Stochastic Dynamics of Gene Switching and Energy Dissipation for Gene Expression

Stochastic dynamics of gene switching and energy dissipation for gene expression are largely unknown, mainly due to the complexity of non-equilibrium mechanisms. Here, based on an important double-deck loop model, the stochastic mechanisms of gene switching and energy dissipation are studied. First, the probability distributions of steady states are calculated theoretically. It is found that the signal can strengthen the choice of gene switching between the “off” and “on” states. Our analysis of energy consumption illustrates that, compared with the synthesis and degradation of proteins, the process of gene switching costs little energy. Our theoretical analysis reveals some interesting insights into the determination of cell state and energy dissipation for gene expression.


DETAILED DERIVATION FROM CHEMICAL MASTER EQUATIONS TO STEADY-STATE DISTRIBUTIONS
Since it is difficult to obtain the theoretical results of chemical master equations, the method of probabilitygenerating functions is used to change chemical master equations into differential equations which may be solved with mathematical skill. Based on this method, two factorial probability-generating functions G i (z) = ∞ m=0 P i (m)z m with i = 0, 1 are introduced. Substituting these into Eq.2: Here, all parameters have been normalized by k 4 as mentioned in Sec.II. It is found from Eq. S1 that e −ak 3 z G i is set as H i for simplicity. From Eq. S2, we can determine that H 0 = −H 1 . This allows us to obtain H 0 from H 1 . In the following, we will only focus on the expression of H 1 , which can be deduced from Eq. S1 as in which A(z) = (k 2 + 1)z − 1, B(z) = ak 2 k 3 z + k off + k on + ak 1 + k 2 + 1, and C = ak 2 k 3 . Until now, it seems difficult to obtain the analytical distribution because the coefficients of Eq. S3 are dependent on the variable z. Another transformation of the form H 1 (z) = e mz W (nz + p) (here, m, n and p are undetermined constants) is introduced in Eq. S3, and hence a solvable differential equation about W is obtained as There are two independent solutions of Eq. S4. One is expressed by the Tricomi function of the form U (α, β; ω). This result is inadmissible because it is required that P 1 (m) → 0 for m→∞. The other solution is expressed by the confluent hypergeometric function of the form 1 F 1 (α, β; ω). The analytical expressions for the probability-generating functions are presented as below Furthermore, based on the relationship between probability distribution and generating function: The analytic results of probability distributions are obtained and shown in the text (i.e., Eq.3 in Sec.II).

THE RESULTS OF NUMERICAL SIMULATIONS
The direct method to verify our analytical results is numerical simulations. The approximate probability distributions of P i (m) obtained by Monte Carlo simulation are given in Fig.S1. Figure S1. The numerical simulated results about probability distribution functions of the gene's states .
The parameters are the same with Fig.3(a) and (b): k 1 = 20, k 2 = 10, k 3 = 1.5, k on = k off = 2. P 0 (m) is the probability of the gene's "off" state indicated with orange curve, P 1 (m) is the probability of gene's "on" state indicated with blue curve. Figure S2. The numerical simulated results about gene's state dominance factor δ which is consistent with that in Fig.4(c). δ will increase with the increasing of the strength of external signal in Mode I. The values of other parameters are listed in Table I.
Moreover, the curve between δ and a which is obtained by Monte Carlo simulation with the same set of parameters in Fig.4(c) is shown in Fig.S2. It is obviously that the trend of δ is similar with the curve in Fig.4(c).

THE COMPARISONS BETWEEN THE TOTAL ENERGY DISSIPATION AND THE ENERGY DISSIPATION IN THE SYNTHESIS-DEGRADATION PROCESS OF ROCK IN MODE I AND II
The Fig.S3 shows the trends of the total energy dissipation (i.e., EP ) and the energy dissipation in the synthesis-degradation process of ROCK (i.e., EP m ) with the increase of the strengthen of external stimulations (i.e., a) in Mode I and II. It is obviously that their trends are consistent as a increases in its respective modes. Specifically, EP and EP m increase simultaneously in Mode I and decrease simultaneously in Mode II with the enhancing of the strengthen of external stimulations (i.e., a). Moreover, the difference between EP and EP m diminishes both in Mode I and Mode II when a increases. Figure S3. The comparisons between the total energy dissipation and the energy dissipation in the synthesis-degradation process of ROCK in Mode I and II. The values of all parameters are listed in Table I.

Frontiers
The values of the main parameters used in calculations are given in Table I. All these parameters have been normalized by the degradation rate of ROCK (i.e., k 4 ). Value The reaction rate of positive control (k 1 ) 20 The reaction rate of negative control (k 2 ) 10 The synthetic rate of ROCK (k 3 ) 1.5 (Mode I) or 5 (Mode II) The basic switch rate of gene to "on" state (k on ) 2 The basic switching rate of gene to "off" state (k off ) 2