The splitting approach is developed for the numerical simulation of genetic regulatory networks with a stable steady-state structure. The numerical results of the simulation of a one-gene network, a two-gene network, and a p53-mdm2 network show that the new splitting methods constructed in this paper are remarkably more effective and more suitable for long-term computation with large steps than the traditional general-purpose Runge-Kutta methods. The new methods have no restriction on the choice of stepsize due to their infinitely large stability regions.
The exploration of mechanisms of gene expression and regulation has become one of the central themes in medicine and biological sciences such as cell biology, molecular biology, and systems biology [
Geometric numerical integration aims at solving differential equations effectively while preserving the geometric properties of the exact flow [
Splitting is one of the effective techniques in geometric integration. For example, Blanes and Moan [
An
In particular, we are concerned in this paper with the following two simple models.
(I) The first model is a one-gene regulatory network which can be written as
(II) The second model is a two-gene cross-regulatory network [
Another model we are interested in is for the damped oscillation of the p53-mdm2 regulatory pathway which is given by (see [
Either the mRNA-protein network ( The system ( The steady state
The most frequently used algorithms for the system (
which we denote as RK4 and RK3/8, respectively.
Splitting methods have been proved to be an effective approach to solve ODEs. The main idea is to split the vector field into two or more integrable parts and treat them separately. For a concise account of splitting methods, see Chapter II of Hairer et al. [
Suppose that the vector field
(i) The method defined by
(ii) The Strang splitting is the following symmetric version (see [
(iii) The general splitting method has the form
Theorem 5.6 in Chapter II of Hairer et al. [
However, in most occasions, the exact flows
For a given genetic regulatory network, different ways of decomposition of the vector field
In order to examine the numerical behavior of the new splitting methods Split(Exact:RK4) and Split(Exact:RK3/8), we apply them to the three models presented in Section
Table
Parameter values for the one-gene network.
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In order to apply the splitting methods Split(Exact:RK4) and Split(Exact:RK3/8), the vector field of the system (
The system is solved on the time interval
One-gene network: average errors for different stepsizes.
Stepsize | RK4 | RK3/8 | Split(Exact:RK4) | Split(Exact:RK3/8) |
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1.2 |
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1.5 |
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2.0 |
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10.0 |
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Then we solve the problem with a fixed stepsize
One-gene network: average errors for fixed stepsize
Time interval | RK4 | RK3/8 | Split(Exact:RK4) | Split(Exact:RK3/8) |
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NaN | NaN |
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NaN | NaN |
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Table
Parameter values for the two-gene network.
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For this system, the decomposition (
The system is solved on the time interval
Two-gene network: average errors for different stepsizes.
Stepsize | RK4 | RK3/8 | Split(Exact:RK4) | Split(Exact:RK3/8) |
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0.1 |
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1.2 |
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1.5 |
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2.0 |
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5.0 |
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Then we solve the problem with a fixed stepsize
Two-gene network: average errors for fixed stepsize
Time interval | RK4 | RK3/8 | Split(Exact:RK4) | Split(Exact:RK3/8) |
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NaN |
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NaN | NaN |
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NaN | NaN |
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Table
Parameter values for the p53-mdm2 pathway.
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For this system, decomposition (
The system is solved on the time interval
p53-mdm2 network: average errors for different stepsizes.
Stepsize | RK4 | RK3/8 | Split(Exact:RK4) | Split(Exact:RK3/8) |
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0.05 |
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0.08 |
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0.10 | NaN |
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0.12 | NaN | NaN |
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5.00 | NaN | NaN |
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Then we solve the problem with a fixed stepsize
p53-mdm2 network: average errors for fixed stepsize
Time interval | RK4 | RK3/8 | Split(Exact:RK4) | Split(Exact:RK3/8) |
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NaN | NaN |
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NaN | NaN |
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NaN | NaN |
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NaN | NaN |
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NaN | NaN |
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In this paper we have developed a new type of splitting algorithms for the simulation of genetic regulatory networks. The splitting technique has taken into account the special structure of the linearizing decomposition of the vector field. From the results of numerical simulation of Tables
We conclude that, for genetic regulatory networks with an asymptotically stable steady state, compared with the traditional Runge-Kutta, the new splitting methods have two advantages. They are extremely accurate for large steps. This promises high efficiency for solving large-scale systems (complex networks containing a large number of distinct proteins) in a long-term simulation. They work effectively for very long time intervals. This makes it possible for us to explore the long-run behavior of genetic regulatory network which is important in the research of gene repair and gene therapy.
The special structure of the new splitting methods and their remarkable stability property (see Appendix) are responsible for the previous two advantages.
The splitting methods designated in this paper have opened a novel approach to effective simulation of the complex dynamical behaviors of genetic regulatory network with a characteristic structure. It is still possible to enhance the effectiveness of the new splitting methods. For example, higher-order splitting methods can be obtained by recursive composition (
The genetic regulatory networks considered in this paper are nonstiff. For stiff systems (whose Jacobian possesses eigenvalues with large negative real parts or with purely imaginary eigenvalues of large modulus), the previous techniques suggested by the reviewer are applicable. Moreover, the error control technique which can increase the efficiency of the methods is an interesting theme for future work.
There are more qualitative properties of the genetic regulatory networks that can be taken into account in the designation of simulation algorithms. For example, oscillation in protein levels is observed in most regulatory networks. Symmetric and symplectic methods have been shown to have excellent numerical behavior in the long-term integration of oscillatory systems even if they are not Hamiltonian systems. A brief account of symmetric and symplectic extended Runge-Kutta-Nyström (ERKN) methods for oscillatory Hamiltonian systems and two-step ERKN methods can be found, for instance, in Yang et al. [
Finally, a problem related to this work remains open. We observe that, in Tables
Stability analysis is a necessary step for a new numerical method before it is put into practice. Numerically unstable methods are completely useless. In this appendix, we examine the linear stability of the new splitting method constructed in Section
The region in the
By simple computation, we obtain the stability function of an RK method
The stability regions of RK4 and RK3/8 are presented in Figure
(a) Stability region of RK4 (left) and (b) stability region of RK3/8 (right).
(a) Stability region of Split(Exact:RK4) (left) and (b) stability region of Split(Exact:RK3/8) (right).
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the anonymous referees for their invaluable comments and constructive suggestions which help greatly to improve this paper. This work was partially supported by NSFC (Grant no. 11171155) and the Fundamental Research Fund for the Central Universities (Grant no. Y0201100265).