Effects of cell-cycle-dependent expression on random fluctuations in protein levels

Expression of many genes varies as a cell transitions through different cell-cycle stages. How coupling between stochastic expression and cell cycle impacts cell-to-cell variability (noise) in the level of protein is not well understood. We analyse a model where a stable protein is synthesized in random bursts, and the frequency with which bursts occur varies within the cell cycle. Formulae quantifying the extent of fluctuations in the protein copy number are derived and decomposed into components arising from the cell cycle and stochastic processes. The latter stochastic component represents contributions from bursty expression and errors incurred during partitioning of molecules between daughter cells. These formulae reveal an interesting trade-off: cell-cycle dependencies that amplify the noise contribution from bursty expression also attenuate the contribution from partitioning errors. We investigate the existence of optimum strategies for coupling expression to the cell cycle that minimize the stochastic component. Intriguingly, results show that a zero production rate throughout the cell cycle, with expression only occurring just before cell division, minimizes noise from bursty expression for a fixed mean protein level. By contrast, the optimal strategy in the case of partitioning errors is to make the protein just after cell division. We provide examples of regulatory proteins that are expressed only towards the end of the cell cycle, and argue that such strategies enhance robustness of cell-cycle decisions to the intrinsic stochasticity of gene expression.

Dynamics of y is not closed and depends to moments yc n , hence in order to have a closed set of equations we add new moments dynamics by selecting ϕ to be yc i d yc 1 dt = k 1 B c 1 + λ n 2 yc n − λ 1 yc 1 , (B.5a) Dynamics of y and yc i , j ∈ {1, . . . , n} are the same as dynamics of x and xc i , j ∈ {1, . . . , n} presented in (3.1b) and (3.3) in the main text, hence x = y and xc i = yc i .
Further, dynamics of xy can be written as In order to have a closed set of equations we add dynamics of xyc i By having a closed set of equations related to xy, in the next step we add dynamics of Intrinsic noise obtained from two-color assay In this section we show that the results obtained here can be derived from two-color assay.
Consider two identical proteins x 1 and x 2 which their dynamics are exactly the same as protein x in the main article. The model describing x 1 and x 2 includes the stochastic events Protein x 1 production: Cell division: Note that the mean burst sizes of x 1 and x 2 are equal to the mean burst size of x. For this model the intrinsic noise can be quantified as Since dynamics of x 1 and x 2 are exactly the same as x, we have x 1 = x 2 = x and x 2 1 = x 2 2 = x 2 . Further in Appendix B we show that xc i = yc i hence In the next we show that x 1 x 2 = y 2 . Time derivative of the expected value of any function ϕ(x 1 , x 2 , c) = ϕ(x 1 , x 2 , c 1 , c 2 , . . . , c n ) for this system can 7 be written as [3] d ϕ( where the propensity function of the events is given by ψ(x 1 , x 2 , c) = ψ(x 1 , x 2 , c 1 , c 2 , . . . , c n ), and ∆ϕ(x 1 , x 2 , c) = ∆ϕ(x 1 , x 2 , c 1 , c 2 , . . . , c n ) is the change in ϕ(x 1 , x 2 , c) when an event occurs. The mean dynamics of x 1 x 2 can be written by choosing ϕ to be In order to have a closed set of equations, we add new moments dynamics by selecting (C.5b) Using the fact that x 1 c i = x 1 c i = yc i , equations (C.4) and (C.5) in steady-state are exactly the same as (B.8) in steady-state. Hence Appendix D

Moments dynamics of z
The random variable z is governed via Further in the time of division, z + is defined as Hence the model by taking into account z contains the following stochastic events Protein production: x and deterministic dynamics of z given in (D.1b). Time derivative of the expected value of any function ϕ(x, z, c) = ϕ(x, z, c 1 , c 2 , . . . , c n ) for this system can be written as [3] where the first term in the right-hand side is contributed from stochastic events and the second one is contributed from (D.1b). The propensity function of the events is given by . . , c n ), and ∆ϕ(x, z, c) = ∆ϕ(x, z, c 1 , c 2 , . . . , c n ) is the change in ϕ(x, z, c) when an event occurs.
By choosing ϕ to be z 2 and z 2 c i , i = {1, . . . , i} we have the following moment dynamics Note that just one of the binary states c i can be 1 at a time, thus z 2 = n i=1 z 2 c i . In order to calculate the terms z 2 c i we need to express the term z 2 c n as the first step.
This term can be calculated by analyzing equation (D.5a) in steady-state (D.6) By using a recursive process we calculate moments z 2 c i : we calculate z 2 c 1 by substituting equation (D.6) in equation (D.5b). Then we use the definition of z 2 c 1 to calculate z 2 c 2 from equation (D.5c) and so on Summing up all the term in equation (D.7) results in z 2 (D.8) Finally, protein fluctuatios level can be written as (D.10)

Appendix E
Optimal value of β From (D.9) it is clear that minimum production noise occurs when β is maximum, and minimum value of partitioning noise happens when β is minimum. β can be written as Note that a 1 > a 2 > . . . > a n ⇒ β ≤ a 1 a n , where equality happens when all k i s are zero except k n . Using the same methodology one can see that minimum of β happens when all the rates are zero except k 1 . The minimum value of β is one.
Appendix F

Cell-to-cell variability in synchronized cells
Statistical moments conditioned on the cell cycle stage C i can be obtained using In order to calculate stochastic variation in protein levels in synchronized cells we need In order to calculate x 2 c n we introduce the moment dynamics of x 2 hence in steady-state By using a similar process used in the previous section we calculate moments x 2 c i (F.5) By having xc i and x 2 c i from (3.5) and (F.5), we can calculate the mean and the noise in synchronized cells. Using (F.1) yields the following conditional mean Further, the protein variability level given that cells are in stage C i is given by ( Gene deactivation: Protein production: where m(t) and p(t) denote mRNA and protein population levels at time t, respectively.
At the end of cell cycle, division occurs and mRNA and protein molecules are partitioned in daughter cells binomially. After each division we select a new cell-cycle time which is correlated to previous cell-cycle times. We add correlation to the new cell-cycle time T i , i ∈ N, by assuming that it is connected to previous cell-cycle time through an Auto Regressive (AR) process where η i s are independent and identical normally distributed random variables η i ∼ N (0, σ η ), T 0 is a constant, and |φ| < 1. For this model the mean and variance of cell-cycle time is Further the cross correlation between two cell cycles which are i cycles apart is φ i .
In the case of transcriptional bursting, burst frequency is gene activation rate, i.e., k on in this model. Hence here we assumed that k on is a function of cell-cycle time. We investigate two scenarios 1) constant gene activation rate 2) synthesis at the end of cell cycle. For constant k on , gene switches between ON and OFF states through the cell cycle.
In the synthesis at the end of cell cycle, we assume that for 75% of T i gene is OFF and k on = 0. In the last 25% of cell cycle time switching occurs and k on is non zero. Further transcriptional bursting is the limit of large k of f and small k on , i.e., genes is OFF most of the time. Here we consider that gene is active for 20% of the cell-cycle time. Further we analyzed the system in both fast and slow switching switching rates.
We use another model in which protein production is modeled deterministic through- Positive correlation in cell-cycle times Negative correlation in cell-cycle times Figure S1: Protein synthesis at the end of cell cycle reduces noise contributed from expression in the limit of slow switching rates and presence of correlated cell cycle times. Noise ratio less than 1 indicates syntheis at the end reduces the noise in comparison with constant production. Noise ratio is less than one for different switching rates and correlation values. For this plot we have assumed both negative correlation of −0.25 [5] and positive correlation of 0.25 between successive cell-cycle times, Mean cellcycle time is 2 hours and noise in cell-cycle times is CV 2 T = 0.05. The mRNA production rate is k m = 50hr −1 and mRNA molecules degrade with rate γ m = 5hr −1 . Protein molecules are translated from mRNA with a rate k p = 25hr −1 . Gene activation rate k on is adjusted to keep mean of protein equal to 150 molecules for all cases. The error bars obtained via bootstrapping by using 20, 000 Monte Carlo simulations. out the cell cycle In the time of division mRNA and proteins are partitioned based on a beta distribution which is the continuous counter part of binomial distribution. The difference between noise levels of these two models give the noise from stochastic expression.
We numerically investigate the models described in (G.1) and (G.4) for correlated cell-cycle times in (G.2). Figure S1 shows the simulation results obtained from 20, 000 Monte Carlo simulations for different switching rates and correlation values. From equation (4.7) in the main article we know that synthesis at the end of cell cycle reduces noise in comparison with constant synthesis. However equation (4.7) obtained for the bursty expression model which is an approximation in the limit of fast switching. Moreover in order to derive (4.7), cell cycle time are assumed to be independent. Simulation results reveal that in the presence of correlated cell-cycle times and by taking into account dynamics of gene and mRNA, synthesis at the end still reduces the noise contribution from stochastic synthesis. This reduction happens even when gene is active for relatively long time, switching is slow, dynamics of mRNA is included and cell-cycle times are correlated.
In summary our analysis reveals that perturbing the assumptions made in this paper to obtain analytic solutions are not changing the fact that synthesis at the end of cell cycle leads to buffering noise contributed from stochastic expression.