Theoretical Research on Dynamics of the Genetic Toggle Switch
Tomohiro Ushikubo1
ushikubo@tbp.cse.nagoya-u.ac.jp
Wataru Inoue2
wataru@sasai.human.nagoya-u.ac.jp
Mitsumasa Yoda1
yoda@tbp.cse.nagoya-u.ac.jp
1
Masaki Sasai1,2,3
sasai@tbp.cse.nagoya-u.ac.jp
Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Furocho, Chikusa, Nagoya 464-8603, Japan 2 Department of Complex Systems, Graduate School of Information Science, Nagoya University, Furocho, Chikusa, Nagoya 464-8601, Japan 3 Institute for Advanced Research, Nagoya University, Chikusa, Nagoya 464-8601, Japan
Keywords: toggle switch, stochastic simulation, gene network
1
Introduction
Recently, small gene networks have attracted intense interest to quantitatively understand the design principle of gene networks. We theoretically examine a simple genetic switch composed of two genes, which is called a toggle switch[1], and study how the noise affects its switching behavior.
2
2.1
Method and Results
The Network Motif and the Model
The toggle switch is a network composed of two genes in which proteins synthesized by one gene represses expression of the other gene (fig.1a). Chemical reactions for each gene are modeled by four processes (fig.1b). First process is the protein production: The protein production rate is g in the repressor-unbound state, and 0 in the repressor-bound state. The synthesized protein acts as a repressor on another gene. Second is the protein degradation with the degradation rate, k. Third is the protein-binding to DNA with the binding rate, hni(ni−1), where ni is the number of the repressor proteins synthesized from the ith gene with i = 1 or 2. Fourth is the protein dissociation from DNA with the dissociation rate, f. In this study we use the value f/h=1000. When the gene switch fluctuates on and off, frequency of the DNA status change is measured by ω= f/k and the representative number of proteins is Xad= g/2k. We calculate the master equation by using the Monte Carlo (MC) simulation with the Gillespie algorithm[2].
protein
a
gene 1 gene 2
b
unbound state
repressor g
operator coding
k
hn(n-1) bound state
f
Figure 1: a, Toggle switch network. Lines connecting each gene to the other show negative regulatory relations. b, Each gene has two states, the active state with the repressor unbound and the inactive state with the repressor bound.
2.2
Phase Diagram
3 2
By drawing the distribution function, P(n1 n2), of numbers of proteins for each value of Xad and ω, we can count the number of maxima of P(n1 n2), that is, the number of stable states of the system. Then, the phase diagram of the number of stable states can be drawn in the Xad -ω plane as in fig.2. In this figure, we can find regions of one to three stable states: Rich structure is found in the region of ω< 1, which is not expected from the analysis of the deterministic equations of reactions. 2.3 Reaction Paths
1
A
D
log(ω)
0
-1
-2
B
1.0 1.2 1.4
-3
C
1.6 1.8 2.0
log(Xad)
Free energy, F(n1 n2), is defined by F ( n1 n 2 ) = − ln P ( n1 n 2 ) and F at the ith local maximum of P(n1 n2) is defined as Fi. The transition state theory predicts that the reaction rate from ‡ ‡ TST the state i to the state j, kijTST is k ij = k ij exp[ − ( Fij − Fi )] ,
‡
Figure 2: Phase diagram of the number of stable states in toggle switch. A is the region where the system has one stable state. B is the region of three stable states. C has two major and one minor stable states D has two stable states
where Fij is free energy at the transition state (saddle point) between the state i and the state j. Then, the ‡ prefactor, kij is calculated from the reaction rate kijMC in the MC simulation by assuming kijTST = kijM. Thus ‡ obtained kij is shown in table 1 for the case that the system has three stable states, where the state 1 is the ‡ ‡ lower minimum in F and the state 2 is the higher minimum in F. This table shows that k12 ≈ k21 for (log ‡ (Xad), log (ω)) = (1.2, -1.0), implying that the transition state is symmetrical along the reaction path but k12 ‡ << k21 for (log (Xad), log (ω)) = (1.8, 2.0), implying that the reaction path is asymmetrical. Table 1: Calculated values of kij . (log (Xad),log (ω)) (1.2, -1.0) (1.8, -3.0) (1.8, 2.0) k (1 to 2) 0.735730 2.121773 2.090087
‡ ‡ ‡
k (2 to 1) 0.745095 1.784990 8.501233
3
Discussion
The genetic toggle switch has been studied by deterministic and stochastic equations, but only cases for one or two stable states have been considered [1,3]. We showed that much richer phenomena including switching among three states are expected when the frequency of the DNA status change is slow. We carried out the energy landscape analyses of the switching rate by using the transition state idea. In further studies validity of the energy landscape picture should be examined by checking whether the prefactor directly calculated from the MC data without employing the transition state theory is consistent with Table 1.
References
[1] Gardner, T. S., Cantor C. R. and Collins, J. J., Construction of a genetic toggle switch in Escherichia coli, Nature, 403:339-342, 2000 [2]Gillespie, D.T., Exact Stochastic Simulation of Coupled Chemical Reactions, J. Phys. Chem, 81(25):2340-2361, 1977. [3] Kepler, T. B. and Elston C., Stochasticity in Transcriptional Regulation: Origins, Consequences, and Mathematical Representations, Biophys. J., 81:3116-3136, 2001.