6C Effects of the cell cycle on stochastic gene expression. 6C.1 Non-equilibrium model of dividing cell populations

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1 6C Effects of the cell cycle on stochastic gene expression In Chap. 6, we mainly focused on the effects of intrinsic molecular noise on gene expression. Here we consider the effects of a major source of extrinsic noise, which arises from the stochastic nature of cell growth and division during the cell cycle. In balanced population growth, the quantities of cellular components double on average during each cell cycle and then halve at cell division. However, individual cells can deviate significantly from the average due to a combination of molecular noise and stochastic growth and cell division. A variety of experimental and theoretical studies have investigated the distribution of proteins across a population of cells under well-controlled conditions, with the goal of understanding the relative contributions of intracellular noise, the random partitioning of proteins at cell division, and the stochastic exponential growth of cells [3, 2, 5, 6, 9, 16, 12, 13, 1, 17, 23, 4, 7]. 6C.1 Non-equilibrium model of dividing cell populations We begin by considering a minimal model of dividing cell populations due to Brenner and Shokef [6], which describes the joint effects of protein synthesis noise and proliferation dynamics. The model uses a separation of time-scales between protein production, which is continuous throughout the cell cycle, and cell division, which occurs during a short fraction of the cell cycle. Another major simplification is that the entire population of cells divides synchronously. The total protein content in a cell is thus taken to evolve according to a discrete map x n+1 = M x n = q n (x n + λ n ), (6C.1) where x n is the protein content of a cell in the nth generation just after cell division, λ n is the amount of protein produced and accumulated in the cell during the nth cell cycle, and q n is the fraction of protein inherited by one of the daughter cells at the end of the nth cell cycle - the other inherits the fraction 1 q n. Suppose that λ n and q n are i.i.d. s drawn from the probability densities ξ (λ) and η(q), respectively. The discrete-time Liouville equation for the evolution of the distribution of proteins across the population of cells at generation n is then given by P n+1 (x) = dq η(q) dλ ξ (λ) dx δ(m (x ) x)p n (x ). (6C.2) Note that the nature of cell division implies the symmetry η(q) = η(1 q). This means that q = 1/2, since 1

2 q = qη(q)dq = qη(1 q)dq = (1 q)η(q)dq = 1 q. Suppose that a steady-state solution exists so that lim P n(x) = P(x), n where P(x) is the (unique) fixed point solution of the Liouville equation. Introducing the moment generating function G(s) = e sx P(x)dx, and using the Liouville equation we have where G(s) = = = dq η(q) dq η(q) dλ ξ (λ) dλ ξ (λ) dq η(q)g(qs)h(qs), dxe sx dx δ(q(x + λ) x)p(x ) dx e sq(x +λ) P(x ) H(s) = e sλ ξ (λ)dλ is the moment generating function of the random variable λ. If q and λ have finite moments then so does x. Moreover, the statistical independence of q and λ means that in the steady-state x k n+1 = qk n(x n + λ n ) k = q k n (x n + λ n ) k = q k In particular, x = q ( x + λ ) = λ, k j= ( ) k x k j λ j. j since q = 1/2. In the case of constant λ and a uniform distribution η(q) = 1, the generating function can be solved explicitly. The Liouville equation reduces to G(s) = G(qs)e sqλ dq, which can be differentiated with respect to s to give 2

3 dg(s) 1 = λ qg(qs)e sqλ dq + qg (qs)e sqλ dq ds Evaluating the second integral using integration by parts yields Hence, qg (qs)e sqλ dq = 1 [qg(qs)e ] sqλ q=1 s 1 G(qs) d ( ) qe sqλ dq q= s dq = 1 s G(s)e sλ 1 [ ] G(qs) e sqλ sλqe sqλ dq. s s dg(s) ds = G(s)e λs G(s). (6C.3) The latter equation can be solved using an integrating factor to give ( s ) G(s) = exp φ(s )ds, φ(s) = 1 [ 1 e λs]. (6C.4) s We have used the identity G() = 1. Taylor expanding φ(s) in powers of λs and integrating shows that s ψ(s) φ(s )ds = k=1 ( λs) k. k!k Finally, the cumulants of the protein distribution P(x) are κ n = ( 1) n dn lng(s) ds n = λ n s= n. (6C.5) Note for s we have φ(s) 1/s, ψ(s) lns and G(s) 1/s. Given the generating G(s), one can obtain P(x) by expressing the inverse Laplace transform using a Bromwich integral and steepest descents for large x: c+i c+i P(x) = e sx G(s)ds = e sx e ψ(s) ds c i c i 1 2π ψ (S ) 1/2 es x+ψ(s ). Here s is the unique solution to the equation x + ψ (s ) =, that is, x = 1 s [1 e λs ], with s <. It follows that, for this example, the tail of the probability density P(x) is non-exponential (since s = s (x)) with P(x) x x. It turns out that the shape 3

4 G(s) (a) s 1 P(x) (b) x Fig. 6C.1: Protein distribution in the case of constant protein synthesis. (a) Generating function G(s) for a uniform distribution η(q) = 1 of the partition fraction q. Asymptotically G(s) 1/x. (b) Corresponding distribution function P(x) (black curve). Also shown is the distribution for a non-uniform η(q) (gray curve), illustrating non-universality. [Redrawn from Brenner and Shokef [6].] of the tail is also sensitive to the choice of the division density η(q) [6]. On the other hand, if significant protein production noise is included, with λ drawn from an exponential distribution, then one finds that the the density P(x) exhibits a similar shape for different choices of η(q) and has an exponential tail. This suggests that it might be possible to distinguish between the two types of noise in experimental data. 6C.2 Contributions of cell growth to stochastic gene expression One major simplification of the above model is that all cells divide synchronously. Asynchronous cell division means that individual cells across the population may be at different stages of the cell cycle when their molecular content is measured, which adds another level of stochasticity. In order to take this into account one must track cell growth. Moreover, if one wants to determine fluctuations in protein concentration rather than copy number, then one has to take into account changes in cell volume during cell growth, which can also be stochastic. A schematic illustration of the various contributions of cell growth to cell-to-cell variability is illustrated in Fig. 6C.2. In this section we carry out a decomposition of the variance in protein copy number into the various sources of stochasticity, following the analysis of Schwabe and Bruggeman [23], see also [13]. 4

5 R = 2, S = protein inherited from mother cell newly synthesized protein R a = 1, R a = 4 stochatic protein synthesis and degradation R T =, S T = 6 q 1-q stochatic partitioning variations in cell division times Fig. 6C.2: Schematic diagram illustrating the chemical kinetic and cell-growth processes contributing to cell-to-cell variability. Sources of stochasticity include (i) fluctuations in the rate of protein synthesis and its regulation, (ii) binomial partitioning of molecules at cell division with partition fraction q, (iii) variability in molecular content of mother cells, and (iv) variability in cell-cycle stage across the population. For a given stage a of a cell cycle, R a denotes the number of molecules inherited from the mother cell that have not yet degraded, and S a is the number of newly synthesized proteins since cell division. Law of total variance In the following, we will make continued use of the law of total variance, also known as Eve s law: if X and Y are random variables on the same probability space, then Var[Y ] = E X [Var[Y X]] + Var X [E[Y X]]. (6C.6) The proof follows from the law of expectation (see also Chap. 11). First, from the definition of variance, Var[Y ] = E[Y 2 ] (E[Y ]) 2. 5

6 Conditioning on the random variable X and applying the law of total expectation to each term gives Var[Y ] = E [ E[Y 2 X] ] (E[E[Y X]]) 2. Now we rewrite the conditional second moment of Y in terms of its variance and first moment: Var[Y ] = E [ Var[Y X] + (E[Y X]) 2] (E[E[Y X]]) 2. Since the expectation of a sum is the sum of expectations, the terms can now be regrouped as Var[Y ] = E[Var[Y X]] + { E[(E[Y X]) 2 ] (E[E[Y X]]) 2} The result follows from observing that the terms in parentheses on the right-hand side give the variance of the conditional expectation E[Y X]. Variance at a fixed cell cycle stage Let a, a T, denote the cell-cycle stage measured with respect to the last cell division. Let X a denote the protein number at stage a, R a the number of molecules inherited from the mother cell that have not yet degraded, and S a the number of newly synthesized proteins since cell division: X a = R a + S a, R = X, S =. (6C.7) We assume that the distribution of non-degraded proteins at stage a is given by a Binomial distribution: P[R a = r R = X ] = X! (X r)!r! p(a)r (1 p(a)) X r, where p(a) is the corresponding survival probability for an inherited molecule. For independent, first-order degradation we have p(a) = e k da where k d is the degradation rate. It follows that R a X = p(a)x, Var[R a X ] = p(a)(1 p(a))x, where we have taken expectation with respect to the Binomial distribution for fixed X. Given that there is also variation in X, and from the law of total variance R a = E[ R a X ] = p(a) X, Var[R a ] = E[p(a)(1 p(a))x ]+Var[p(a)X ] = p(a)(1 p(a)) X + p(a) 2 Var[X ]. 6

7 Combining these various results, we see that and X a = p(a) X + S a (6C.8) Var[X a ] = p(a)(1 p(a)) X + p(a) 2 Var[X ] + 2p(a)Cov[X,S a ] + Var[S a ]. (6C.9) The next step is to determine the variation in the number of proteins X in a daughter cell just after cell division. This will depend on two factors: (i) the variation in the number of proteins X T in a mother cell just before cell division, and fluctuations due to randomly partitioning the molecules between the two daughter cells with partition ratio q. Conditioning on X T and q, which are independent random variables, we apply the law of total variation twice. First, Var[X q] = E[Var[X q,x T ]] + Var[E[X q,x T ]], with expectation taken with respect to X T. Since it follows that X q,x T = qx T, Var[X q,x T ] = q(1 q)x T, Var[X q] = q(1 q) X T + q 2 Var[X T ]. Applying the law of total variation a second time, with expectation now taken with respect to q, Var[X ] = E[Var[X q]] + Var[E[X q]] = q(1 q) X T + q 2 Var[X T ] + Var[q] X T 2 ( ) ( ) 1 1 = 4 Var[q] X T + Var[q] X T Var[q] Var[X T ]. (6C.1) We have used the fact that q = 1/2 (see Sec. 6A.1) and E[X q] = q X T. Equation (6C.1) establishes that fluctuations in the partitioning probability q enhances Var[X ]. We can express the population level variance Var[X a ] given by equations (6C.9) and (6C.1) solely in terms of the variance in protein synthesis Var[S a ], its correlations with X, the variance in the partition distribution Var[q], and the survival probability p(a). First, setting a = T in equation (6C.8) we have X T = p(t ) X + S T. Since X T = 2 X (doubling at cell division), we see that X = S T 2 p(t ) = X T /2. (6C.11) 7

8 Second, setting a = T in equation (6C.9) gives Var[X T ] = p(t )(1 p(t )) X + p(t ) 2 Var[X ] + 2p(T )Cov[X,S T ] + Var[S T ], and substituting this into the second line of (6C.1) yields Var[X ] = q(1 q) X T + Var[q] X T 2 1 q 2 p(t ) 2 (6C.12) + q2 (p(t )(1 p(t )) X + 2p(T )Cov[X,S T ] + Var[S T ]) 1 q 2 p(t ) 2. Equations (6C.9), (6C.11) and (6C.12) yield the desired result. In the above analysis, we have not specified the particular mechanism underlying the variation in the partition ratio q. This issue is explored by Huh and Paulsson [13], who consider some kinetic models of fluctuations in q due to various mechanisms such as differences in the available volumes of the daughter cells, and molecular clustering. In order to calculate the variance due to cell division, they introduce an effective birth death process that captures the exact segregation statistics even though it need not describe the precise physical process underlying partitioning. Here we will illustrate the basic approach by considering the simpler case of unbiased independent partitioning, where each protein has.5 probability of being assigned to either daughter cell, that is, q = 1/2. This could be physically realized by several mechanisms: molecules rapidly diffusing between the two halves during cell division; proteins being synthesized at the same rate in either half of the mother cell but not diffusing or degrading; molecules binding to either mitotic spindle at cell division (see Sec. 8.2) with equal probability. Irrespective of the particular model, the statistical partitioning error can be realized by Ehrenfest s urn model in the stationary state. The latter is described by a birth-death process for the number of molecules L in one daughter cell given a total of X molecules in the mother cell. The reactions are L X L L + 1, L L L 1, that is, each molecule (L on one side and X L on the other) switches sides at a constant rate taken to be unity. The associated birth-death master equation for the probability distribution P(L,t) is, for fixed X, dp(l, t) dt = (X L + 1)P(L 1,t) + (L + 1)P(L + 1,t) XP(L,t), which is identical in form to the master equation for a two-state ion channel (see Sec. 3.3). The stationary solution is thus the Binomial distribution P s (L) = q L (1 q) X L X! (X L)!L!, with q = 1/2 (deterministic partition ratio) and E[L X] = qx, Var[L X] = q(1 q)x. It follows from the law of total variance that 8

9 Var[L] = 1 4 X Var[X]. This agrees with equation (6C.1), since Var[q] =. Variance due to the distribution of cell cycle stages So far we have determined the population variance at a specific cell cycle stage a, given some unspecified model of stochastic protein synthesis S a. However, there is an additional source of variance due to the fact that, across the population, cells are at different stages of the cell cycle. Making the identifications X a = E[X a], Var[X a ] = Var[X a], and again applying the law of total variation, we have that the total variance in protein synthesis across the population of cells is Var[X] = E[Var[X a ]] + Var[ X a ], (6C.13) where expectation is now taken with respect to the distribution u(a) of cell cycle stages. The latter can be specified in terms of the distribution f (τ) of interdivision times τ, under the assumptions that the cells divide asynchronously and population growth has reached a stationary state [19]. In particular, all extensive quantities grow exponentially so that the number of cells N(t) is given by N(t) = e µt N, where µ is the growth rate and N is the number of cells at time t =. In the case of binary division, the rate at which new cells are generated is then given by 2µN e µt. We can relate the age distribution u(a) to f (τ) as follows. The number of cells younger than some fixed age a is by definition a N(a) = N u(y)dy. An alternative way to calculate N(a) is to note that these cells are precisely those that formed during the time-interval ( a,) and do not divide until a time t >. The rate of formation of such cells at time t ( a,) is γ(t) = 2µN e µt f (y)dy. t It follows that N(a) = γ(t)dt a Differentiating both sides with respect to a then gives (6C.14) 9

10 u(a) = N 1 γ( a) = 2µe µa a f (y)dy. (6C.15) In the analysis of Ref. [23], it is assumed that the interdivision time is the same for all cells, τ = T, that is, f (τ) = δ(τ T ). It follows that u(a) = 2µe µa a T, (6C.16) and is zero otherwise. Moreover, since the population doubles over a time interval of size T, we have µ = ln2 T. Example: zero degradation (k d = ) and constant rate of synthesis. In the case of zero degradation, p(a) = 1 for all a T. Moreover, the synthesis of proteins is given by a Poisson process with Hence S T = Var[S T ] = k s T. X T = 2 X = 2 S T = 2k s T, and from equation (6C.12) with Cov[X,S a ] =, Var[X ] = q(1 q) X T + Var[q] X T 2 + Var[S T ] 1 q 2 = k st (3 4Var[q] + 16Var[q]k s T ). 3 4Var[q] Similarly, equation (6C.9) reduces to Hence, Finally, so that Var[X a ] = Var[X ] + Var[S a ] = Var[X ] + k s a. T E[Var[X a ]] = u(a)[var[x ] + k s a]da = Var[X ] + k s (µ 1 T ). X a = X + S a = k s (T + a), 1

11 Var[ X a ] = k 2 s T u(a)[a a ] 2 da = ks 2 (µ 2 2T 2 ). Hence, from equation (6C.13), the total variance is Var[X] = k s µ + k2 s (µ 2 2T 2 ) + (k st ) 2 16Var[q] 3 4Var[q] (6C.17) 6C.3 Stochastic exponential growth of single bacterial cells The above two sections focused on growth and division at the population level, for which all extensive quantities grow exponentially under balanced growth conditions. This is irrespective of how the size of an individual cell changes in time. That is, the observation of population growth is not sufficient to determine the the growth law at a single cell level. Both linear and exponential growth laws have previously been considered [8], and linear protein synthesis was assumed in the above models. However, recent advances in imaging techniques has allowed more precise measurements of the stochastic growth of individual Caulobacter cresentus bacterial cells [15], revealing that mean sizes grow exponentially in time. Moreover, the size distributions collapse to a single curve when rescaled by their means. This universal behavior of fluctuations during the growth of single bacterial cells has also been accounted for theoretically by Iyer-Biswa et al. [14] using a minimal microscopic model. In the following we describe this model in more detail, since it also provides an illustrative example of a cyclic biochemical process. Iyer-Biswas et al. consider a stochastic version of a simple kinetic model introduced by Hinshelwood [11]. The latter assumes that the cell components controlling cell growth are linked via a cycle of autocatalytic reactions, whereby each chemical X N X 1 X 2 k 2 X 1 X 1 X 1 X 2 k 3 X X 2 X 2 X 3 X 3 k 1 X N + X N X N X 1 Fig. 6C.3: Hinshelwood model for exponential growth, showing the autocatalytic cycle, in which each species activates production of the next, and the corresponding reactions. 11

12 species catalyzes the production of the next, see Fig. 6C.3. The stochastic Hinshelwood cycle (SHC) consists of N species {X 1,X 2,...,X N }. The mean rate of production of X j is taken to be k j x j 1, 1 j N, where x j is the copy number of X j, and X X N. The reaction scheme is X j 1 k j x j 1 X j 1 + X j. (6C.18) (We will use X i to denote the chemical species and the stochastic copy number of the given reactant.) The corresponding master equation for the probability distribution P(x,t), x = (x 1,x 2,...x N ) is N P t = k j x j 1 [P(x 1,...,x j 1,...x N ) P(x 1,...,x j,...x N )]. j=1 (6C.19) with P(x 1,...,x j 1,...x N ) = if x j =. Multiplying both sides by x i and integrating with respect to x, we see that all terms on the right-hand side vanish except when j = i, for which d X i dt = k i = k i X i 1. x i x i 1 [P(x 1,...,x i 1,...x N ) P(x 1,...,x i,...x N )]dx In vector form with µ j (t) = X j (t), we have dµ(t) dt where K is a cyclic matrix of period N, that is, and I N is the N N identity matrix. Consider the eigenvalue equation which can be iterated to give = Kµ(t), (6C.2) K N = k 1 k 2...k N I N, Ku = λu, K N u = κ N u = λ N u, where κ is the geometric mean of the reaction rates k j : It follows that the m-th eigenvalue is κ N = k 1 k 2...k N. λ m = κe 2πim/N, 12

13 and the q-th component of the corresponding eigenvector u m is u (q) m = λ q m The eigenvalue with the largest real part is λ N = κ, and this will dominate the asymptotic dynamics with relaxation time-scale κ 1. In particular, suppose that we expand the first moment vectors in terms of the eigenvectors u m : so that N m=1 ċ m (t)u qn = µ(t) = N n=1 N m=1 q p=1 k p. c m (t)u m, λ n c n (t)u nq, U qm = u (q) m Since the eigenvalues are distinct, the matrix of eigenvectors U is invertible so that ċ m (t) = λ m c m (t), c m (t) = e λ mt c m (), and the eigenvalue expansion takes the form µ j (t) = N m=1 e λ mt U jm c m () = N m,i=1 e λ mt U jm U 1 mi µ i() Hence, in the asymptotic limit (t κ 1 ), ( N ) µ j (t) UiN 1 µ i() e κt U jn, i=1 (6C.21) which implies that the mean copy number of all the reactants evolve asymptotically as the single exponential e κt. It also follows that µ j (t) µ r (t) = U jn U rn, which is independent of initial conditions. An interesting interpretation of the geometric mean κ can be obtained by assuming that each individual reaction rate is given by the Arrhenius form (see Sec. 3.3): k i = A i exp( E i /k B T ), where E i is the activation energy of reaction i. Then 13

14 κ = (k 1 k 2...k N ) 1/N = (A 1 A 2...A N ) 1/N exp Aexp( E/k B T ), ( E ) E N Nk B T where E is the arithmetic mean of the elementary activation energies. Hence, assuming that each reaction step has an energy barrier that is of the order of a typical enzyme reaction, then so does the effective growth rate. This is consistent with experimental measurements of growth rates in single bacterial cells [15]. The next step is to determine the asymptotic behavior of the covariance matrix C(t) with C i j = X i X j X i X j. A basic result from the theory of chemical master equations with transition rates that are linear in the copy number is that the covariance matrix satisfies the same Ricatti equation (2.2.22) as the corresponding OU process obtained using a system size expansion. That is, where dc(t) dt = KC(t) + C(t)K T +Θ T, (6C.22) Θ i j = δ i j µ j (t). It can be shown that in the asymptotic limit [14], C i j (t) U in U jn e 2κt N r=1b r µ r (), (6C.23) with the coefficients b r dependent on the rates but not the initial conditions. Comparing equations (6C.21) and (6C.23), one finds that the rescaled covariance Ĉ i j (t) = Y i (t)y j (t) Y i (t) Y j (t), Y i (t) = X i(t) µ i (t), takes the asymptotic form C i j (t) ( N ) 2 N UlN 1 µ l() b r µ r (), l=1 r=1 (6C.24) In particular, Var[X i (t)/µ i (t)] constant. Since the rescaled covariance is independent of i, j and time, it follows that the random variables Y i (t) are perfectly correlated. This only occurs if the random variables Y i (t) are linearly related. We conclude that although the means µ i (t) grow exponentially in time, the rescaled random variables X i (t)/µ i (t) have the same time-independent probability density in the asymp- 14

15 totic limit, which is also observed experimentally [15]. This, in turn, implies that the n-th order moment of X i varies as e nκt. Supplementary references 1. Amir, A.: Cell size regulation in bacteria. Phys. Rev. Lett (214) 2. Avery, S. V. : Microbial cell individuality and the underlying sources of heterogeneity. Nat. Rev. Microbiol (26) 3. Berg, O. G.: A model for the statistical fluctuations of protein numbers in a microbial populations. J. Theor. Biol. 71, (1978). 4. Braun, E.: Rep. Prog. Phys (215). 5. Brenner, N., Farkash, K., Braun, E. : Phys. Biol (26) 6. Brenner, N., Shokef, Y.: Nonequilibrium statistical mechanics of dividing cell populations. Phys. Rev. Lett (27) 7. Brenner, N., Newman, C. M., Osmanovic, D., Rabin, Y., Salman, H., Stein, D.L.: Universal protein distributions in a model of cell growth and division. Phys. Rev. E 92, (215). 8. Cooper, S.: Distinguishing between linear and exponential cell growth during the division cycle: Single-cell studies, cell-culture studies, and the object of cell cycle research. Theor. Biol. Med. Model. 3 (26) Friedman, N., Cai, L., Xie, X. S.: Linking stochastic dynamics to population distribution: An analytical framework of gene expression. Phys. Rev. Lett. 97, (26). 1. Gomez, D., Marathe, R., Bierbaum, V., Klumpp, S.: Modeling stochastic gene expression in growing cells. J Theor Biol 348, 1-11 (214). 11. Hinshelwood, C. N.: On the chemical kinetics of autosynthetic systems. J. Chem. Soc (1952). 12. Huh, D., Paulsson, J.: Non-genetic heterogeneity from stochastic partitioning at cell division. Nat Genet (211). 13. Huh, D., Paulsson, J.: Random partitioning of molecules at cell division. Proc. Natl. Acad. Sci. USA (211). 14. Iyer-Biswas, S., Crooks, G. E., Scherer, N. F., Dinner, A. R. Universality in stochastic exponential growth. Phys Rev Lett (214). 15. Iyer-Biswas, S., Wright, C. S., Henry, J. T., Lo, K., Burov, S., Lin, Y., Crooks, G. E., Crosson, S., Dinner, A. R., Scherer, N. F.: Scaling laws governing stochastic growth and division of single cell bacteria. Proc. Natl. Acad. Sci. USA (214). 16. Klumpp, S., Zhang, Z., Hwa, T.: Growth rate-dependent global effects on gene expression in bacteria. Cell (29) 17. Klumpp, S., Hwa, T.: Bacterial growth: global effects on gene expression, growth feedback and proteome partition. Curr. Opin. Biotechnol (214) 18. Marguerat, S., Bahler, J.: Coordinating genome expression with cell size. Trends Genet., (212). 19. Painter, P. R., Marr, A. G.: Mathematics of microbial populations. Annu. Rev. Microbiol. 22, (1968). 2. Salman, H., Brenner, N., Tung, C.-K., Elyahu, N., Stolovicki, E., Moore, L., Libchaber, A., Braun, E.: Phys. Rev. Lett. 18, (212). 21. Schmoller, K. M., Skotheim, J. M.: The biosynthetic basis of cell size control. Trends Cell Biol. (215) 22. Schwabe, A., Rybakova, K. N., Bruggeman, F. J.: Transcription stochsticity of complex gene regulation models. Biophys J (211). 23. Schwabe, A., Bruggeman, F. J.: Contributions of cell growth and biochemical reactions to non-genetic variability of cells. Biophys J (214). 24. Taheri-Araghi, S., Bradde, S., Sauls. J. T., Hill, N. S., Levin, P. A., Paulsson, J., Vergassola, M., Jun, S.: Cell-size control and homeostasis in bacteria. Curr Biol 25, (214) 15