Deterministic and Stochastic Descriptions of Gene Expression Dynamics

Abstract

A key goal of systems biology is the predictive mathematical description of gene regulatory circuits. Different approaches are used such as deterministic and stochastic models, models that describe cell growth and division explicitly or implicitly etc. Here we consider simple systems of unregulated (constitutive) gene expression and compare different mathematical descriptions systematically to obtain insight into the errors that are introduced by various common approximations such as describing cell growth and division by an effective protein degradation term. In particular, we show that the population average of protein content of a cell exhibits a subtle dependence on the dynamics of growth and division, the specific model for volume growth and the age structure of the population. Nevertheless, the error made by models with implicit cell growth and division is quite small. Furthermore, we compare various models that are partially stochastic to investigate the impact of different sources of (intrinsic) noise. This comparison indicates that different sources of noise (protein synthesis, partitioning in cell division) contribute comparable amounts of noise if protein synthesis is not or only weakly bursty. If protein synthesis is very bursty, the burstiness is the dominant noise source, independent of other details of the model. Finally, we discuss two sources of extrinsic noise: cell-to-cell variations in protein content due to cells being at different stages in the division cycles, which we show to be small (for the protein concentration and, surprisingly, also for the protein copy number per cell) and fluctuations in the growth rate, which can have a significant impact.

Appendix

1.1 A.1 Typical Values of the Parameters

Estimates of typical parameter values in the model organism E. coli are summarized in Table 1. Most of these can, for example, be estimated from the data of Ref. [43]. A few of them require additional comments: (i) In E. coli proteins are typically stable, i.e. β _p ≈0. So far, no complete survey of protein stability has been made, but the total cellular protein mass was found to be stable [31] and early proteomics studies (2d-gels) also indicated that almost all proteins covered by their approach were stable [37]. Nevertheless, some proteins are known to be unstable and, in these cases, β _p can be of the order of 1 min^-1. (ii) Genes are typically present as a single copy in the genome. This means that the gene copy number per cell is 1 before the gene is replicated and 2 after replication. Average gene copy numbers are between 1 and 2, except at fast growth with doubling times T<60 min, where rounds of DNA replication overlap and the gene copy numbers can be larger [7, 12]. (iii) The cell volume doubles over the division cycle and its average value depends on the growth conditions [7]. The value given in the table should be taken as an order or magnitude estimate.

Table 1 Typical parameter values for E. coli cells

1.2 A.2 Models with Stochastic Protein Synthesis and Stochastic Division

A general method for solving processes involving different rules of protein synthesis and cell division has been described in Ref. [8]. This method allows us in most of the cases to find averages and standard deviation of the protein number. We will describe the method briefly here following [8]. Let P _n be the protein content in the n ^th generation immediately after the cell division. Let λ _n be the amount of protein produced and accumulated till the cell division time in generation n and q _n be the fraction of protein inherited by the daughter cell at the time cell division. Then one can write

(17)

The protein generation as well as division can be taken from some distributions. If these distributions admit finite moments then in the steady-state the distributions of λ and q become independent and hence one can write

(18)

From here one can get all the moments for P, in particular 〈P〉=〈λ〉. Let us consider an example where we add protein with rate α in between every two cell divisions and where the protein number is divided deterministically into half at every cell division after every T time. In this case the synthesis of protein follows a binomial distribution giving 〈λ〉=δλ ²=αT and the division fraction is given by a delta function δ(q-1/2) with 〈q〉=1/2 and 〈q ²〉-〈q〉²=0. Thus Eq. (18) gives 〈P〉=〈λ〉 and $\langle P^{2} \rangle= \frac{1}{3}(2\langle\lambda\rangle^{2} + \langle\lambda^{2} \rangle)$. After some algebra one finds $\eta^{2} = \frac{\langle P^{2} \rangle- \langle P \rangle^{2}}{\langle P^{2}\rangle} = \frac{1}{3\langle P_{0} \rangle}$ which is one of the cases discussed in the main text.

1.3 A.3 Distribution of Protein Number and Concentration Due to Variation over the Division Cycle

The distribution of the protein number discussed in Sect. 5.1 is obtained by inverting the time-dependence of the protein copy number, P(t) to obtain t(P) and a transformation of variables in the age distribution from t to P, which leads to

(19)

Specifically, for the constant age distribution that describes averages over a single lineage, this leads to $\varPhi (P)=\frac{d}{dP} t(P)$. As a consequence, the result for an arbitrary age distribution can be rewritten as

(20)

i.e., the distribution of protein number in a single lineage weighted with the age distribution of the corresponding inverse.

The distributions for the concentrations are obtained in an analogous fashion, but the calculation is technically more involved as the concentration is not a monotonic function of time (see, e.g. Fig. 1). We thus split the functions p _lin(t) and p _exp(t) into piecewise monotonic functions and determine the distributions for these separately. The concentration for linear cell growth, p _lin(t), is monotonic in the intervals [0,t _x ] and [t _x ,T], and for p _exp(t) we have three intervals [0,t _x ], [t _x ,t _max] and [t _max,T], where t _max is the time where p _exp(t) is maximal. The complete distributions Φ(p _lin(t)) and Φ(p _exp(t)) are then obtained by adding up the distributions from the respective intervals. The distributions for the concentrations Ψ(p), are again obtained for the corresponding intervals, weighted with the age distribution and summed up to yield the full distribution.