanism for achieving size homeostasis. We next consider a diﬀerent class of models where size-based regulation is at the level division rather than growth.

2.3 Size-dependent division rate In contrast to growth rate control, many organisms rely on size-dependent regulation of division rate for size homeostasis [2,3, 79–81]. To analyze this strategy, we consider the SHS continuous dynamics (2.2) with a timer-dependent growth rate α(τ ), and a division rate f(v, τ ) that now depends on size. The theorem below provides

18 Cell cycle time

Partitioning in cell size Mean cell size Stochastic variability

Noise in partitioning and cell cycle time

Figure 2.2: Stochastic variation in cell size (blue) and mean cell size (green) as a function of CV 2 (noise in inter-division time) and CV 2 (error in partitioning Ti β of volume among daughters) for linear cell growth and a timer-based division mechanism. The mean cell size is dependent on CV 2 but in- Ti 2 dependent of CVβ . Fluctuations in cell size increase more rapidly with CV 2 than with CV 2 . β Ti

19 suﬃcient conditions on f(v, τ ) for size homeostasis.

Theorem 4: Let there exist a non-decreasing function g(τ ) and p > 0 such that

f(v, τ ) ≥ g(τ )vp. (2.28)

Moreover, the division rate for a suﬃciently small cell size f0(τ ) := limv→0 f(v, τ ) satisﬁes

D R Ti E R x α0(y)dy − f(0y)dy e y=0 > 2,Ti ∼ f0(x)e y=0 . (2.29)

Then, for the SHS given by (2.2) and (2.3)

l lhα(τ )i p 0 < lim hvl(t)i < , (2.30) t→∞ hg(τ )i(1 − hβli) for l ∈ {1, 2,... }.

Proof of Theorem 4: Consider a newborn cell with a suﬃciently small size at time t = 0. Then, based on Theorem 1, the mean size will grow over successive generations (and not go extinct) iﬀ (2.29) holds. Based on the Dynkin’s formula for (2.2)-(2.3), the time evolution of moments is given by

dhvli = lα(τ )vl − f(v, τ )vl 1 − βl (2.31) dt Using (2.28), the fact that α(τ ) is a non-increasing function, while g(τ ) is a non- decreasing function,

dhvli ≤l hα(τ )i vl − hg(τ )i vl+p 1 − βl (2.32) dt

l+p l+p l l Finally, using v ≥ v in (2.32) result in (2.30) at steady state.

Next, we show that diﬀerent known strategies for size-dependent regulating of inter-division times are consistent with Theorem 4. A common example of size- dependent division is the “sizer strategy”, where a cell senses its size, and divides when

20 a critical size threshold is reached [24,82–84]. Such as strategy can be implemented by

v p f(v, τ ) = (2.33) v¯ wherev ¯ and p are positive constant. A large enough p corresponds to division events occurring when size reachesv ¯. In contrast to the sizer strategy, many bacterial species use an “adder strategy”, where a cell divides after adding a ﬁxed size from birth [6,9, 10, 19]. In the case of exponential growth (constant growth rate α), the adder strategy can be implemented by

v (1 − e−ατ )p f(v, τ ) = . (2.34) v¯

A large enough p would correspond to cells adding a ﬁxed sizev ¯ between cell birth and division [22]. Both these division rates are consistent with the form of f required for size homeostasis in Theorem 4. We investigate the ﬁrst two moments of v in more detail for the sizer strategy. Using (2.31) for a constant growth rate α and division rate (2.33) results in the following moment dynamics

d vl = lα vl − v¯−p vl+p 1 − βl . (2.35) dt

Let µ = hvi , hv2i · · · vL T be a vector of moments up to order L, where L is the order of truncation. Using (2.35), the time evolution of µ can be compactly written as

dµ T = a + Aµ + Cµ,¯ µ¯ = vL+1 ··· vL+p (2.36) dt for some vector a, matrices A and C, andµ ¯ is the vector of higher order moments. Note that nonlinearities in the division rate lead to the well known problem of moment closure, where time evolution of µ depends on higher-order momentsµ ¯. Moment closure techniques that expressµ ¯ ≈ θ (µ) are typically used to solve equations of the form (2.36). Here, we use closure schemes based on the derivative-matching technique [85– 87], that yield analytical expressions for the steady-state moments. For example, L = 2

21 in (2.36) (second order of truncation) results in the following steady-state mean and coeﬃcient of variation squared of cell size

p+1 1 2 2p ! p 1 1 3 − CVβ 4 p p 2 hvi ≈ 2 α v¯ ,CVv ≈ 2 − 1, (2.37) 4 3 − CVβ respectively. Intriguingly, (2.37) shows that the mean cell size decreases with increasing

2 magnitude of partitioning error CVβ . While the results from (2.37) are qualitatively consistent with moments obtained via Monte Carlo simulations, a much higher order of truncation is needed in (2.36) to get an exact quantitative match (Fig. 3).

Approximation: 2nd order 20th order Simulations

Figure 2.3: Stochastic variation in cell size (blue) and mean cell size (green) as a 2 function of CVβ (error in partitioning of volume among daughters) for exponential cell growth and sizer-based division mechanism. The mean 2 cell size decreases with increasing CVβ , while noise in cell size increases with it. Results are shown for a 2nd (dashed) and a 20th (solid) order moment closure truncation, and compared with moments obtained by running a large number of Monte Carlo simulations. Errors bars show 95% conﬁdence estimates.

Here we have used a phenomenological SHS framework to model time evolution of cell size (Fig. 8.1). The model is deﬁned by three features: a growth rate α(v, τ ), a

22 division rate f(v, τ ), and a random variable β ∈ (0, 1) that determines the reduction in size when division occurs. A key assumption was that α and f are monotone functions: with increasing size and cell-cycle progression, the growth rate decreases, and propensity to divide increases. Our main contribution was to identify suﬃcient conditions on α and f that prevent size extinction and also lead to bounded moments (Theorems 2 and 4). In essence, these conditions require the growth (division) rate to decrease (increase) with cell size in a polynomial fashion. We also analyzed two strategies for size homeostasis: i) Linear growth in size with timer-controlled divisions and ii) Exponential growth in size with size-controlled divisions. Analysis reveals that in the former strategy, the mean cell size is independent of volume partitioning errors at the time of mitosis. In contrast, the mean cell size decreases with increasing partitioning errors for size-controlled divisions. Moreover, stochastic variations in cell size are found to be highly sensitive to partitioning errors for both strategies (Fig. 2 and 3). This suggests that cells may use mechanisms to minimize volume mismatch among daughter cells. In summary, theoretical tools for SHS can provide fundamental understanding of regulation needed for size homeostasis. Future work will focus on coupling cell size to gene expression, and understanding how concentration of a given protein is maintained in growing cells [38,40,88,89].

23 Chapter 3

A MECHANISTIC STOCHASTIC FRAMEWORK FOR REGULATING BACTERIAL CELL DIVISION

Recurring cycles of growth and division of a cell is a ubiquitous theme across all organisms. How an isogenic population of exponentially growing cells maintains a narrow distribution of cell size, a property known as size homeostasis, has been extensively studied, e.g., see [1,3, 83, 90] and references therein. From a phenomenological standpoint, recent experiments reveal that diverse microorganisms achieve size homeostasis via an adder principle [7–10]. As per this strategy, cells add a constant size from birth to division regardless of their size at birth [6,91]. Interestingly, the size accumulated by a single cell between birth and division exhibits considerable cell-to-cell differences, and these differences follow unique statistical properties. For example, in a given growth condition, the added size is drawn from a fixed probability distribution independent of the newborn cell size. Moreover, the distribution of the added size normalized by its mean is invariant across growth conditions [8]. Here, we explore biophysical models that lead to the adder principle of cell size control and provide insights into its statistical properties. To realize the adder principle mechanistically, a cell needs to somehow track the size it has accumulated since the previous division and trigger the next division upon addition of the desired size. One biophysical model proposed to achieve this assumes a protein which begins to get expressed right after cell birth at a rate proportional to instantaneous volume (size). The cell grows exponentially over time and division is triggered when protein copy numbers reach a critical threshold after which the protein is assumed to degrade (Fig. 3.1a) [6,9, 21]. Such copy number dependent triggering of cell division could potentially be implemented via the localization of protein into

24 compartments whose volume does not change appreciably with the cell volume [62]. Moreover, the synthesis and the degradation of the protein in this model are used in broad sense; they could as well be activation of timekeeper proteins in size dependent manner, and deactivation after triggering of division. While this deterministic model results in a constant size added from cell birth to division [6, 21], it remains to be seen how noise mechanisms can be incorporated in this model to explain statistical fluctuations in cell size. A plausible source of noise could be the inherent stochastic nature of protein expression that has been universally observed in prokaryotes and eukaryotes [92–96]. Such stochasticity in protein synthesis is amplified at the level of individual cells, where gene products are often present at low molecular counts. Considering noisy expression of the timekeeper protein, one can formulate cell- division time as a first-passage time problem: an event (division) occurs when a stochastic process (protein copy numbers) hits a threshold for the first time (Fig. 3.1b). Ex- ploiting this first-passage time framework, we derive an exact analytical formula for the cell-division time distribution for a given newborn cell size. Consistent with data, these results predict that the mean cell-division time decreases with increasing cell size at birth, and the randomness (quantified by coefficient of variation squared) in the cell-division time increases with newborn cell size. Intriguingly, analysis of the model further shows that the distribution of the volume added from cell birth to division is always independent of the newborn cell size. Finally, we find that the distributions of added volume and cell division time have scale invariant forms: distributions in different growth conditions collapse upon each other after rescaling them with their respective means. We discuss potential candidates for the timekeeper protein and deliberate upon model modifications that result in deviations from the adder principle.

Consider a newborn cell with volume Vb at time t = 0. Its volume at a time t after birth is given by V (t) = Vb exp(αt), where α > 0 represents the growth rate. After cell birth, the timekeeper protein begins to get transcribed at a rate r(t) = kmV (t), where km is the transcription rate in the concentration sense. Note that this scaling of protein synthesis with instantaneous cell volume is essential for preserving gene

25 product concentrations in growing cells. In the stochastic formulation, the probability of a transcription event occurring in an inﬁnitesimal time interval (t, t + dt] is given by r(t)dt. Assuming short-lived mRNAs, each transcript degrades instantaneously after producing a burst of protein molecules [97–102]. Stochastic expression of the timekeeper protein is compactly represented by the following biochemical reaction:

r(t) ∅ −−→ Bi × P rotein, (3.1) where r(t) = kmV (t) can be interpreted as the burst arrival rate and Bi, i ∈ {1, 2, ···}, are identical and independent random variables denoting the size of protein bursts with mean b := hBii. The burst size represents the number of protein molecules synthesized in a single mRNA lifetime and typically follows a geometric distribution [98, 100, 102–105]. However, to allow a wide range of protein accumulation processes to be covered by equation (3.1), we assume that Bi follows an arbitrary non-negative integer-valued distribution. One example of such a mechanism could be to consider a protein A whose concentration is constant throughout the cell cycle. This protein is stochastically converted to an active form A∗ at a rate proportional to the number of molecules of A. In essence, this can be thought of as production of A∗ in bursts which takes place at a rate proportional to the cell volume. Let x(t) denote the number of timekeeper molecules in the cell at time t after birth. Assuming a stable protein with no active proteolysis, we have x(t) = Pn i=1 Bi, x(0) = 0, where n is the number of bursts (transcription events) in [0, t]. Cell division occurs when x(t) reaches a threshold X and the protein is degraded (or deactivated) thereafter. Given this timing mechanism, cell-division time can be math- ematically represented as the ﬁrst-passage time (FPT )

FPT := inf {t : x(t) ≥ X | x(0) = 0} . (3.2)

This first-passage time framework assumes that cell division occurs upon precise attainment of X protein molecules. In principle, one could generalize equation (3.2) by defining a monotonically increasing function h(x) that defines a probabilistic rate

26 Figure 3.1: Proposed molecular mechanism to realize adder principle of cell size control. (a) An exponentially growing rod-shaped cell starts synthesizing a timekeeper protein after its birth. The production rate of the protein scales with the cell size (volume). When the protein’s copy number at- tains a certain level, the cell divides and the protein is degraded. (b) Stochastic evolution of the protein copy numbers is shown for cells of three different sizes at birth. The threshold for triggering cell division is assumed to be 50 molecules. The distribution of the first-passage time (generated via 1, 000 Monte Carlo realizations) for each newborn cell volume is shown above the three corresponding trajectories. The first- passage time distribution depends upon the newborn cell size: on average, the protein in a smaller cell takes more time to reach the threshold as compared to the protein in a larger cell. of cell division at time t given x(t) molecules. Interestingly, analysis reveals that the average size added from birth to division is invariant of the newborn cell size Vb iff

h(x) = 0 for x < X, h(x) = ∞ for x > X (3.3)

(see Supplementary Information (SI), section S1). Thus, a sharp threshold, where cell division cannot be triggered before attainment of a precise number of molecules seems to be a necessary ingredient of the adder principle.

27 3.1 Distribution of the cell-division time given newborn cell size Here we derive the distribution of the cell-division time (FPT ) for a given newborn cell size Vb and investigate how its statistical moments depend on Vb. We begin by ﬁnding the distribution of the minimum number of burst events N required for x(t) to reach the threshold X. In particular,

( n ) n ! X X N := inf n : Bi ≥ X =⇒ P rob (N ≤ n) = P rob Bi ≥ X . (3.4) i=1 i=1

Given a speciﬁc form for the distribution of Bi, the corresponding distribution for N can be obtained using equation (3.4). For example, if Bi is geometrically distributed, then the probability mass function of N is given by

n + X − 2 1 n−1 b X f (n) := P rob (N = n) = , n ∈ {1, 2,...}, N n − 1 b + 1 b + 1 (3.5) where b represents the mean burst size [106,107]. Having determined the number of bursts needed for cell division, we next focus

th on the timing of burst events. Let Tn represent the time at which n burst event takes place. If the burst arrival rate in equation (3.1) were constant, then the time intervals between bursts would be exponentially distributed, resulting in an Erlang distribution for Tn. However, in our case this rate is time varying (due to dependence on cell volume), the arrival of bursts is an inhomogeneous Poisson process. Employing the distribution for the timing of the nth event, and using the fact that FPT is same as the time at which the N th burst event occurs, the probability density function of FPT is obtained as

∞ ∞ n−1 X X (R(t)) f (t) = f (t) f (n) = r(t) exp(−R(t))f (n), (3.6) FPT Tn N (n − 1)! N n=1 n=1 Z t k V R(t) := r(s)ds = m b eαt − 1 , (3.7) 0 α

(see SI, section S2). One can note that fFPT (t) is dependent on the newborn cell size

Vb through the function R(t).

28 Figure 3.2: Both model prediction and data show increase in the noise in timing as newborn cell size increases. (a) Model prediction for noise (coeﬃcient of variation squared, CV 2) in division time as computed numerically using equation (3.7) . The model parameters used are: transcription −1 rate km = 0.13 min , threshold X = 65 molecules, growth rate α = 0.03 min−1, and mean burst size b = 5 molecules. The distribution of protein burst size Bi is assumed to be geometric. For details on how these parameter values were estimated, see SI, section S6. (b) Experimental data from [90] for Escherichia coli MG1655 also shows increase in cell division time noise as newborn cell size increases. Single-cell data was categorized in one of the four bins (1−2.8 µm, 2.8−4.5 µm, 4.5−6.3 µm, and 6.3−8 µm) depending upon newborn cell sizes. CV 2 of division time with 95% conﬁdence interval (using bootstrapping) for each bin is shown (more details in SI, section S6).

This FPT distribution qualitatively emulates the experimental observations that the mean cell division time decreases with increasing cell size at birth (see SI, section S6). Intuitively, a larger newborn cell expresses the protein at a higher rate as compared to a smaller cell. Hence, the time taken by the protein to reach the prescribed molecular threshold is shorter in larger cells. Analysis of equation (3.7) also predicts that the noise (quantified using the coefficient of variation squared, CV 2) in cell-division timing increases with increasing Vb, and we confirmed this behavior from published data (Fig. 3.2). The noise behavior can be understood from the fact that a small newborn cell requires more time for cell division. This allows for efficient time averaging of the underlying bursty process resulting in lower stochasticity in FPT .

29 3.2 Distribution of the volume added between divisions

(a) (b) Simulations Size

Time (minutes) (c) (d) Experimental data Mean size added Mean size added

Time (minutes)

Figure 3.3: The proposed mechanism results in added cell size distribution being independent of the newborn cell size. (a) The cell volume grows exponentially (shown for three diﬀerent newborn cell sizes) until the timekeeper protein reaches a critical threshold. (b) The size added to the newborn cell size also grows exponentially until division takes place. For three diﬀer- ent newborn cell sizes, the distribution of the the added volume comes out to be same. (c) The added size generated via simulations is plotted against the newborn cell size in range 2 − 3.5 µm for 10, 000 cells. The cells are further binned in 13 uniformly spaced bins (number of cells per bin > 100). The dashed line shows the mean of the added volume, which is independent of the newborn cell size. (d) Data from [8] showing the added size versus newborn cell size for Escherichia coli NCM3722 grown in Glucose as carbon source. Cells were categorized into bins according to their newborn cell size (number of cells per bin > 100). For each bin, the circle shows mean of the added size whereas the error bar represents the standard deviation of the added size. It can be seen that the mean added cell size (shown by dashed line) is independent of the newborn cell size (also see Fig. 2D in [8]).

Having derived the distribution for the cell-division time (FPT ), we determine

30 the volume added by a single cell from birth to division (denoted by ∆V ). Since

αF P T volume grows exponentially, ∆V is related to FPT as ∆V = Vb e − 1 . Using the distribution of FPT from equation (3.7) yields the following probability density function for ∆V

n−1 ∞ kmv X km kmv f (v) = α exp − f (n) (3.8) ∆V (n − 1)! α α N n=1

(see SI, section S3). One striking observation is that f∆V (v) is independent of the initial volume Vb (as illustrated in Fig. 3.3). This is in agreement with experimental observations that the histograms of the added volume for diﬀerent newborn cell sizes are statistically identical [8]. Next, we investigate how statistical moments of ∆V depend on model parameters, in particular, the growth rate α.

Mean volume added between divisions Using equation (3.8), the average volume added is obtained as

∞ Z v=∞ X α n α h∆V i = vf (v)dv = f (n) = hNi . (3.9) ∆V k N k v=0 n=1 m m Here hNi represents the mean number of protein burst events from cell birth to division, which depends on the threshold X and the form of the burst size distribution. For example, if the protein bursts Bi are geometrically distributed with mean b, then using equation (3.5) α X h∆V i = + 1 . (3.10) km b These formulas reveal a linear dependence of ∆V on α, in agreement with data from Pseudomonas aeruginosa [9]. It turns out that the dependency of ∆V on α can vary among bacterial species. For instance, Caulobacter crescentus exhibits an added volume independent of α, whereas this relationship is thought to be exponential in case of Escherichia coli [7,8]. Studies connecting cellular growth rates to gene expression parameters have shown that α primarily aﬀects the transcription rate, with mRNA translation and stability being largely invariant across growth conditions [108, 109].

31 Thus, if the transcription rate km is a linear function of α, then ∆V becomes independent of α. Next, we discuss a slightly different model formulation that results in exponential dependency of ∆V on α. So far we have considered that the timekeeper protein observes time from cell birth to division. In principle, the timekeeping could be for some other important event in the cell cycle. Consider a scenario where the initiation of DNA replication takes place when sufficient timekeeper protein has accumulated per origin of replication [6, 13, 110, 111]. The corresponding division event is assumed to occur with a constant delay of T after an initiation. The delay T here is the C + D period, where C represents the time to replicate the DNA and D denotes the time between DNA replication and division [112, 113]. As growing bacterial cells are known to regulate the number of DNA replication forks as a function of growth rate, we assume that the threshold for the timekeeper proteins changes accordingly. More specifically, if there are θ origins of replication, the number of timekeeper protein molecules required to be accumulated for the next initiation event are θX. The above assumption is consistent with the understanding that all origins of replication fire almost synchronously. Further, the timekeeper molecules are assumed to get degraded (deactivated) after initiation and a new set of timekeeper molecules are produced for the next initiation. Upon a division event between two successive initiations, the partitioning errors in the timekeeper protein are assumed to be negligible. In this alternative formulation, the average volume added between two consecutive initiation events for each origin of replication is approximately same as ∆V obtained in equation (3.10) (see SI, section S3). Moreover, the average volume added between successive division events is now given by [13]

h∆V ∗i ≈ h∆V i eαT . (3.11)

Recall from equation (3.10) that h∆V i depends linearly on α. Thus, the expression in equation (3.11) suggests two diﬀerent regimes of how h∆V ∗i depends upon α. For small values of α, α exp(αT ) ≈ α, i.e., the mean added volume increases linearly with

32 the growth rate. In the regime where α is large, the exponential term dominates. This implies that if α is small, it may not be possible to distinguish whether the underlying mechanism accounts for volume added between two division events or two initiation events as the data will show a linear dependence of the average added volume with changes in α [9]. Notice that a pure exponential relationship between h∆V ∗i and α can also be obtained if km is a linearly increasing function of α. For this particular case, the volume accounted by each origin of replication h∆V i becomes invariant of the growth rate, consistent with previous works [13, 114]. In summary, depending on the underlying assumptions, the model captures a variety of relationships between the average volume added from cell birth to division and α. It is noteworthy that in the above setup, dependency of the time T = C + D on growth rate or cell size has been neglected even though there is evidence that D usually depends upon both growth rate and cell size [17]. We have done so for simplicity as incorporating this would not change the fact that an exponential dependency can be generated between ∆V and α by having the protein account for two other events in the cell cycle. We next investigate higher order moments of ∆V in the original model formulation, where the timekeeper protein accounts for timing between division events.

3.3 Higher order moments of added volume We can use the distribution of ∆V computed in equation (3.8) to get insights

2 into its higher-order statistics such as coeﬃcient of variation squared (CV∆V ) and skewness (skew∆V ). For example, when the protein production occurs in geometric bursts

2 3 2 2 b + 2bX + X 2 (b + 3b X + 3bX + X) CV∆V = , skew∆V = (3.12) (b + X)2 (b2 + 2bX + X)3/2

(see SI, section S3). Note that ∆V is always positively skewed, consistent with previous understanding [91]. Moreover, both CV 2 and skewness are independent of the growth rate α. It turns out an even more general property is true: an appropriately scaled jth order moment of ∆V , i.e., h∆V ji / h∆V ij is independent of α, in spite of the underlying

33 distribution of the burst size. This arises from the fact that the distribution of ∆V can be written in the following form

1 v f (v) = g (3.13) ∆V h∆V i h∆V i for some function g (see SI, section S3). This form implies that f∆V (v) is scale invariant: the shape of the distribution across different growth rates is essentially the same, and a single parameter h∆V i is sufficient to characterize the distribution of ∆V [115]. This property was seen in experiments [8, 21, 116], where the histograms for ∆V/ h∆V i in different growth conditions collapse upon each other (Fig. 3.4).

Figure 3.4: Collapse of added cell size in different growth conditions upon rescaling by respective mean values. (a) Using data from [8] for Escherichia coli NCM3722, the added size is plotted versus the newborn cell size for different growth conditions. The mean added size (shown by circles) for each growth condition is different for a given newborn cell size. Cells were categorized into bins according to their newborn cell sizes (number of cells per bin > 100). The error bars represent the standard deviation of the added volume of cells in each bin. (b) The added size data for different growth conditions collapse upon rescaling them by their means in the respective growth conditions (also see Fig. 2D in [8]).

Interestingly, the above invariance property is not limited to the distribution of the added volume ∆V . As the distributions of the cell size at birth, and cell size at

34 division are generated by weighted sums of random variables drawn from the distribution of ∆V , they naturally inherit the scale-invariance property [8] (see SI, section S4). Furthermore, the distribution of the cell-division time also has the scale invariance property (see SI, section S5), which is in agreement with previous works [117,118]. It is now well understood that several prokaryotes, such as, Escherichia coli, Caulobacter crescentus, Bacillus subtilis and Pseudomonas aeruginosa employ an adder mechanism for size homeostasis [7–10]. In this work, we studied a simple molecular mechanism for realizing the adder principle that consists of a timekeeper protein expressed at a rate proportional to cell volume up to a critical threshold. Our work shows that stochastic expression of this protein is suﬃcient to explain the statistical properties of the cell-division time and the size added from cell birth to division. Key model insights are as follows:

• Distribution of the volume added from birth to division is independent of the newborn cell volume, a hallmark of the adder principle (Fig. 3.3).

• The distributions of key quantities such as the added volume, division time, volume at birth and division are scale invariant.

• The noise in cell-division time increases with increasing newborn cell size (Fig. 3.2).

An important point to note is that if variation in ∆V is indeed a result of noisy gene expression, then ∆V for successive cell-cycles should be independent. Indeed, data shows a weak correlation between the volume added for mother and daughter cells [7,8]. This result also argues that extrinsic ﬂuctuations in parameters that exhibit strong memory between mother and daughter cells cannot account for the statistical ﬂuctuations in ∆V .

35 Chapter 4

PART 2: OPTIMALITY IN HOST-VIRUS SYSTEMS

Life traits of virus are strikingly variable, ranging from highly infectious and virulent to less virulent and chronic. Unveiling the mechanisms behind these diﬀerent viral strategies of host explotation remains a key challenge in biology. The classic theory of parasite evolution shows that nature will select the virus that maximizes the basic reproductive ratio (R0). This quantity represents the number of secondary infections resulting from one infected host. We can compute it by understanding the dynamics behind host-virus interactions. Let T be the target cell population available in the environment. Assuming this population is near steady state, its dynamics can be described by the ODE

˙ T = λ − dT T

. Under unconstrained conditions, if a virus V and a target cell meet, the former will infect the later. Infected cells (I) will actively produce viruses until its death. This phenomena is modeled as

˙ T = λ − dT T − βT V (4.1) ˙ I = βT V − dI I (4.2) ˙ V = bI − dV V, (4.3) where β is the adsorption rate, b is the number of virus produced by an infected cell.

β and b might be interpreted as the infectivity of the virus. The death rates dI and dV represent the death rate of the infected cell and virus, respectively. dI can be referred

T = λ/dC ,I = 0,V = 0.

36 Our system will leave this state if and only if

λ β b R0 = > 1, (4.4) dC dI dV i.e., if the number of secundary infections is greater than 1.

Clearly, virus infectivity (β, b) acts to increase R0, whereas virulence (dI ) decreases it by reducing the infection period. With this in mind, if the objective is to maximize R0 (number of secondary infections), virus might attemp to evolve to infinte infectivity (β, b) and zero virulence (dI ) as suggested by Equation (4.4). Experimental work had shown contradicting results, supporting the existence of both moderated infectivity and virulence. Through the following chapters we propose scenarios in which moderated infectivity and virulence might be advantageous to the virus through several fitness definitions including the above discussed. There is evidence suggesting that viruses with intermediate (moderated) adsorption rate are more adequate under structured host populations where virus-host local interactions cannot be discarded. For instance, the local interaction between debris of the lysed cell and newborn virus has been suggested as explanation for intermediate evolved traits. Chapter 2 introduces a model that includes local interactions between bacterial cell debris and newborn viruses (bacteriophage). We found that these interactions may in fact explain the observed intermediate adsorption rate. We show this by exploring single and competition dynamics of different virus strains. How this moderated infection mechanisms are implemented in more complex setups like human-virus interactions? In chapter 3 we study the infection by Human Imunnodeficiency Virus. It has been shown that the infection of CD4+ T Cells by HIV happens by two distinct mechanisms: cell-free transmission by free viruses (large infectivity), and cell-cell transmission (intermediate infectivity) in which viral particles are transmitted directly across a tight junction or synapse between an infected and an uninfected cell. We show that synapse-forming HIV (moderated) provides an evolutionary advantage relative to non synapse-forming virus given a specific scenario.

37 Is moderate virulence of current virus an evolutionary advantage? Chapter 5 explores the way in which virus evolved to moderate temperateness in terms of the propensity of a virus to enter lysogeny. Assuming scenarios with recurrent cycles of good and bad conditions, we studied how multiplicity of infection (MOI) - the ability of a virus to infect an already infected cell - drives the stochastic decisions of entering or not into lysogenic mode. We found that temperate virus might use the lysogenic path to protect themselves from extended periods of detrimental conditions. In HIV infections, populations of quiescently infected cells form a ”latent pool” (moderated virulence) which causes rapid recurrence of viremia whenever antiviral treatment is interrupted. A ”cure” for HIV will require a method by which this latent pool may be eradicated, by increasing the virulence of the virus. Current eﬀorts are focused on the development of drugs that force the quiescent cells to become active. Previous research has shown that cell-fate decisions leading to latency are heavily inﬂuenced by the concentration of the viral protein Tat. In chapter 6, we explore therapeutic alternatives using combination of traditional and Tat based drugs.

38 Chapter 5

OPTIMAL ADSORPTION RATE: IMPLICATIONS OF THE SHIELDING EFFECT

Since the evolution of parasites and pathogens is important to human [119], agri- cultural, and wildlife systems, there is a mature theory that focuses on how infection mechanisms may evolve. Given that viruses are the most abundant and simple entities on the planet, they are frequently used as models for studying parasite-pathogen evolution. In particular, parameters like replication, lysis time, adsorption among others have been suggested as possible knobs used by viruses to drive infection [120–123]. Furthermore, adsorption rate has been proposed as one factor that the viruses may easily tweak to gain maximum advantage from bacterial populations [124–126]. The classic theory of parasite evolution states that natural selection will maximize the number of secondary infections resulting from infecting a susceptible host [127]. One way of doing so is by evolving the adsorption rate, which is the probability of a virus infecting a bacteria upon direct contact. Under unconstrained environments, the classical theory will predict that the virus will evolve to maximum (inﬁnite) adsorption rate. However, experiments show the unexpected emergence of virus with moderate or intermediate adsorption rates [125,128]. How and under which conditions this intermediate rate evolves remains to be poorly understood. From the biological standpoint, several theories has been proposed. The occurrence of intermediate adsorption rates may be explained by the presence of host spatial structure [129]. This spatial structure have importance to the local virus-host interactions that can constrain the viral evolution into higher adsorption rates. Presumably, the viral strains with high absorption rates tend to create a shielding eﬀect in which the local availability of healthy bacteria is reduced, producing more virus-infected cell

39 interactions, which lead to zero new viruses [127]. In other words, high adsorption rate can be a curse for viruses since it can lead to premature discard if attached to debris produced by cell lysis [123]. Spatial structure in the susceptible bacteria has been modeled using deterministic [130–132], and stochastic approaches [133–137]. Specifically, these models address the problem of virus-bacteria coexistence rather than explaining why intermediate adsorption rate may arise. To the best of our knowledge, there is only a recent work which studies optimality on adsorption rate using agent-based modeling and modeling explicitly bacterial structure [123]. Specific debris modeling has been also used for studying host-pathogen coexistence without studying theoretical aspects of the occurrence of optimal adsorption rate under shielding effects conditions [138,139]. In this chapter, we propose an approach for modeling local interaction effects through the inclusion of debris as a form of shielding. We use ODE tools to study the impact of debris on the virus fitness. Specifically, we assume that newborn viruses are locally surrounded by debris from the lysed cell. The effective burst size of an infected cell is related with the propensity of a newborn virus to attach to lysed cell material. If virus escapes from this local interaction it will find susceptible bacteria and create new progeny. Our simple yet effective model shows the existence of optimal virus fitness in single and competition assays that cannot be explained by classic parasite-host models. This chapter is organized as follows: first classic virus-bacteria interaction models are introduced, and its implications for viral fitness are discussed. Then we describe how local interactions can be included into the classic model, i.e., how debris is included in the model. Later, we discuss single virus dynamics implications. Furthermore, competition between strains is explored. Finally we present conclusions and further work.

5.1 Traditional virus dynamics model We consider ﬁrst a basic model for virus dynamcis introduced by [140]. Let C, I, and V be respectively the number of healthy, infected bacteria, and virus particles.

40 Figure 5.1: Dynamics of the shielding eﬀect. Virus attach to a healthy bacteria, injects its genetic material, infected cells produce new viruses, and then the infected bacteria lysis producing debris and newborn viruses. The newborn viruses can either attach to the debris (shielding eﬀect) or escape from it and infect new susceptible bacteria.

The dynamics of these species is given by the ODE system

˙ C =λ − dC C − rV C (5.1) ˙ I =rV C − dI I (5.2) ˙ V =BdI I − rV C − dV V. (5.3)

Healthy bacteria reproduces at a rate λ and dies at rate dC . Virus attacks bacteria at rate rV C, where r is the adsorption rate. Once the infected bacteria dies (with latent period 1/dI ), it releases a burst of viruses of size B. Free virus may die at the rate dV . It is easy to check that the steady state virus level of system (2.1-2.3) is

λr (B − 1) − d d lim V = C V . (5.4) t→∞ dV r

Eq. (2.4) can be seen as one way to measure virus fitness. Note that, for the virus to obtain maximum fitness, adsorption rate should be infinite ( maximum virus load at steady state is λ (B − 1) /dV ). Alternatively, the baseline reproduction ratio may be used to represent the fitness of the virus. Recall that the uninfected steady state C = λ/dC , I = 0, and V = 0 is unstable point (meaning the infection will take place) if

Bλr R0 = > 1, (5.5) dC dV

41 where R0 is the baseline reproduction ratio and can be interpreted as the number of viruses produced per infection. R0 can be used to infer the evolutionary outcome of the system (2.1-2.3). For instance, we can infer from (2.5) that the virus should evolve towards inﬁnite adsorption rates in order to get the maximum ﬁtness.

5.2 Modeling the shielding effect Consider the basic lytic cycle of a virus depicted in Fig 2.1. A given virus attach to a healthy bacteria, and injects its genetic material. Then, infected cells produce new viruses during the latent period. The lysis of the infected bacteria releases debris and newborn viruses. The newborn viruses can either attach to the debris (shielding effect) or escape from it and infect new susceptible bacteria. To model the shielding effect, we included two new species into the system (2.1-2.3). First, we assumed that, once the virus is released after infected cell lysis

(Vs), this virus will be surrounded by cell debris (X). Cell lysis will release B virus copies and q debris. The newborn virus will either attach to the debris and become inactive, or escape at rate df and be able to reach suceptible bacteria. We assume that after escaping, the shielding eﬀect disappears due to fast debris dynamics. The virus dynamics of this new system can be written as

˙ C = λ − dC C − rV C (5.6) ˙ I = rV C − dI I (5.7) ˙ Vs = bdI I − rXVs − dV Vs − df Vs (5.8) ˙ X = qdI I − dX X − rXVs (5.9) ˙ V = df Vs − rV C − dV V, (5.10)

where dX is the debris degradation rate. Assuming that the dynamics of the debris

(X) and the shielded virus (Vs) are assumed to be faster than free virus, susceptible

42 bacteria, and infected cell, this is

bdI I − rXVs − dV Vs − df Vs =0 (5.11)

qdI I − dX X − rXVs =0, (5.12)

the shielded virus Vs and the debris levels are given by

¯ bdI I ¯ qdI I Vs = ¯ , X = . (5.13) (df + dV ) + rX dX ¯ Replacing Vs with Vs in (2.10), and recalling that we assumed faster dynamics for X and Vs interactions, the burst size B can be rewritten in terms of the adsorption rate (r) as df B (r) = b ¯ . (5.14) df + rX where b is the maximum burst size that can be obtain from infected cells. Now we can replace the constant burst size in system (2.1-2.3) by the adsorption-dependent burst size (2.14). The burst size function (2.14) can be seen as the eﬀect of the spatial structure in the system (2.1-2.3). For instance, small escape rates of the virus (df ) can be interpreted as debris clusters formed after cell lysis that protects susceptible bacteria from being infected by the newborn viruses. Larger escape rates imply no debris surrounding new viruses, allowing newborn viruses to attack bacteria once cell lysis occurs.

5.3 Single virus dynamics The adsorption-dependent burst size given by (2.14) provides new properties to system (2.1-2.3). In particular, the maximum steady state virus load is only possible if the adsorption rate is given by √ df d d r∗ = C V√ . (5.15) p ¯ ¯ bλdf X − X dC dV Fig. 2.2 shows the eﬀect of constant and adsorption-dependent burst size in the steady state virus load. Instead of inﬁnite adsorption rate, optimal virus load requires a

43 finite and intermediate adsorption to avoid virus waste produced by the shielding effect. Moreover, note that the fitness defined by R0 becomes finite when adsorption- ¯ dependent burst size is used, given by λbdf /(dC dV X).

Figure 5.2: Effect of constant and adsorption-dependent burst size in the steady state virus load. Instead of infinite adsorption rate, optimal virus load requires a finite and intermediate adsorption to avoid virus waste produced by the shielding effect.

Although single virus dynamics show emerging properties promoted by the shielding eﬀect, it is not clear how these properties will promote the evolution of the virus to moderate adsorption rates. Next we will study the implications of having adsorption-dependent burst size in competition between diﬀerent viral strains.

5.4 Competition dynamics To study competition between diﬀerent strains we built a two-strain ODE system by extending (2.1-2.3). We assume that there is an established virus V (which is at steady state). A diﬀerent virus strain Vm (invader) is introduced to the environment

44 at small copy number. The dynamics of the expanded system is given by

˙ C =λ − dC C − rV C − rmCVm (5.16) ˙ I =rV C − dI I (5.17) ˙ V =B (r) dI I − rV C − dV V (5.18) ˙ Im =rmCVm − dI Im (5.19) ˙ Vm =B (rm) dI Im − rmCVm − dV Vm, (5.20)

where Im, and rm are the infected cell level and the adsorption rate of the new viral strain introduced into the environment. Next, we derive the condition for the mutant virus Vm to invade the established virus V . Consider the dynamics (described in Eq. (2.16-2.20)) of a resident and an invader virus with adsorption rates r and rm, respectively. Additionally, let the resident virus be at its steady state level, i.e.,

d C¯ = V (5.21) (B (r) − 1)r d d + λr − λrB (r) I¯ = C V (5.22) dI r − B (r) dI r (B (r) − 1)λ d V¯ = − C (5.23) dV r

We are interested in the local stability of the steady state solution C = C¯, I = I¯, ¯ V = V , Vm = 0, and Im = 0. To this end, we compute the Jacobian   J 0 JC,¯ I,¯ V,¯ 0,0 =   , (5.24) 0 Jm where J is the matrix containing the partial derivatives of the dynamics of the resident virus, and   dV rm − (B(r)−1)r − dV B (rm) dI Jm =   (5.25) dV rm (B(r)−1)r −dI contains the partial derivatives of the mutant dynamics. Each element of Jm represents an increase or decrease in the number of infections and viruses. For instance, viruses

45 and infected cells die by degradation (dV ) and lysis events (dI ), respectively. Alterna-

dV rm tively, the number of viruses increase by B (rm) dI . The expression (B(r)−1)r represents the number of mutant viruses required to produce new infections in presence of the established strain. Note that the stability of the system is dictated by the eigenvalues of the matrix

Jm. We use a simple yet powerful approach to study this eigenvalues. As proposed by [141], we decompose Jm into the matrices   0 B (rm) dI F =   , (5.26) 0 0   dV rm (B(r)−1)r + dV 0 V =   . (5.27) dV rm − (B(r)−1)r dI

F represents the alternatives available to produce a mutant virus. The processes in which a mutant virus die are included in V . Given the matrices F and V , the stability of the aforementioned system is dictated by the following theorem.

Next-generation theorem [141]: The maximum real part of all eigenvalues of Jm = F − V is greater than 0 if and only if

ρ FV −1 > 1, (5.28)

−1 −1 where ρ (FV ) is the maximum absolute value of all eigenvalues of matrix FV Applying this theorem to eq. (2.26), it is straightforward to show that

B (r ) r ρ FV −1 = m m . (5.29) (B (r) − 1) r + rm

Therefore, the mutant virus will invade if and only if

B (r ) r m m > 1. (5.30) (B (r) − 1)r + rm

Note that, when B (r) = B (rm) = b, Eq. (30) reduces to

b r m . (5.31) (b − 1)r + rm

46 In this extreme case the mutant virus will invade when rm > r, i.e, a virus should evolve to its maximum adsorption rate. The above result can be easily extended to study burst size in the form given by Eq. (2.14). Let the burst size be defined by (2.14) and assume for simplicity that the debris X¯ is constant. Then the optimal mutant adsorption rate required for invasion is given by s d r(b d − d − r X¯) r∗ = f f f . (5.32) m ¯ ¯ 2 df X + rX ∗ Note that rm increases as the maximum burst size b becomes larger. This can be explained intuitively as follows. A larger burst size implies that there are more newborn viruses escaping from debris. Once virus escapes, it is better to have a larger adsorption rate (rm) that allows infecting more bacteria. Eq. (2.32) also shows an inverse relationship between the optimal adsorption ∗ ¯ rate (rm) and the amount of debris (X). Virus might evolve to mutants with moderate (intermediate) adsorption rates when the bacterial debris (X¯) in the vicinity of the newborn viruses increases. A moderate strain will have a lower chance of attaching to bacterial debris, and hence a higher chance of escaping from the debris and reaching more healthy bacteria. We studied the system (2.16-2.20) with shielding effect (adsorption-dependent) and without it (adsorption-independent) burst size. In the adsorption-dependent case, we set the adsorption rate of the established virus (r) to an intermediate value. For the invader virus we studied two cases, one where its adsorption rate is lower than the established virus adsorption rate (rm < r), and the other when the invader adsorption rate is larger (r < rm). Results are shown in Fig. 2.3. When the shielding effect is included in the system, any mutant adsorption rate (rm) larger or smaller than the established virus (r) will not invade the system. In contrast, in the absence of shielding effect, if the invader has a larger adsorption, invasion will take place as expected. The classic theory of parasite evolution states that viruses will evolve to infinite adsorption rates in order to attain its maximum reproductive ratio R0. However,

47 Figure 5.3: When the shielding effect is included in the system, any mutant adsorption rate (rm) larger or smaller than the established virus (r) will not invade the system. In contrast, in the absence of shielding effect, if the invader has a larger adsorption, invasion will take place as expected. Pa- rameters used were: b = 100, dC = 0.1, dV = 0.001, λ = 100, df = 0.01, −4 −6 −3 X = 1000, r = 10 , dI = 1, lower adsorption rate rm ∈ {10 , 10 }. experiments shows that this may be not the case, and intermediate adsorption rates are the natural selection in evolution. Moderate adsorption rate might arise in situations where healthy bacteria is unavailable or have poor quality as proposed by [142]. Despite the body of research that attempts to explain why moderate adsorption rates exists, current models are unable to explain the existence of optimal adsorption rate under competition situations. This chapter presents the first attempt to explain why virus may evolve into moderate instead of infinity adsorption rates. Our approach uses classic ODE analysis

48 to study the effect of cell debris (shielding effect) on virus fitness under both, individual and competition scenarios. We included this shielding effect by assuming that the burst size is a decreasing function of the adsorption rate. This assumption is valid as long as the interaction between the newborn virus and the cell debris produced is fast enough to produce a net decrease in the viral burst size. We found that this adsorption-dependent burst size produces a finite basic reproductive ratio, enables moderate adsorption rate virus to maximize the steady state virus load, and allows establishment of viruses with moderate adsorption rates when competing with larger adsorption rate strains. This result expose the importance of virus-bacteria local interactions on the study of traits evolution. Until now we explored a burst size which is a decreasing function of the adsorption rate. One question raises after this study: which class(es) of burst size functions warranties optimal adsorption rates in both individual and competition scenarios. Ad- ditionally, we considered a deterministic adsorption rate, however, adsorption rate may be a distribution (as shown in experiments [143, 144]) in which natural selection will choose moderate adsorption instances. How this selection arises is also an intriguing open question. Furthermore, debris is not the only mechanism that may produce the shielding effect. Other alternatives include multiplicity of infection [145], presence of resistant bacteria, or high density of infected cells in the neighborhood of the newborn virus. In this chapter we modeled the lysis process using rate equations. In reality, lysis is an exact delayed process. Moreover, optimal lysis time is required to ensure virus establishment [146]. Future work includes studying the role of both shielding effect and lysis time on viral fitness.

49 Chapter 6

THE EFFECT OF MULTIPLICITY OF INFECTION ON THE TEMPERATENESS OF A BACTERIOPHAGE: IMPLICATIONS FOR VIRAL FITNESS

Since the evolution of parasites and pathogens is important to human [119], agri- cultural, and wildlife systems, there is a mature theory that focuses on how infection mechanisms may evolve. Given that viruses are the most abundant and simple entities on the planet, they are frequently used as models for studying parasite-pathogen evolution. In particular, parameters like replication, lysis time, and adsorption among others have been suggested as processes used by viruses to drive infection [120–123]. Bacteriophages are viruses that infect bacteria. Their natural environment is challenging, characterized by fluctuating host cell populations and other sources of stress to the phage [147]. In this situation, a phage has two courses of reproductive action: lysogenization or initiation of the lytic cycle (see Fig. 1). Lysogenization is a means for the phage to lie dormant inside of a bacterial host by integrating viral nucleic acid into the genome of the host cell [148, 149]. A phage thus integrated is called a prophage. While the prophage remains latent, it does not impede the host cell in any way. The bacterium will continue thriving and propagating, copying and transmitting the prophage into its progeny. This allows the phage to reproduce without exposing itself to the detrimental effects of the outside environment. The lysogen has the ability to maintain its current state of latency or undergo induction. When induction occurs, prophage DNA is cut off from the bacterial genome and coat proteins are produced via transcription and translation of the phage DNA for the regulation of lytic growth. In the lytic cycle, the genome of the phage is inserted into the cytoplasm of the bacterium. The DNA resides separately from that of the genetic material of the host.

50 Replication of the phage begins and, once many phage components have been created, new phage are produced. Over time, the phage will begin to accumulate within the host cell. This eventually results in the lysing of the bacterium and the release of the free phages. A phage that has the ability to enter lysogeny is called a temperate phage, and its temperateness can be deﬁned as the propensity of the phage to enter a lysogeny. The idea of temperateness has been greatly deliberated and analyzed [150, 151]. However, the main questions still stands: what advantage does a phage derive by being able to switch between the lysogenic cycle and the lytic cycle?

Virus

Lysogen Viral Host Integrated genome genome viral genome

Bacteria

Lytic

Figure 6.1: Plasticity of the temperate phage. Under the lysogenic cycle, the cell can either undergo the lytic cycle with a probability of 1 − φ or integrate the prophage into the genome of the host cell with a probability of φ.

One mechanism phages might use to regulate temperateness is by means of the multiplicity of infection (MOI) – the ability of a phage to infect an already infected cell. To gain a more concrete understanding of the behavior of the temperate phage, we analyzed how MOI may affect the probability of entering the lysogenic cycle. Math- ematical modeling of a multitude of fluctuating environments [130–137] allow us to theoretically and quantitatively understand the adaptive nature of fixed and plastic latency. In this chapter, we explore the advantages of choosing the lysogenic path under

51 fluctuating environments. Additionally, we explore the additional advantages provided by MOI. The chapter is organized as follows: first we describe the modeling approach used to represent the dynamics of the different species under fluctuating environments. Then we explore the lysogeny-lytic decision made by the phages. We study the impact of MOI on the overall fitness of the phage and present some conclusions.

6.1 Model We modeled the fluctuating environments using a similar approach as in [142]. Consider a situation where the environment fluctuates between ”good” and ”bad” conditions. The ”good” conditions allow for an environment in which the phage is proliferating efficiently through a replenishing population of bacteria. The ”bad” conditions result in the entirety of the free phage population decreasing to 0. This would be consistent with an environment of extreme host scarcity. The phage has no control over its environment. However, it does have the ability to infect a host with a probability k > 0 assuming the ”good” conditions. Then, with a probability of φ, the lysogenic cycle can be induced. This allows the phage to reproduce after a period of dormancy by incorporating it’s own DNA into the genome of the host. Furthermore, the phage has a probability of (1 − φ) to enter the lytic cycle. This causes the bacteria to lyse and, in turn, create new free phage. These new phage are represented by BV where B is the phage burst size per infection. Fluctuations in the free phage and lysogen populations during the ”good” condition are represented by

(1-φ) k φk V −−−→ BV, V −−→ L, (6.1)

During the ”bad” condition, the free phage die out completely and the lysogen population decreases at a ﬁxed rate. The change of lysogen population under ”bad” conditions is given by L −−→∅α , (6.2) where α > 0 is the degradation rate of the lysogens. Once the ”bad” condition ends, all lysogens release their phage (BL), and a new ”good” condition starts. This does

52 not imply the existence of a sensor in the lysogens that recognize the good condition. This is a simpliﬁcation, without loss of generality, since the contribution of new free phage will be proportional to the lysogen population at the beginning of the good time. Additionally, the exponential growth via the lytic pathway rapidly overwhelms any residual contribution of remaining lysogens from the previous round. The dynamics of such phenomena can be represented by the hybrid system depicted in Fig. 5.2a. The ellipses represent the good and bad environment situations. Arrows represent birth, death or environmental switching events. Free phage is created in bursts of ﬁxed size B at a rate (1 − φ)k v, where v represents the current number of phage within the system. Alternatively, new infections might choose the lysogenization path, which happens at a rate φ k v, adding an extra lysogen to the system. A timer variable τ keeps track of the elapsed time in the good condition until time τg is reached. The rate in which it takes to cycle through one iteration of good conditions is given by

δ(τ − τg). After this time, the environment switches to bad conditions, resetting the free phage population and the timer τ to zero. In bad conditions, death solely occurs at a rate αl. Once (τ = τb), the environment returns to good conditions,τ is reset to zero, and initial conditions are set to a free phage population proportional to the lysogen population at the end of the bad time, with lysogen population set to zero.

6.2 Why do Bacteriophages display Temperateness? To study the above hybrid system, an equivalent model can be constructed by assuming the deterministic counterpart (see Fig. 5.2b). Let τg and τb be the time spent in the good and bad environment, respectively. Starting with a single copy, the free phage and lysogen count at the end of the good environment are given by

φ ekB(1−φ)τg − 1 v (τ ) = ekB(1−φ)τg , l (τ ) = . (6.3) g g B(1 − φ)

Note, that when the probability φ of becoming a lysogen is 0, the lysogen count is zero. For example, a completely lytic virus will produce zero lysogens. Although, if

(φ = 1), the average lysogen count is k τg. Fig. 5.3 shows the lysogen population at

53 a) b)

Figure 6.2: Hybrid systems describing the fluctuating environment that lysogenic capable phage might face. a) The environment dynamics is described by one birth-death processes and a pure death process. Under good environment situations (g = 1), phage might produce bursts of B free phage particles at a rate (1 − φ) k v. Alternatively, the virus might go lysogenic. A timer τ keeps track of the elapsed time in each condition. Once time spend under good conditions is τg, the environment switches to bad conditions by resetting τ, g, and v to zero. Under bad conditions, lysogenic cells die at a rate αl. When the time spent in the bad conditions is complete (τ = τb), the environment switches back to good conditions by resetting g = 1, v = Bl, l = 0, and τ = 0. b) Deterministic version of the hybrid system on part a). the probability range of φ ∈ [0, 1]. Note that phage fitness is optimized for a value between 0 and 100% chance of lysogeny. Next, we speculate how a phage can spread over multiple rounds of good-bad conditions. Since the free phage is wiped out completely during bad conditions, the only way for the phage to propagate is through the lysogenic cell. Therefore, we are interested in the conditions that allow for the lysogenic cells to thrive. At the end of the first round, the lysogenic count is given by the equation 6.3

φ ekB(1−φ)τg − 1 l = e−ατb . (6.4) 0 B(1 − φ)

54 Subsequent rounds (n > 1) are described by

φ ekB(1−φ)τg − 1 l = l e−ατb (6.5) n (1 − φ) n−1 !n 1 φ ekB(1−φ)τg − 1 = e−ατb . (6.6) B (1 − φ)

The lysogenic population will grow unbounded when

φ ekB(1−φ)τg − 1 > eατb , (6.7) (1 − φ) i.e., when the amount of phage (in form of lysogens) per infection during the good condition is larger than the average lost per infection during the bad condition. Now, the question is how this optimal lysogen count reﬂects on the phage population in subsequent rounds of good-bad environments.

Reproduction rate k = 0.20 Lysogen count (avg) k = 0.15 k = 0.10

Probability of becoming lysogen probability population Optimal lysogenic Optimal lysogenic Viral reproduction rate

Figure 6.3: Eﬀects of lysogenic probability on phage infection. a) Population count vs lysogenic probability for several infection rates. Note that average lysogenic count is optimized for a value between 0 and 100% chance of lysogeny. b) The optimal chance of lysogeny decreases as the phage reproduction rate increases.

55 Since the free phage count is zero during the bad conditions, the lysogen concentration at the end of this cycle is given by

φ ekB(1−φ)τg − 1 l (τ + τ ) = e−ατb . (6.8) g b B(1 − φ)

At the end of the next good condition, the free phage concentration can be written as

φ ekB(1−φ)τg − 1 v (τ + τ + τ ) = e−ατb ekB(1−φ)τg . (6.9) g b g (1 − φ)

Fig. 5.4 shows the free phage concentration proﬁle for the nominal parameter values. Note that, similar to the lysogen count, intermediate lysogen probabilities produce maximum free phage counts. Virus count (avg)

Probability of becoming lysogen

Figure 6.4: Free phage population count vs chance of lysogeny. Population count is optimized at a value between 0 and 100 % chance of lysogeny.

6.3 Probability of survival of a lysogen Is there an optimal lysogen count that characterizes the survivability of the lysogen cells after the end of a good-bad environment sequence? To answer this question we model the dynamics of free phage and lysogens as deterministic. Under bad conditions, we assumed that lysogens dynamics obey a pure death process. Let the hybrid system in Fig. 5a represent the dynamics of the free phage and lysogen species at the end of the bad conditions. Since the dynamics of the lysogen species during the bad

56 conditions results in the degradation of the lysogen count, the probability of having x lysogens can be written as   x0 x −x −ατbx −ατb 0 P (x, τb) =   e 1 − e , (6.10) x where x0 is the lysogen count at the end of the good condition. The probability that at least one lysogen survives the bad conditions is then

Ps = 1 − P (0, τb) . (6.11)

The term h = 1 − e−ατb in equation (6.10) represents the probability of extinction for one single lysogen during bad conditions. This probability is dictated by the ratio between the average lifetime of the lysogen and the length of the bad environment. The larger the ratio between these two, the larger the probability of extinction for one single copy. From equation (6.10) when the probability of extinction is close to 0, the survival of one single lysogen will suﬃce to preserve this virus across multiple rounds of good and bad conditions. For extinction probabilities close to 1, virus should compensate by getting the maximum proﬁt out of the good conditions, which is only achieved at moderated lysogenic probabilities. Note that when viral reproduction rate is large enough to produce lysogen counts >> 1, the probability that at least one lysogen survives the bad conditions is 1 for φ < 1. Fig. 6.5b shows the probability distribution

Ps.

6.4 The Eﬀect of Multiplicity of infection (MOI) Consider the expanded version of the model in Fig. 5.2b where the dynamics of the lytic cell are slow. We represent the transient dynamics by the set of ODEs

l˙ =φ k v + φ a i v (6.12)

i˙ =(1 − φ) k v − dii − φ a i v (6.13)

v˙ =B di i (6.14)

57 sdfie ntrsof terms in defined is hg sual or-neta led netdlsgn h term the lysogen, infected already an re-infect to unable is phage the with ( lysogens infections become might new cells the These as cells. probability infected same already the to phage the of qain 51)ad(.3.FrMIealdpaew aidteasrto ae( rate adsorption the varied we phage MOI-enabled For (5.13). and (5.12) equations by given is cells lytic the of rate death where 6.5: Figure i eepoe iutoswt n ihu O ffcs oti n,we the when end, this To effects. MOI without and with situations explored We ecie h aea hc h neto nege h yi aha.The pathway. lytic the undergoes infection the which at rate the describes h xicinpoaiiyo igelsgn h oe h uvvlprob- larger survival the the prob- lower Additionally, this the lysogen, that tempered. single ability. Note a is of conditions. phage probability bad extinction the the the probability when of Survival optimal end b) is the ability process. at death cells bad pure lysogenic under a dynamics of as Population mod- modeled are ODEs. are of conditions conditions system good deterministic a under as dynamics eled Population a) environment. yrdsse eciigtesicigdnmc ftego n bad and good the of dynamics switching the describing system Hybrid a i.e., ,

Lysogen survival k

= probability ac Probability of becoming lysogen ofbecoming Probability extinction probability (h) probability extinction Single lysogen where 0.999 0.998 0.997 d i Parameter . φ 58 .Nt httepaerpouto rate reproduction phage the that Note ). c safie ubro elh atracells. bacteria healthy of number fixed a is a eemnsteasrto rate adsorption the determines v i a φ iapasfrom disappears a k ) and kept constant the phage reproduction rate (k). Since k = a c, an increase in phage adsorption implies a reduction in the constant number of bacteria in the environment. Fig. 5.6 shows the effects of MOI for different adsorption rates. The blue solid curve represents the non-MOI version of each phage adsorption rate. Note that by adding MOI, the survival probability increases. Additionally, the optimal lysogeny probability decreases. As the phage becomes aggressive, the optimal lysogenic probability reduces and the optimal survival probability increases.

Adsorption rate a = 10-3 a = 10-4 w/o MOI survival Probability of lysogen

Probability of becoming lysogen

Figure 6.6: Eﬀects of the multiplicity of infection (MOI) on the survival probability. The optimal chance of lysogeny decreases for MOI enabled phages compared with phages without MOI. The blue solid curve represents the non-MOI version of both MOI enabled phage (a = 10−4,a = 10−3)

When the availability of the bacterial population is held constant, classical theory of parasite evolution states that aggressive phage strains will be selected for in their environment. In this case, the probability of becoming lysogenic should be close to zero. However, the role of lysogeny on viral ﬁtness under these conditions has yet to be fully uncovered. In this chapter, we observed a series of dynamic environments with constantly changing conditions. These conditions oscillate between those where the bacterial populations is readily available to the phage, followed by periods where there is no bacterial population to infect. During such periods, the free phage has a null chance of survival.

59 We found that under such scenarios, lysogenic cells play a key role in the preser- vation of the phage for subsequent generations. Moreover, only temperate phages (those with intermediate lysogenic probabilities) maximize phage population in good conditions, lysogenic population at the end of bad conditions, and virus survival probability. How temperate the phage should be depends on the phage reproduction rate, which depends on how aggressive the phage is (adsorption rate). The more aggressive the phage is, the lower the required probability of becoming lysogenic and therefore, the larger the optimal survival probability. Using this framework, we also studied the impact of the multiplicity of infection mechanism implemented by some phage species. We found that in some cases, MOI might double the probability of lysogen survival, and reduce the optimal probability of becoming a lysogen (φ) by a factor of 10.

60 Chapter 7

CONDITIONS FOR INVASION OF SYNAPSE-FORMING HIV VARIANTS.

Human Immunodeficiency Virus (HIV) is a human retrovirus that infects certain immune cells, primarily the CD4+ helper T cells and the macrophages. Untreated infection can lead to the eventual collapse of the immune system, resulting in severe immunodeficiency and death due to opportunistic infections [152]. Over 34 million people are infected, with as many as 2.5 million new infections each year [153]. The lifecycle of the HIV virus has been extensively studied. The virus particle consists of two positive single-stranded RNA copies of the viral genome, together with the functional HIV enzymes Vif, Vpr, Nef, Protease, Integrase, and Tat, are enclosed in a protien capsid. This capsid is in turn encased within a lipid-bilayer envelope studded with the viral glycoproteins gp41 and gp120. The envelope protein gp120 has a high affinity for the cell marker CD4, which is a characteristic marker of the helper-T cells and the macrophages [154], and to a co-receptor, either the CCR5 or CXCR4 transmembrane proteins on the target cells [155, 156]. Binding of gp120 to these two receptors results in a conformational change which exposes an active site of the viral protein gp41. The exposed gp41 mediates fusion of the viral membrane with the host cell membrane, which decapsulates the virus [157]. The viral membrane and all integrated proteins, including gp41 and gp120, becomes part of the host cell’s membrane. Once the virus has entered the host cell, viral reverse-transcriptase creates a DNA copy of the viral RNA, a step that may be interrupted by the presence of reverse transcriptase inhibitors. The viral DNA is transported to the cell nucleus, and integrated into the host-cell chromosomes by the viral integrase enzyme. This step can

61 be interrupted by the presence of integrase inhibitors. The integrated HIV genome is then expressed by the normal cellular RNA transcription machinery, although a hair- pin structure in the evolving HIV RNA transcript can result in aborted transcription unless the viral protein Tat is present. The viral RNA transcript is transported to the cytosol, where ribosomes transcribe several non-functional super-proteins, which must be cleaved by the viral protease enzyme into their functional forms. The virus products aggregate on the cell surface and form viral particles. These viral particles bud off of the surface of the host cell and are released into the surrounding fluid, able to infect other cells. However, prior to the formation of mature, budding HIV particles, there is an accumulation of gp41 and gp120 complexes on the surface of the infected cell. In the same manner in which these viral proteins mediate the binding of and fusion of viral envelopes to the target cell, they are also capable of allowing the infected cell to bind to an uninfected target cell and partially fuse membranes, resuling in the formation of a viral synapse. Formation of the synapse allows for the direct transmission of viral particles between the infected and the uninfected cell [158–166]. This has been observed experimentally, and it has been shown that as many as several hundred virus particles can be transmitted across a single synapse [167]. It has been suggested that the cell-cell transmission pathway provides an evolutionary advantage, either by allowing the virus to evade the host immune response [163], to overwhelm antiviral drug activity [168, 169], or simply by a more efficient mode of infection. Previous modeling work considering the potential benefits of cell-cell transmission has shown that the optimal number of viruses transmitted by a synapse is small [170, 171]. This previous work did not explicitly consider the competition between synapse-forming and non-synapse forming viruses, as we do here. The previous work was also based on nominal parameters, where we use parameter values identified from experimental patient data [172]. We present a novel model of HIV virus dynamics that accounts for transmission by both the cell-free and cell-cell pathways. We analyze the stability of the stationary

62 points of this model, and derive bifurcation conditions. We show that the steady-state viral loads in a synaptic transmission model increase as a function of the probability of successful infection given a cell-entry event. We further show that the steady-state virus load increases with synaptic multiplicity of infection when the fraction of total burst size is small, but begins to decrease with increasing multiplicity of infection once a relatively small fraction of the burst size is reached. We show that a synapse forming virus variant will successfully invade against an established non synapse-forming virus when the muliplicity of infection as a fraction of total burst size is less than a critical value determined primarily by the ﬁtness of the non synapse-forming virus. The chapter is organized as follows. Section II introduces the basic model of HIV dynamics as developed by Perelson et al. [173–175]. Section III introduces the model of synapse-forming virus, develops stability conditions on the stationary points of the model, and explores the eﬀect of varying muliplicity of infection and probability of successful infection on the steady-state virus loads. Section IV introduces a competition model between synapse-forming and non synapse-forming virus, and develops a bifurcation condition for the stability of the stationary point with no synapse-forming virus. Section V summarizes the results, and discusses implications for HIV treatment and future work.

7.1 HIV Model The free virus transmission mechanism is described using the extensively studied model introduced by [173]. In this model the behavior of uninfected, infected cells and

63 Parameter Value Units Biological meaning Parameter Value Units Biological meaning 2 cells 3 copies λ 7 × 10 µL×day Uninfected birth rate k 2 × 10 cell×day Copies of virus per cell 1 −6 mL dT 0.1 day Uninfected death rate βf 2 × 10 copies×day Rate of uninfected-virus interaction 1 −5 µL dI 1 day Infected cells death rate βs 10 cells×day Rate of infected-uninfected interaction 1 dV 23 day Virus death rate

Table 7.1: Parameter values for simulations on this chapter. All parameters values except βs were taken from [172]. Rate of infections by free virus pathway is assumed to be 20 times greater than rate of infections by synaptic pathway.

HIV virus is given by

˙ T = λ − dT T − βf TVf (7.1a) |{z} |{z} | {z } T-cell T-cell Free Virus Production Death Infection ˙ If = βf TVf − dI If (7.1b) | {z } |{z} Free Virus Infected Infection Cell Death ˙ Vf = k If − dV Vf . (7.1c) |{z} | {z } Free Virus Free Virus Production Death

Here T (t), If (t) and Vf (t) represent uninfected, infected cells and virus population at time t, respectively. The rate of production of uninfected cells is represented by λ.

Death rate of uninfected, infected cells and virus are dT , dI and dV . k represents the number of free virus particles produced per infected cell per time unit. The infection rate is given by βf . Table 8.1 shows these parameters and experimental values for them obtained from [172]. If we assume that virus copies die at high rate greater than infected cells death rate, this is dV >> dI , then virus population can be assumed in steady state, which k means Vf = If . Thus Equation (8.2) reduces to dV

˙ k T = λ − dT T − βf TIf (7.2a) dV ˙ k If = βf TIf − dI If . (7.2b) dV

64 In order to know if there is a successful infection in a person, steady state analysis of Equation (7.2) is performed. Above Equation has two steady state solutions:

λ T = , If = 0, (7.3) dT which means there is not infection in the host, and

dT dV βf kλ − dI dT dV T = , If = , (7.4) βf k dI βf k meaning virus infection succeed. Determining if local stability take place at point represented by Equation (7.3) is equivalent to determine if infection will take place by means of the free virus transmission mechanism. The eigenvalues at Equation (7.3)

λkβf are −dT which is always negative, and −dI + which could be positive or negative. dV dT Let λkβf R0f = (7.5) dV dI dT be the basic reproductive ratio of HIV infection by means of the free virus pathway. If

R0 > 1 then Equation (7.3) is unstable and the infection will take place.

7.2 Modeling Synaptic Virus Equation (8.2) describes transmission by free pathway. However that is not the only way of HIV transmission. Also infection replication may happen by direct interaction between cells, called synaptic transmission. When cells interact with others, sometimes form channels known as synapses. When infected and uninfected cells form synapses, virus copies coming from the former infect the later.

We modify Equation (8.2) in order to model synaptic transmission. Let Vs (t) and Is (t) be population of synaptic virus and cells infected with synaptic virus at time t, respectively. Also let s be the synaptic size, which is the amount of virus sent through a given synapses. Besides free virus infection βf TVs, we add an extra production of infected cells given by p (s) βsTIs. Here βs is the rate of interaction between infected and uninfected cells. Function p (s) is the probability that an uninfected cell will

65 get infected by receiving s virus particles through synapses, given there is interaction between infected and uninfected cells. Probability p (s) is deﬁned as

p (s) = f (s) σ (s) , (7.6) where σ (s) is the probability that uninfected and infected cell form synapses given there is interaction. f (s) is the probability that sending s viruses through given synapses leads to an infection, and can be any monotonically increasing function on s. If we assume the probability that s copies infect a cell as a binomial distribution, with each virus copy having probability r of successful infection then

f (s) = (1 − (1 − r)s) , (7.7) i.e. f (s) is the probability that at least one of s virus has successful infection given there is synapses. There are two possible scenarios for synapses formation: infected-uninfected and infected-infected synapses. The former leads to an infection with probability p (s).

Therefore there is a reduction of s σ (s) βsTIs virus copies that cannot be used in further infections. The other scenario arises because there is no discrimination mechanism that leads infected cells to form synapses with uninfected cells only, thus infected- infected interactions also should occur. Infected-infected synapses lead to a waste of

2 s σ (s) βsIs virus copies that does not produce any additional infection, because both cells are already infected. Figure 7.1 shows all three possible synaptic virus pathways: free virus transmission, infected-uninfected and infected-infected virus transmission through synapses. Using the synaptic mechanism illustrated above and including it in (8.2) leads

66 (a) (b)

(c)

Figure 7.1: Synaptic virus mechanism. A synaptic virus has the capability of infect cells by means of free pathway (a) and also through synapses formation (b). In the free pathway (a), infected cells produce RNA strings (red lines) using virus information stored in its genome (blue and red line), encapsules them (blue and red concentric circles) and send this capsids outside the cell. Uninfected cells absorbe them releasing RNA virus strings (opened blue circle) which integrates with cell’s DNA (blue line). Synaptic interactions may occur between infected and uninfected (b) or infected-infected cells (c). The virus copies in (b) sent through synapses are not used in the infection of other cells. to ˙ T = λ − dT T − βf TVs − p (s) βs TIs (7.8a) | {z } Synaptic Infection ˙ Is = βf TVs − dI Is + p (s) βs TIs (7.8b) | {z } Synaptic Infection ˙ Vs = k Is − s σ (s) βs (T + Is) Is −dV Vs. (7.8c) | {z } Reduction67 of Virus Production If we assume synaptic virus copies die at a greater rate than infected cells s k (dV >> dI ), then virus is in steady state (Vs = 1 − σ (s) βs (T + Is) Is) and k dV Equation (7.8) reduces to

˙ T = λ − dT T (7.9a) s k − 1 − σ (s) βs (T + Is) βf TIs k dV

− p (s) βs TIs ˙ s k Is = 1 − σ (s) βs (T + Is) βf TIs − dI Is (7.9b) k dV

+ p (s) βs TIs, which have two stationary points, one of them being the uninfected state

λ T = , Is = 0. (7.10) dT

Infection will occur (this point is unstable) if

s λ λ R0s = 1 − σ(s)βs R0f + p (s) βs > 1. (7.11) k dT dI dT

The other stability point is not diﬃcult to calculate, however is not included here due space limits. Figure 7.2 illustrates the eﬀect of small fraction of virus particles sent through

k synapses. Here B = a represents the burst size of the cell which is the number of s virus particles that an infected cell produces over its lifespan. B is the synaptic size s as a fraction of this burst size. For small values ( B < 0.1) steady state level of virus load increase more than 20%. Increasing the probability of infection of a single virus r increases speed of infection growth. In the other hand Figure 7.3 shows that increasing s B close to 1 (s close to burst size) leads to a decreasing behavior of virus load. Table 8.1 shows parameter values used for all simulations on this chapter.

68 Virus Copies 13 500

13 000

12 500

12 000

11 500

11 000 r 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Figure 7.2: Steady state behavior of virus load with diﬀerent fractions of synaptic s size B , with probability r of successful infection. Increasing virus load s when B < 0.1 is observed. Fast growth is observed for r close to 1. Burst size used on this simulations is B = 2 × 103. Probability of forming synapses used is σ (s) = 1

Virus Copies 13 500

13 000

12 500

12 000

11 500 s 11 000 0.02 0.04 0.06 0.08 0.10B

Figure 7.3: Steady state behavior of virus load for diﬀerent synaptic size fractions s B , with probability r = 0.1 of successful infection. Steady-state virus s s load decreases with B when B > 0.1, and drops below the non synapse- s forming equilibrium for B ∼ 1. Burst size used on this simulations is B = 2 × 103. Probability of forming synapses is σ (s) = 1

7.3 Competition Model In order to determine if synaptic virus outperform free virus mechanism, we propose a new competition model between synaptic and non-synaptic virus. Merging

69 (8.2) and (7.8) into a new competitive system leads the new model

˙ T = λ − dT T − βf T (Vs + Vf ) − p (s) βs TIs (7.12a) ˙ If = βf TVf − dI If (7.12b) ˙ Vf = k If − dV Vf (7.12c) ˙ Is = βf TVs − dI Is + p (s) βs TIs (7.12d) ˙ Vs = k Is − s σ (s) βs (T + Is + If ) Is − dV Vs. (7.12e)

Note that in Equation (7.12a) βf T (Vs + Vf ) represents the total infection rate by the free virus transmission for both non-synaptic and synaptic virus. In this model, cells infected by synaptic virus (Is) are able to establish synaptic interaction with uninfected

(T ), cells infected by free (If ) and synaptic virus (Is).

In order to study stability of the competition model we assume dV >> dI , thus Equation (7.12) reduce to

˙ T = λ − dT T (7.13a) k s − βf T 1 − σ (s) βs (T + Is + If ) Is + If dV k

− p (s) βs TIs ˙ k If = βf TIf − dI If (7.13b) dV ˙ s k Is = 1 − σ (s) βs (T + Is + If ) βf TIs − dI Is k dV

+ p (s) βs TIs.

Determining if synaptic outperforms free mechanism is equivalent to ﬁnd out if given there is infection by means of the free virus, adding copies of synaptic virus implies the extinction of the free virus. Thus, assuming that free virus is present and at steady state (Equation (7.4)), the stability analysis of Equation (7.13) reduces to determining

70 s B 1.0 Free Virus 0.9 0.8 outperforms 0.7 0.6 Synaptic Virus 0.5 outperforms 0.4 R0 f 1 2 3 4 5 6 7 8 9 10

Figure 7.4: Bifurcation diagram showing regions where synaptic virus outperforms free virus. The larger basic reproductive ratio of the free virus R0f is, the smaller the region where the synaptic virus outperforms gets. whether ˙ ∂Is βsσ (s) 2 = 2 (dV dI kf (s) − dV dI s ∂Is βf k

+ dV dI dT s − 2k sIsβf ) β σ (s) − s λs (7.14) k is positive (unstable) for Is = 0. This is satisﬁed if

s f (s) < . (7.15) B 1 − dT /dI + R0f dT

Figure 7.4 shows a bifurcation diagram that represents the region of values, for

s synaptic size fraction B and basic reproductive ratio of free virus R0f , where synaptic outperforms free virus. Note that the boundary in Equation (7.15) does not depend on the reproductive ratio of synaptic virus R0s Figure 7.5 shows competition simulations between synaptic and non-synaptic virus. In this chapter, we have introduced a novel model of HIV dynamics accounting for both the cell-free and cell-cell mechanisms of viral transmission. We explicitly consider the increase in likelihood of infection due to multiple cell entry events, the reduced production of free virus by cells forming synapses, and the loss of virus due

71 Virus Copies Virus Copies 30 000 25 000 40 000 20 000 30 000 15 000 20 000 10 000 5000 10 000 0 Days 0 Days 300 400 500 600 700 1000 1200 1400 1600 s s (a) R0f = 2, B < 0.91 (b) R0f = 5, B < 0.71

Virus Copies

50 000 40 000 30 000 20 000 10 000 0 Days 2000 2500 3000 3500 s (c) R0f = 10, B < 0.52

Figure 7.5: Dynamic behavior of competitive model built in this work. On this simulations three values of basic reproductive ratio for the free virus R0f are shown with their respective conditions defined by Equation (7.15). Population of free virus and infected cells with free virus are in steady state (Equation (7.4)) for all three cases. There is no infected cells with synaptic virus at time 0. One particle of synaptic virus is present at t = 0. The fraction of synaptic size is s/B = 0.2. In all three cases synaptic virus outperforms free virus regardless increasing value of R0f . to synapse formation between infected cells. We derived the stationary points of these models, and evaluated local stability under a variety of parametric conditions. When realistic parameter values identified from clinical data were used, our model showed that steady-state viral load of a synapse forming virus increased monotonically with the probability of successful infection r. Steady-state viral load also increased with the number of viruses transmitted per synapse s until this reached a threshold measured as a fraction of total burst size B of approximated 1-2% for the parameters used in this study. This is consistent with the results reported in [170], and reflects the fact that once the increased probability of successful infection begins

72 to saturate, additional viruses transmitted via the cell-cell pathway are ´’wasted”, in that they reduce the number of viruses transmitted by the cell-free pathway without signiﬁcantly increasing the probability of success of the cell-cell pathway. However, by using realistic paramter values, our model makes it clear that 1-2% of the burst size is approximately 20-40 viruses, which is on the same order of magnitude as the observed transmission numbers for synapses in vitro [167]. Figure 7.3 illustrates the fact that a

s much higher penalty is paid (measured in viral load at equilibrium) for values of B that are smaller than optimal rather than for values larger than optimal. It is feasible that the synaptic muliplicity of infection s has evolved to a value larger than the optimum due to decreased sensitivity of the viral ﬁtness around this level. When the synapse-forming HIV variant was considered in competition with a non synapse-forming variant, we were able to derive conditions for the invasion of a synapse-forming HIV variant against an established non synapse-forming variant. This condition was best expressed as an upper-bound on the number of viruses transmitted

s via a synapse as a fraction of total burst size B . The upper bound was most sensitive to the ﬁtness of the cell-free virus R0f , and was as high as 90% for R0f ∼ 2, which is consistent with values measured during chronic infection [172, 176], to as low as 50% for R0f ∼ 10, which is consistent with values measured during acute infection [177]. It is important to remember that these are not the optimal muliplicities of infection; as discussed previously, the optimal multiplicity of infection as a fraction of burst size is on the order of 1%. As mentioned in the introduction, synapse forming is a known transmission strategy of HIV. We have developed a model that shows an evolutionary advantage for synapse forming virus, and which predicts multiplicities of infection for the cell- cell transmission pathway that are consistent with those observed in experiment. This model will serve as the basis for future work investigating the impact of cell-cell transmission on viral persistence during suppressive therapy.

73 Chapter 8

OPTIMAL MULTI-DRUG APPROACHES FOR REDUCTION OF THE LATENT POOL IN HIV

Human Immunodeficient Virus (HIV) infection is a wide-spread chronic illness, affecting over 34 million people, with as many as 2.5 million new infections each year ([153]). Untreated infection results in the progressive depletion of the helper T-cell population, and the resulting immunodeficiency leads to death by opportunistic infection ( [152]). The advent of multi-drug approaches to treating HIV infection, known as combination antiretroviral therapy (cART), has resulted in HIV infection becoming a chronic, manageable disease. The durability of viral suppression in some patients has led to hopes that eradication of the virus and a “cure” for the disease may be possible ( [178]). When cART is interrupted, rapid viral rebound occurs in almost all patients, regardless of the duration of virus suppression ( [179]). This is most commonly attributed to the ability of HIV to infect cells without entering active replication. If the infected cell then takes on a memory phenotype, the latently infected cell can persist for decades without triggering an immune response, but may re-activate at any time, triggering a rebound of active infection ( [180–182]). The decision point where an infected cell either becomes actively infected or remains quiescent is stochastically determined by a feed-forward process in the transcription of HIV RNA from integrated viral DNA ( [183,184]). When RNA polymerase binds to the HIV promoter region, transcription usually fails to complete unless the HIV viral protein Tat binds to the emerging complex ( [185–187]). Once a single RNA transcript is completed, large numbers of Tat are produced and this binding becomes

74 certain. Conceivably, a drug that acted in a similar manner to Tat could bias this decision and prevent the formation of latently infected cells, as illustrated in Figure 1.

Cell’s DNA

Cell+Viral DNA

Lysis Free Virus

Release of Viral RNA New TAT Infection New Extracellular Lysis Latent TAT Infection

Figure 8.1: The eﬀect of additional Tat on viral latency. Following cell entry, reverse transcription, and integration, an infected cell becomes either latent or actively infected depending on whether intracellular Tat binds the early transcription product. Exogenous sources of Tat bias this toward active infection.

Current approaches to the eradication of the viral reservoir formed by these latent cells have focused on a so-called “shock-and-kill” strategy, where immune-stimulating agents trigger latently infected cells to begin active production of the virus. This allows the cells to become targets of the immune system for killing, or to be killed by the cytopathic effects of the budding virus ( [178, 188–193]). Commonly discussed agents include interferon-type drugs, which are generalized immune activators ( [194]). All of the “shock-and-kill” strategies are predicated on the assumption that successful virus replication is essentially halted by the background regimen of cART, and that the formation of new quiescently infected cells during the viral burst following activation is essentially impossible. Recent work,however, has shown that anatomical reservoirs with limited antiviral activity may serve as sanctuary sites, and permit significant amounts of efficient viral replication during apparently effective cART treatment ([195–198]). This so-called Cryptic Viremia may create a condition where reservoir flushing can actually increase the size of the latent reservoir. In Section 2 of this chapter, we introduce a new model of HIV infection dynamics that incorporates the activity of two potential drugs targeting the latent reservoir.

75 The first drug is an interferon-like drug which acts by increasing the activation rate of latently infected cells. The second drug is a Tat analog, which acts by decreasing the likelihood of infection events leading to the formation of latently infected cells. We find the bifurcation points for this model where interferon therapy along is not able to clear the latent reservoir. In Section 3, we simulate the behavoir of the model using model parameter values derived from the existing literature, and use a simple optimal control formulation to explore the usefulness of a hypothetical second drug, the Tat analog, in the presence and absence of cryptic viremia. We show that, while drugs that increase the activation rate are sufficient to clear the reservoir when cryptic viremia is absent, drugs that inhibit the establishment of latent infection must be used together with the activation rate enhancing drugs to achieve clearance when cryptic viremia is present.

8.1 HIV Model Our HIV dynamic model is based on the extensively studied model of HIV infection ﬁrst introduced by [173]. In this model the behavior of uninfected cells, infected cells and HIV virus is modeled by the equations:

˙ T = λ − dT T − β T V (8.1a) |{z} |{z} | {z } T-cell T-cell Free Virus Production Death Infection ˙ I = β T V − dI I (8.1b) | {z } |{z} Free Virus Infected Cell Infection Death ˙ V = k I − dV V . (8.1c) |{z} | {z } Free Virus Free Virus Production Death

Here T (t), I (t) and V (t) represent uninfected cells, infected cells and virus population at time t, respectively. The rate of production of uninfected cells is represented by λ.

The death rates of uninfected cells, infected cells and free virus are dT , dI and dV . The rate of virus production by infected cells is given by k, which can be reduced by the activity of protease inhibitors. The infection rate is given by β, which can be

76 reduced by the activity of reverse-transcriptase inhibitors and integrase inhibitors. For the purpose of simplicity, we will model the activity of the background cART regimen as a reduction in β. To include latent reservoir, we add latent cell dynamics, as in [199].

0.8 7000 0.7 1.6 60 1.4 6998 0.6 1.2 0.5 1 40 6996 0.4

T−cells 0.8 Virus load 0.3 Active infection 0.6 Latent infection 6994 20 0.2 0.4 R = 0.999 0 0.1 R = 0.5 0.2 0 6992 0 0 0 0.001 0.01 0.1 1 10 100 1,000 0.001 0.01 0.1 1 10 100 1,000 10,000 Time (days) Time (days) (a) (b)

Figure 8.2: Reservoir Behavior without Treatment. Cryptic viremia (R0 = 0.999) provides a mechanism for the maintenance of a steady-state latent reservoir. Without cryptic viremia (R0 = 0.5), the reservoir decays without intervention. Results are shown for uninfected T-Cells (a), Actively Infected T-Cells (b), Free virus (c), and latently infected T-Cells (d).

˙ T = λ − dT T − β T V (8.2a) |{z} |{z} | {z } T-cell T-cell Free Virus Production Death Infection ˙ I = (1 − ρ) β T V − dI I + αL (8.2b) | {z } |{z} |{z} Active Infected Reactivated Infection Cell Death Cells 1 L˙ = ρβ T V − αL (8.2c) | {z } 2 Latent |{z} Net Infection Clearance ˙ V = k I − dV V , (8.2d) |{z} | {z } Free Virus Free Virus Production Death

77 with L (t) as the latent cell population, ρ as probability that a new infected cell becomes latent, and α as the rate of reactivation of an latent cell. Table 8.1 shows all parameters used in the model and experimentally-derived values for them obtained from [172].

Parameter Value Units Biological meaning λ 7 × 102 cells Uninfected birth rate µL × day 1 dT 0.1 Uninfected death rate day 1 dI 1 Infected death rate day 1 dV 23 Virus decay rate day k 2 × 103 copies Virus copies per cell cell × day β 2 × 10−6 mL Infection Rate copies × day α 0.001 1 Reactivation rate day ρ 0.001 - Latency Probability

Table 8.1: All parameters values were taken from [172].

In this new model, we assume that the net clearance of the latent pool is half

1 of the reactivation rate of a latent cell. The actual ratio is unknown, as the term 2 αL in Equation 6.2c is the net eﬀect of reservoir cell activation, reservoir cell division, and reservoir cell death, while αL in Equation 6.2b represents activation only. The assumption that the net loss of reservoir cells to an activation process is less than the net production of actively infected cells is consistent with the experimental observation that the latently infected reservoir is remarkably stable despite relatively high levels of ongoing activation ( [200]). The steady state value for the latent pool is 2ρ ((1 + ρ) kβλ − d d d ) L = I T V (8.3) (1 + ρ) αkβ Note that the latent pool decays to zero if and only if 1 R < , (8.4) 0 (1 + ρ) where kβλ R0 = (8.5) dI dT dV

78 R0 =0.5 R0 =0.5 Virus Load Latent infection 60 1.4 h=0 h=0 1.2 50 h=0.001 h=0.001 h=0.009 1.0 h=0.009 40 0.8 30 0.6 Virus load Virus

20 Latentinfection 0.4

10 0.2

Time:Days 0 Timev :Days 4 v 0.1 1 10 100 1000 0.01 1 100 10 Time (days) Time (days) (a) (b)

R0 =0.999 R0 =0.999 Virus load Latent infection 1.4 150 h=0 h=0 1.2 h=0.001 h=0.001 h=0.009 1.0 h=0.009

100 0.8

0.6 Virus load Virus

Latentinfection 0.4

0.2

0.0 Timev :Days Timev :Days 4 5 1 10 100 1000 104 105 1 10 100 1000 10 10 Time (days) Time (days) (c) (d)

Figure 8.3: Eﬀect of Interferon Therapy Alone Interferon therapy accelerates the decay of the reservoir in the absence of cryptic viremia (a) and (b), but does not change the outcome. In the presence of cryptic viremia (c) and (d), interferon therapy reduces the the steady-state reservoir level, but does not eradicate it. The application of interferon also creates a signiﬁcant transient burst of viremia in the presence of cryptic viremia. is the infectivity ratio of the virus during cART therapy. The baseline probability of a cell becoming latently infected ρ has been experimentally estimated at about 0.001

([200]), so a stable steady-state value of the latent pool can only occur when R0 is very close to 1. We have shown, however, that sanctuary site dynamics can enforce exactly this condition in the presence of cryptic viremia ( [195]), the presence of which has been experimentally veriﬁed ( [201]). In this chapter, therefore, we will consider both the condition where cryptic viremia is not present (Inequality 4 holds), or where cryptic viremia is present (Inequality 6.4 is violated). In all cases, we will assume that suppressive therapy is present, and that R0 < 1.

79 We also want to model the activity of our two reservoir-targeting drugs. The interferon-like drugs enhance the activation rate of quiescently infected cells. While it is possible that the drugs will asymmetrically affect the various factors that play into the meaning of α in Equations 6.2b and 6.2c, the simplest assumption is that the drug affects them equally, and this effect can be modeled by replacing α in Equations 6.2b and 6.2c with (α + η(t)), where η(t) is the administration of the interferon-like drug. Modeling the effect of a Tat-analog drug on the probability of latency is slightly more complex. If we assume that the likelihood of any given Tat molecule present in the cell is an independent binary random process, and that one Tat molecule is transmitted with each successful infection event, then the effect of adding a Tat analog at a concentration that increases the average number of Tat molecules per cell by λ(t) can be written as:

∞ X 1+n ρ(t) = 1 − 1 − ρ Pλ(t) (n) (8.6) n=0

where Pλ(n) is a Poisson distribution with mean λ (t). As a practical matter, λ(t) = 1 is suﬃcient to render new latent infections vanishingly improbable. The dynamics of this model in the absence of any reservoir targeted therapy (λ(t) = η(t) = 0) is shown in Figure 6.2, using the parameter values shown in Table 1, with β chosen such that R0 is either equal to 0.5 (no cryptic viremia) or 0.999 (cryptic viremia). It is clear that when R0 = 0.999, the latent reservoir reaches a steady-state of approximately 1.4 latently infected cells per µL of whole blood, consistent with levels observed in patients. This also enables the maintenance of a steady-state viral load, which is also observed in patient on long-term cART therapy. Conversely, when

R0 = 0.5, the latent reservoir continues to decay slowly toward zero, with a half-life of approximately 5 years. When we consider the eﬀect of Interferon-like therapy alone, the behavior of the model is shown in Figure 6.3. When R0 = 0.5, the administration of interferon has no measurable eﬀect on the virus load, which decays uniformly in all conditions. The

80 addition of the drug does have a measurable eﬀect on the decay rate of the latently infected population, though not on the outcome; the latent pool decays toward zero regardless of the value of η(t).

The behavior is signiﬁcantly diﬀerent when R0 = 0.999. Under these conditions, the administration of the interferon-like drug results in a measurable transient increase in the viral load. The latent cell population decreases following that addition of η, but reaches a new steady-state level rather than decaying toward zero. Even when η = 0.009 (equivalent to a 10-times increase over the baseline activation rate), the new steady-state latent cell population is still approximately 20% of the baseline latent cell population. This implies that interferon-type therapies along may be unsuccessful in clearing the latent reservoir when cryptic viremia is present.

h=0.001 h=0.009 Latent infection Latent infection 1 1

R0 =0.999,tat=0 0.01 R0 =0.999,tat=0 R0 =0.999,tat=1 0.01 R0 =0.999,tat=1 R0 =0.5,tat=1 -4 R0 =0.5,tat=1 10 10-4 Latentinfection Latentinfection -6 10 10-6

-8 10 Timev :Days -8 v 4 4 4 10 Timev :Days 1000 2000 5000 1 ¥ 10 2 ¥ 10 5 ¥ 10 1000 1500 2000 3000 5000 7000 10 000 Time (days) Time (days) (a) (b)

Figure 8.4: Eﬀect of Interferon Therapy with additional Tat The addition of a background Tat analog drug dramatically changes the outcome when interferon is applied. In the presence of cryptic viremia, interferon therapy is now able to eradicate the viral reservoir. Application of a relatively low dose of interferon (a) results in a slower decline in the reservoir when compared to a higher dose of interferon (b).

In order to address this limitation of interferon-like therapies, we consider the addition of a Tat-like drug together with the interferon-like drug. As we mentioned previously, the eﬀect of additional Tat on the likelihood of latency formation is so dramatic that an administration of λ(t) = 1 is suﬃcient to reduce the likelihood of latency by several orders of magnitude. Simulations of the model behavior with λ(t) =

81 1 are shown in Figure 4. The addition of Tat eliminates the steady-state behavior of the latent reservoir, which now converges exponentially to zero, at a rate that depends weakly on the level of applied interferon-like drug.

8.2 Optimal Control and Simulations In order to further explore the synergism between interferon-like and tat-like drugs in reservoir clearance, we explored a simple constrained optimal control formulation of reservoir clearance in the presence of cryptic viremia (R0 = 0.999). While it is true that only the administration of Tat allows for the exponential decay of the latent reservoir in this case, stochastic extinction may eliminate the reservoir if it is reduced to a sufficiently low concentration for an extended period of time. To simulate the goal of stochastic extinction, we formulated an optimization problem where latent cell trajectories were constrained to drop below 1% of their initial concentration by the end of the first year of therapy, and to remain below the 1% level between year 1 and year 2 of therapy. We consider fixed-dose schedules of Tat and interferon, and minimize the application of interferon. Figure 5(a) shows the results of this optimization for three conditions. When Tat is applied alone (η(t) = 0), the problem was infeasible no matter how large λ(t) was allowed to grow. While the administration of Tat causes the latent reservoir to decay in this case, the half-life is measured in decades. When interferon is applied alone (λ(t) = 0), the constraint is achievable with a minimum administration of η(t) = 0.238, or an effect 240 times the baseline. If Tat is administered λ(t) = 1, the constraint is achievable with an interferon dose of only η(t) = 0.025. Furthermore, the achieved trajectory continues to decay exponentially toward zero throughout year 2. The administration of interferon causes a migration of latently infected cells into the active infection compartment, and the administration of Tat prevents the migration of actively infected cells back into the latent pool. This asymmetry of action enables dynamic treatments, with short-term applications of interferon followed by long-term administration of Tat. This may be desirable due to the poorly tolerated side-effects of

82 interferon therapy. To explore this possibility, we modified the optimal control problem to minimize a weighted average of the pulse height and pulse width of a single applied interferon pulse during a constant application of Tat. The trajectories for two different weighting values are shown in Figure 5(b). Both a short, high intensity application and a longer, lower intensity application of interferon are able to meet the desired treatment constraints. This implies that dynamic therapeutic schedules may be of value, and should be explored further in future work. Recent findings suggest that cryptic viremia in sanctuary sites may be common in treated HIV patients. These findings have significant implications for reservoir- flushing approaches to “cure” HIV infection. In this chapter, we have demonstrated that reservoir-flushing approaches using interferon-like treatments may be incapable of clearing the reservoir in the presence of cryptic viremia. We have shown that, in the presence of cryptic viremia, additional drugs that reduce the probability of latent infection may be necessary to successfully clear the pool of latently infected cells. We further demonstrated through simulation that a significant synergism between these two drugs exists when applied in the presence of cryptic viremia. The results in this chapter are preliminary, and many known effects have been neglected for the sake of simplicity. It would be premature to engage in a more complete exploration of the optimal control problem, as no Tat analog drug with the properties described in this chapter yet exists. This is primarily because the phenomenon of cryptic viremia has only recently been recognized, and in the absence of cryptic viremia, such a drug would have no measurable effect. Indeed, the fastest decay of the latent reservoir was always observed in the case where cryptic viremia was absent, indicat- ing that cryptic viremia, rather than latency formation, may be the better target for therapy. We have modeled the HIV infection dynamics as a single, well-mixed compartment with a uniform R0 throughout. In fact, cryptic viremia is inherently a spatially heterogeneous phenomenon, with local regions where R0 > 1, and larger regions where

R0 < 1, resulting in average behavior with R0 appearing to be very slightly less than

0 10

−2 10

−4 10 Tat and Interferon Tat only Interferon Only

Latent Infected Cells (per µL) −6 10 Target Concentration (1%)

0 100 200 300 400 500 600 700 800 900 1000 200 150 100 50 ( per mL) Viral Load 0 100 200 300 400 500 600 700 800 900 1000 300 200 100 (times baseline) Interferon 0 0 100 200 300 400 500 600 700 800 900 1000 Time (Days) (a)

0 10 Low Interferon Cost High Interferon Cost Target Concentration (1%) −1 10

−2 10

Latent Infected Cells (per µL) −3 10

0 100 200 300 400 500 600 700 800 900 1000

200 150 100 50 ( per mL) Viral Load 0 100 200 300 400 500 600 700 800 900 1000 100 80 60 40 (times 20 baseline) Interferon 0 100 200 300 400 500 600 700 800 900 1000 Time (Days) (b)

Figure 8.5: Optimal Reduction of Latent Reservoir When constant-dosage schedules are considered (a), the constrained optimization problem is infeasible when only Tat analog is administered. Application of Tat together with interferon allows the constraint to be achieved with less than 1/10 the dose of interferon when compared to interferon alone. When pulsed administration of interferon is considered (b), Tat administration enables short-dose schedules of the interferon to achieve the constraint.

84 one ( [196]). In future work, we will explore a spatially compartmentalized version of the model to determine how such spatial heterogeneity would change the eﬀect of cryptic viremia on the reservoir ﬂushing approaches.

85 Chapter 9

DISCUSSION

Despite considerable leapfrogging due to single-cell technologies shown in chapter 1 and the insights provided by our frameworks in chapters 2 and 3, we are far from a full grasp over size regulation and expression homeostasis. For example, coordination of critical cell-cycle events with division cycle in various organisms, and their mechanistic underpinnings still remain unresolved. Identifying how cell-size control is applied on genome replication in prokaryotes is also closely linked to gene product concentration homeostasis since replication time of a gene of interest affects concentration profiles of its products over the cell-cycle. Given the physiological relevance of concentration homeostasis, could it be that size control over replication timing is exerted to ensure a constant concentration? Speculating even further, is it possible that the size control strategies are a mere consequence of maintaining gene product concentrations? A theory driven experimental investigation in these issues would certainly be useful. Further, in context of size homeostasis, how mammalian cells achieve size homeostasis is beginning to unfold only now. Mechanistic insights into how growth rate is regulated in these cells are critically required. Since size control is intimately tied with activation of competence in bacteria upon usage of antibiotics [56], a better understanding of size homeostasis would go a long way in therapeutics and drive new targets for drugs development for tuberculosis [12,57]. Regarding concentration homeostasis, the relationship between mRNA concentration homeostasis and protein concentration homeostasis is not established as such; specifically it is not known whether both are simultaneously exhibited. It can be argued that protein concentration homeostasis can be achieved regardless of mRNA concentration homeostasis by changing translation and/or protein degradation rates with cell size. Simultaneous measurements of both mRNA and

86 protein numbers of a given gene of interest would shed light into this issue. Another question of interest is to ask why some organisms (particularly prokaryotes) do not seem to have dosage compensation while others do. Additional insights are expected to be developed as these issues are investigated in organisms other than the current model ones [58]. In chapter 5, we argue that bacterial debris is detrimental for strains with aggressive infectivity. Our hypothesis proposes that this detriment is due to debris produced by lysed (dead) cell remains. Is cell debris actually present in nature? In practice, bacterial debris is hard to detect due to its size, which ranges between phage (nm) and bacterial size (µ m). Recent experiments using biofilm-forming bacteria suggest that uninfected cells in the center of the biofilm are surrounded by (possible) dead cells (debris) on the biofilm surface after phage exposition. This dead cells (debris) may cause a sink for phages at the surface that further protects uninfected cells at the biofilm center. However, determining if dead cell debris is actually a sink for phages remains elusive An alternative to phage waste due to dead cell debris is the possibility of re- sistance emergence. This hypothesis states that biofilm communities may be able to produce bacteria strains that are invisible to the phage specie that is attacking the community. This alternative explanation is rule out since current experiments show that, although mutation exists, a significant number of bacteria wild types still survive to the phage’s attack. Phage waste may be produce due to the multiplicity of infection being larger in the external layers of the biofilm community where the cell population is mostly infected cells. Whether this is a significant sink for phages remains unknown. Although we modeled bacterial debris produced by individual lysed bacteria, our model can be easily adapted to biofilm communities. For instance, we may assume that instead of modeling a community of bacterial cells, we have a set of bacterial communities being attack by a phage strain. Therefore, the effective burst size of a given bacterial community is given by how aggressive is the virus infection. As observed in the experiments by [], larger adsorption rates are detrimental for viral production

87 since most of the new progeny is attached to cell debris (or already infected cells). The rate of scape of the virus from a given community allows the virus to visit other non- infected communities, increasing the chances of invade the environment. Our model suggest an optimal foraging strategy in which the virus select the proper adsorption rate which produce maximum host-exploitation. Additionally, we predict similar optimal foraging strategies in bacterial community environments less elaborated like clumped bacteria. The results obtained from the free and synaptic pathway model developed in chapter 7 suggest several avenues of future research. We have treated multiplicity of infection as multiple independent trials in this work; future work will consider more general formulations for the function f(s), including synergistic and antagonistic behavior between the multiple virions invading the cell. We have considered all cells to have the same viral production rate and the same death rate regardless of multiplicity of infection. The authors of [170] suggest that virus production rate may scale with multiplicity of infection; future work will include this possibility as well. Viral fitness is known to decrease dramatically from acute to chronic phase infection, and our model shows that this significantly changes the optimal multiplicity of infection. Fur- thermore, the decrease in viral fitness is almost certainly due to an increased immune response [177], and several authors have suggested that cell-cell transmission evades certain immune system mechanisms [163]. The implications of this for the evolution of synapse formation rates over the course of infection will be investigated in future versions of the model. This last consideration may be related to the emergence of syncytium-inducing variants of the virus (where membrane fusion becomes so extreme as to result in the formation of huge non-functional multinucleate cells), which has been associated with disease progression in the final stages of immune collapse into AIDS [202]. Our synaptic pathway model for HIV infection captures the impact of this transmission mode to the infected cell population which are actively productive. However, this virus is capable of create quiescent or latent cells which are not affected by any

88 antiviral treatment. These latent cells are persist for decades before stochastically switching to a fully activated virus producing infected cell. This situation constitute a major barrier in eradicating the virus from patients. How synaptic transmission mode may aﬀect the latent population is still a unknown. Research in this direction had been proposed in a recent paper submitted. We found that synaptic transmission may provide an evolutionary advantage by means of modulation of how often the cells become latent.

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