Noise Propagation in Gene Expression in the Presence of Decoys

bioRxiv preprint doi: https://doi.org/10.1101/2020.04.01.020032; this version posted April 2, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-ND 4.0 International license.

Noise propagation in gene expression in the presence of decoys

Supravat Dey1, and Abhyudai Singh 2

Abstract— Genetically-identical cells can show remarkable by identical promoters [2]. While correlated fluctuations intercellular variability in the level of a given protein which represent the extrinsic noise, the uncorrelated fluctuations is is commonly known as the gene expression noise. Besides a measure of intrinsic noise [2]. intrinsic fluctuations that arise from the inherent stochasticity of the biochemical processes, a significant source of expression Here, we investigate the role of genomic decoy binding noise is extrinsic. Such extrinsic noise in gene expression on the gene expression noise in the presence of a dynamic arises from cell-to-cell differences in expression machinery, external disturbance. The genomic decoys are the numerous transcription factors, cell size, and cell cycle stage. Here, we consider the synthesis of a transcription factor (TF) whose high-affinity binding sites where a transcription factors (TFs) production is impacted by a dynamic extrinsic disturbance, and bind without any direct functional consequences [21]. In systematically investigate the regulation of expression noise by functional binding, transcription factors bind to the specific decoy sites that can sequester the protein. Our analysis shows site of a gene and directly regulate the expression by acti- that increasing decoy numbers reduce noise in the level of the vating or inhibiting the transcription process. The binding free (unbound) TF with noise levels approaching the Poisson limit for large number of decoys. Interestingly, the suppression of transcription factors to the decoy sites, on the other of expression noise compared to no-decoy levels is maximized hand, alters the abundance of the transcription factor and at intermediate disturbance timescales. Finally, we quantify indirectly regulates the expression of target genes. The role the noise propagation from the TF to a downstream target of decoys in the stochastic gene expression were studied protein and find counterintuitive behaviors. More specifically, experimentally using synthetic circuits in Saccharomyces for nonlinear dose responses of target-protein activation, the noise in the target protein can increase with the inclusion cerevisiae and in various non-oscillatory circuits [22]–[27] of decoys, and this phenomenon is explained by smaller but and oscillatory circuits [28], [29] theoretically. However, the more prolonged fluctuations in the TF level. In summary, our effect of decoys in the presence of external disturbance has results illustrates the nontrivial effects of high-affinity decoys not been addressed. in shaping the stochastic dynamics of gene expression to alter cell fate and phenotype at the single-cell level. To investigate the role of decoys on the gene expression noise in the presence of a dynamic external perturbation, we I.INTRODUCTION formulate a simple stochastic model for the gene expression, schematically shown in Fig. 1. The transcription factor, Genetically identical cells can show remarkable variability whose production rate is subject to an external perturbation, in the level of a gene product (mRNA/protein) which is reversibly binds to unoccupied decoy sites. Assuming small commonly known as the gene expression noise [1]–[9]. fluctuations in molecular copy numbers, we linearize asso- Depending on the situation, the gene-expression noise can ciated binding terms around the means and solve the first be detrimental or beneficial [10]–[14]. For example, large and second-order moment dynamics at the steady-state. We noise can cause defects in developing embryos [15]. Gene quantify the noise in free TFs using the squared coefficient expression noise can drives different cell-fates of genetically of variation and obtain analytical formulas for noise levels identically cells [12]. Importantly, it can enhance phenotypic in the fast binding/unbinding limit. Our results show that the diversity, crucial for the survival of an organism in a popu- addition of decoy sites reduces the extrinsic noise component lation under fluctuating environments [13], [14]. of the free TF level. Finally, using stochastic simulations, we There are two components of the cell-to-cell variability in investigate the noise for a downstream protein. Interestingly, gene expression — intrinsic and extrinsic [2], [3], [6], [7], we observe that decoys can enhance or buffer noise in the [15]. The intrinsic noise arises from the inherent stochasticity target protein depending on the nature of the dose response of biochemical reactions associated with mRNA/protein proof the target protein activation. ductions and degradations involving few molecules. There are several cell-specific factors such as, cell cycle stage, Symbols and Notation: At a given time t, the number cell size, abundance of expression machinery and global of molecules for the species associated with the external factors that are the extrinsic sources of expression noise [16]– disturbance is denoted by x(t), the molecular number of free [20]. The extrinsic and intrinsic noises can be quantified by and bound TFs by yf (t) and yb(t), and the number of target experiments with two-color reporter genes that are regulated protein by z(t). The molecular count of a species takes a random non-negative integer value i.e., x(t), yf (t), yb(t), z(t) ∈ 1 Department of Electrical and Computer Engineering, University of {0, 1, 2, 3, ...}. For a stochastic process, we use angular Delaware, Newark, DE 19716, USA [email protected] brackets h·i and h·i to denote the transient and steady state 2 Department of Electrical and Computer Engineering, Department of Biomedical Engineering and Department of Mathematical Sciences, Uni- expected values respectively. The total noise in the molecular versity of Delaware, Newark, DE 19716, USA [email protected] counts of a free TF is quantified by the square coefficient bioRxiv preprint doi: https://doi.org/10.1101/2020.04.01.020032; this version posted April 2, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-ND 4.0 International license. variation at the steady-state {1, 2,...}. Similarly, when a degradation event occurs the 2 levels are reset as hy i − hy i CV 2 = f f . (1) y 2 x → x − 1. hyf i (3)

Similarly, we quantify noise in external disturbance and The probabilities of these rests occurring in the next inﬁnites- 2 2 target protein count using CVx and CVz , respectively. imal time interval (t, t + dt) are II.MODELING GENEEXPRESSIONINTHEPRESENCEOF P rob(x → x + i) = k α(i)dt (4a) EXTRINSICDISTURBANCESANDDECOYSITES x P rob(x → x − 1) = γ xdt, (4b) We formulate a simple stochastic model schematically x shown in Fig. 1, where the synthesis of TF is subject respectively, where γ is the decay rate. For this system the to an extrinsic disturbance. This disturbance biologically x mean and noise in molecular copy number x(t) at steady corresponds to ﬂuctuations in the abundance of enzymes, state is given by expression machinery, or other global factors connected to cell size/cell-cycle stage. As discussed in further detail below, 2 kxhBxi hBxi + hB i we phenomenologically model this disturbance as a bursty hxi = and CV 2 = x , (5) γ x birth-death process. x 2hBxihxi respectively, where hB i is the average burst size [44], [45]. Bound TFs Free TFs x If hBxi follows a geometric distribution with mean hBxi, then the second equation in (5) reduces to

2 hBxi Decoy sites CVx = (6) ∅ hxi External disturbance and the magnitude of disturbance fluctuations increases with increasing hBxi. The speed of the fluctuations is given by the auto-correlation function Promoter Gene hx(τ)x(0)i − hx2i −γxτ Fig. 1. Model schematic of TF expression impacted by Rx(τ) = 2 = e (7) 2 upstream extrinsic noise. The synthesis of a transcription hx i − hxi factor (TF) from a gene is modeled as a simple birth- [43], [44]. Thus, the degradation rate γ is a measure of death process. An upstream disturbance affects the stochastic x the speed or timescale of disturbance fluctuations. In other dynamics of TF copy numbers by making the synthesis rate words, the fluctuations become fast as γ → ∞, and become itself a random process. There are N decoy binding sites in x slow as γ → 0. Next, we describe how this disturbance the genome that can sequester the TF. A free TF molecule x impacts the synthesis of a TF. reversibly binds to an unoccupied decoy site with relatively fast binding/unbinding kinetics compared to the timescales of disturbance and TF turnover. B. Effect of extrinsic noise in the absence of decoy First, we discuss the effect of extrinsic noise on the A. Modeling extrinsic disturbance as a bursty process stochastic synthesis of the TF. The dynamics of TF is as- sumed to follow a simple birth-death process with production Increasing evidence shows that expression of gene prod- rate that linearly depends on the disturbance, and a constant ucts in single cells occurs in intermittent bursts [30]–[37], decay rate γy. Analogous to (4), but assuming no bursting, and bursty birth-death processes have been commonly used the probability of resets in the levels of the TF occurring in to capture fluctuations in the levels of these gene products the next infinitesimal time interval (t, t + dt) are [38]–[43]. Borrowing this framework, we model the extrinsic disturbance as a chemical species that is produces in bursts x P rob(yf → yf + 1) = ky dt (8a) as per a Poisson process with rate kx, and the species is hxi subject to degradation at a constant rate. Let x(t) denote the P rob(yf → yf − 1) = γyyf dt, (8b) level of the disturbance at time t. Then, whenever a burst event occurs the level is reset by where yf (t) denotes the level of the TF at time t. To analyze the coupled random processes (8) and (4) we use the frame- x → x + Bx (2) work of moment dynamics for discrete-state continuous-time where the burst size Bx is an independent and identically Markov processes as described in [46]. More specifically, the m1 m2 distributed random variable with P rob(Bx = i) = α(i), i ∈ time evolution of any arbitrary function φ(x, yf ) = x yf bioRxiv preprint ihbnigadubnigrates unbinding sites decoy and unoccupied to binding binds with reversibly TF free decoy The of sites. presence the in noise extrinsic of Effect C. ffe n on F hn h rbblt frst in resets of probability the Then, TF. bound and from free protected of are rate decoys constant to [27]. at bound [25], degrades [23], TFs degradation molecule that assume TF we free a While disturbance slow very 1/ a For very a the intrinsic. perturbation, For is noise. external part extrinsic the fast second to the due contribution and additional process, of birth-death Poissonian part first The (as TF the of level squared) the variation of in coefficient its noise by following quantified time the of the moments reveals second-order obtain and to first and generator the all extended of above evolution this use We for when level noise the by normalized of noise value the particular when used: a observed for is minimum behavior becomes noise large optimal against For An plotted disturbance. is external disturbance. counts slower extrinsic TF a to for due higher noise is expression against gene plotted buffers is binding Decoy 2. Fig. et eepn h oe oinclude to model the expand we Next, e admprocesses random Let hy hy was notcertifiedbypeerreview)istheauthor/funder,whohasgrantedbioRxivalicensetodisplaypreprintinperpetuity.Itmade A m y f ,y dhφ(x, f Noise in free TF count f 1 i hB i 0.2 0.4 0.6 0.8 m , ovn hs oetdnmc tsteady-state at dynamics moment these Solving (t). + = 0 1 x dt CV i 2 k γ 10 Number ofdecoys (N) 50 = y y { ∈ fetra disturbance external of Increasing speed 1 f x , 2 doi: )i +hk . and 0, +hk 10 , +hγ = 1, https://doi.org/10.1101/2020.04.01.020032 hxi N 2 y CV hγ x ...} 2, hxi y CV x o ifrn auso pe fteetra itrac (γ disturbance external the of speed of values different for X i=0 x y 50, = 10 ∞ y 2 x f 3 ,y [φ(x, y ,y [φ(x, [φ(x 2 [φ(x steitiscniedet the to due noise intrinsic the is y sgvnb , by given is = f 10 γ (t) hy γ x Intrinsic 4 − x + f z f hy f h os spurely is noise the ∞, → i for }| y 1, and ,y i, 1) + 1 10 f − 50 = i { 5 k f f 1) N b ) ) y − + available undera b , − − − and 10, = (t) γ CV z ,y φ(x, γ B ,y φ(x, y φ(x, ,y φ(x, y x N Extrinsic and 1, = eoetelevel the denote k x

2 Noise in free TF count → u γ and 100, γ eo binding decoy }| respectively. , x 0.2 0.4 0.6 0.8 f x f f f hsmnmmvleo h os erae ihincreasing with decreases noise the of value minimum This . )]i. Speed ofexternal disturbance (γ 0 1 γ )]i )]i )]i 10 0, + y -2 γ CV y { k CC-BY-ND 4.0Internationallicense 10 d N . ; y h os prahst h oso ii for limit Poisson the to approaches noise The 1000. 2 x(t) this versionpostedApril2,2020. (10) 1. = -1 h os prahstePisnlmt B h os ntefree the in noise The (B) limit. Poisson the approaches noise the , γ (9) = y , 10 sites Increasing decoy 0 xeddgnrtrfrayabtayfunction arbitrary any for generator extended fuocpe eosrsligi the represent is in resulting resets decoys two unoccupied last of that the Note and decoys. molecule, to TF binding/unbinding free a of tion hy setting by obtained is moments of evolution φ(x time and prxmto 4][1 htudrteasmto fsmall of to assumption in exploited the fluctuations noise under linear are that well-known [48]–[51] the schemes moments, use approximation we order closure Here of results moments. higher evolution moment compute rate on time typically – binding depends and dynamics moments nonlinear unclosed order the of lower that problem out the in turns It [46]. in y eetefis w eesrpeettesnhssaddegrada- and synthesis the represent resets two first the Here f 10 f k (t) ,t (t, +hk i (y rob P (y rob P (y rob P (y rob P b dhφ( f 1 (N y , and and b + +hk f x,y 10 +hk − y dt y , k +hk f dt) hy 2 b f +hγ y (N y y b f f f f ,y u b b b = ) x b )y y ierzstenniert as nonlinearity the linearizes i, b (t) → → → → x y y 10 r sfollows as are P )i b ) − f y f y hxi ,y [φ(x, 3 y x obn 1)wt 4 ilstefollowing the yields (4) with (11) Combing . f y y y y C = ≈ i=0 ∞ x orsodn odfeeteet occurring events different to corresponding y ,y [φ(x, f f f f x m and b ,y [φ(x, hγ k [ ) 1; + − − = 1) +

1 Noise in free TF count 0 = [φ(x b . y ,y φ(x, x (y The copyrightholderforthispreprint(which 1; )= 1) f m y f x Noise in TF count without decoys b b 2 0.8 0.9 .1, A h os ntefe Fcounts TF free the in noise The (A) y y y 1, + f [φ(x hy y rudterrsetv envalues mean respective their around + f b b 1 − 10 b m Speed ofexternal disturbance (γ f → → y 1, + γ k f and 1, ,y i, i 3 -2 y y y 1, − − + y hxi for y y b f f x b b N y 1, − 10 y 1, y b y , dt − = 1) + ) b f dt -1 m ) 1) hy b − h os buffering noise The 0. = .Tenieat noise The 10). b )= 1) f ) − 1 f y , 1) + − − ,y φ(x, m , 10 hy − i ,y φ(x, nonlinear b ,y φ(x, ,y φ(x, N decoy sites Increasing 0 ) 2 k k − m , − − u b (N f 10 dt. ,y φ(x, b f y ,y φ(x, y , 3 ihy b 1 y , f f γ φ(x { ∈ y , b y , x − N stenumber the is )]i b f idn rate binding )]i >γ >> b 10 b Parameter . y i). )]i, f )]i f 0, f b 2 y , )y y , y , 1, N b f f b )]i y 10 dt ...} 2, )]i x y , (11d) (11b) (11c) (11a) (C) . ) (12) (13) 0 = 3 b ): bioRxiv preprint doi: https://doi.org/10.1101/2020.04.01.020032; this version posted April 2, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-ND 4.0 International license.

Bound TFs Free TFs Target Proteins A I. Linear ∅ Decoy sites Target activation External disturbance II. Step

Target activation Target Promoter Gene Gene TF level B C D 1 1.2 1.8 II. Step 0.8 1.6 II. Step 1 0.6 1.4

simulation I. Linear 0.4 analytical 0.8 1.2

0.2 1 I. Linear Normalized noise in free TF noise in Normalized

0 1 2 3 4 5 protein target noise in Normalized 0 1 2 3 0 1 2 3 10 10 10 10 10 10 10 10 10 10 noise propagation Normalized 10 10 10 10 Number of decoys (N) Fig. 3. Target protein noise from stochastic simulations shows a distinct behavior depending on linear and step-like regulation. (A) A schematic diagram of the model for studying noise in the downstream gene expression. Free TFs activate the expression of the target gene: production rate is kzg(yf ). We consider two cases: (I) Linear target activation: g(yf ) = yf , and (II) Step-function target activation: g(yf ) = Θ(yf − hyf i), where Θ is the Heaviside step-function, i.e., Θ(x) = 0 when x < 0 and Θ(x) = 1 when x > 0. (B) The results of the noise in free TF counts obtained from running a large number of Monte Carlo simulations using the Stochastic Simulation Algorithm [47] and analytical formula (15) are in agreement. 2 (C) The noise in the target protein (normalized by CVz (N = 0)) plotted against N. While for the linear regulation, the noise in the target protein decreases, for the step-like regulation noise in the target protein increases as a function of N. (D) Noise propagation deﬁned as the ratio of free TF noise level to the target protein noise level increases as function of N. However, the noise propagation is higher for the step-like dose response. For this plot parameters taken as hBxi = 50, kx = 1, ky = 50, γx = γy = γz = 1, kd = 1, and kz = 1.

Using these linearized rates in place of the nonlinearity (kb → ∞, ku → ∞ and kd = ku/kb being finite), kbybyf in (12) , one can again derive the time evolution of the moments and solve them to get approximate analytical Intrinsic Extrinsic formulas for the TF noise levels. Given the space constraints, z}|{ z }| { 2 1 2 hyf iγy we omit the proof and only present the main results. CVy = + CVx , (15) hy i Nf(1 − f)γ + hy i(γ + γ ) Solving the first order moment dynamics at the steady- f x f x y state, we obtain the expressions for the mean free TF and bound TF counts, where f = hybi/N is the fraction of bound TFs. Note that decoy binding directly affect the extrinsic noise part. The noise decreases monotonically to the Poisson limit 1/hyf i ky Nhyf i as N → ∞. Fig. 2(A) shows how the noise decays with N hyf i = and hybi = , (14) for various values of the speed of the extrinsic (disturbance) γy hy i + k f d 2 2 fluctuations γx. For a given N, CVy = 1/hyf i + CVx for 2 γx << γy and CVy = 1/hyf i for γx << γy (see Fig. 2(B)). where, kd = ku/kb is the dissociation constant. Please note If we plot the noise in free TF normalized with that of N = 0 that hyf i is independent of the decoy sites as has been shown as a function of γx, the noise show a minima for a specific before in the absence of disturbance [25]. Solving the second value of γx (see Fig. 2(C)). In essence, the noise suppression order moment dynamics, we obtain the following formula for ability of decoys is highest at intermediate timescales of the noise in the free TF in the limit of fast binding/unbinding disturbance fluctuations. bioRxiv preprint doi: https://doi.org/10.1101/2020.04.01.020032; this version posted April 2, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-ND 4.0 International license.

III.NOISEINTHE TARGETPROTEIN 3000 revealing that the decay of Ry becomes slower for Having quantified the effect of decoys on TF expression larger decoy abundances, explaining the enhancement in the noise, we next study noise propagation from the TF to a noise propagation seen in both the linear and step-like dose downstream target gene. Towards that end, free TF molecules responses (Fig .3). Why is the noise propagation higher activate the production of a target protein with synthesis for the step-like dose response? Note that for a step-like dose response, noise propagation to the target protein only rate kzg(yf ) (Fig. 3(A)). We study two specific cases: (I) depends on the speed of fluctuations in yf (t), i.e., how Linear activation, g(yf ) = kzyf and (II) Step-like activation fast TF levels go below the threshold and bounce back. (activation only occurs when y > hy i), g(y ) = Θ(y − f f f f In contrast, for a linear dose response, noise propagation hy i), where Θ is the Heaviside step-function, i.e., Θ(x) = 0 f depends both on the speed of fluctuations and the magnitude when x < 0 and Θ(x) = 1 when x > 0. As in the case of the of fluctuations in the free TF level. As the fee TF noise levels TF, assuming a non-bursty production of the target protein decrease with increasing N (Figs. 2 and 3), they buffer the results in the following probability of resets noise propagation in the linear case, but not in the step-like

P rob(z → z + 1) = kzg(yf )dt (16a) nonlinear case.

P rob(z → z − 1) = γzzdt. (16b) 1 for the birth and death of the target protein, where z(t) denotes the level of the target protein, and γz is the target protein’s degradation rate. An important point to mention is 0.8 that only the TF binds to decoys. A key focus here is to quantify the extent of noise propagation from the TF to the target protein as a function of decoy numbers for different 0.6 dose responses g(yf ). We study this system of four nonlinearly coupled random processes x(t), yf (t), yb(t) and z(t) by running a large 0.4 number of Monte Carlo simulations using the Stochastic Slow decay Simulation Algorithm [47]. Quantification of steady-state level noise levels for different species are shown Fig. 3. 0.2 Increasing Auto-correlation in free TF in Auto-correlation First, we show that the analytical result of noise in the free decoy sites TF (15) as obtained using the linear noise approximation matches nicely with the simulations (Fig. 3(B)). The noise 0 1 2 3 4 2 in the target protein CVz shows a counterintuitive behavior. time delay Given that decoys buffers noise in free TF, one might expect the same behavior in the target protein noise. However, we Fig. 4. Decoy slow the speed of the fluctuations in the see two opposite role of decoys depending on the profiles levels of the free TF. Auto-correlation function Ry for free of the activation. For a linear dose response, decoys buffer TF as a function of time for N = 0, 1000, and 3000. The noise in the target protein, but enhance noise for the step-like decay of Ry becomes slower for larger N. Parameter used: dose response (Fig. 3). hBxi = 50, kx = 1, ky = 50, γx = γy = 1, and kd = 1. How does the net noise propagation to the target protein behave? The net propagation of noise from TF to target 2 2 protein can be measured by CVz /CVy . It is interesting ONCLUSION 2 IV. C to see that although CVz shows a distinct behavior, the propagation of noise shows enhancements for both the cases A significant portion of gene expression noise is extrinsic (Fig. 3(D)). However, it should be noted that noise propaga- and can propagate downstream to target proteins with im- tion is significantly higher for the step-like dose response. portant consequences for cellular functioning. We have in- To better understand these results we decided to focus vestigated the role of decoy binding sites on the propagation on the timescale of fluctuations of the free TF level in of extrinsic noise to downstream proteins. For this, we have the presence of decoys. We computed the auto-correlation formulated a stochastic model where a dynamical extrinsic function of the free TF level using stochastic trajectories at disturbance modulates the synthesis rate of a TF, and free steady-state as per TF molecules can bind to genomic decoy binding sites that are present in a fixed number. 2 hyf (τ)yf (0)i − hyf i We have obtained analytical formula of noise in free TF Ry(τ) = 2 , (17) 2 counts in the limit of fast binding/unbinding, assuming small hy i − hyf i f fluctuations around the mean. We have observed that free TF A slower decay of Ry(τ) enhances noise propagation but a noise levels decrease as a function of the total number of faster decay reduces noise propagation [25], [52]. In Fig. 4, decoy sites, and approach the Poisson limit for large decoy we plot auto-correlation function for N = 0, 1000, and abundances (Fig. 2). Interestingly, the noise suppression bioRxiv preprint doi: https://doi.org/10.1101/2020.04.01.020032; this version posted April 2, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-ND 4.0 International license.

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