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GODEL Gene ordinary differential equation library

In GODEL, the evolution of gene product concentrations is assumed to be governed by Boolean-Like ODE models that are briefly introduced below. We also provide a simple example illustrating the main features of GODEL.

Boolean-like models of genetic regulatory networks

In the context of Boolean networks, the activation status of gene i, with $i=1,\ldots, n$, is encoded by a binary variable X_i that is 1 if the gene is active and 0 otherwise. The laws governing the activation of gene i are captured by a Boolean rule $T_i(X_1,\ldots, X_n)$. In practice, T_i depends only on the subset of variables $X_1,\ldots, X_n$ corresponding to the genes that control the expression of gene i.

For defining quantitative dynamics of gene expression that incorporate the logical structure of Boolean networks, we follow an approach inspired by [1] and consider ODE models of the form

$\dot x_i=g_i(x)-\gamma_i(x)x_i$

where $i=1,\ldots, n$ denotes the i-th of n genes, $x_i\geq 0$ denotes the concentration of the corresponding product, $x=(x_1,\ldots, x_n)$, $g_i(x)\geq 0$ is a synthesis rate and $\gamma_i(x)\geq 0$ is a degradation function.

In order to account for the discrete regulatory logics, functions b_i(x) and γ_i(x) are typically a combination of switch-like (e.g. sigmoidal) regulatory functions describing the effect of protein j on the expression of gene i and the synthesis of the corresponding protein.

According to [1], each variable X_i is replaced by a monotone, nondecreasing sigmoidal function $\sigma^+:[0,+\infty)\to [0,1]$ of the concentration x_i. Given any two functions τ(x) and τ'(x) representing the Boolean expressions T(x) and T'(X), $\neg T(X)$ is replaced by 1 − τ(x) and $T(X)\wedge T'(X)$ by $\tau(x)\cdot\tau'(x)$. In particular, $\neg X_i$ is represented by σ ⁻ (x_i) = 1 − σ ⁺ (x_i).

As an example, b(x) = σ ⁺ (x₁)σ ⁻ (x₂) and $b(x)=1-\left(1-\sigma^+(x_1)\right)\left(1-\sigma^-(x_2)\right)$ correspond to the Boolean formulas $X_1 \wedge \lnot X_2$ and$X_1 \vee \lnot X_2$, respectively.

Further details on this class of models can be found in [2], [3] and [4].

GODEL features

GODEL offers pre-defined functions for sigmoids commonly used for modelling gene dynamics, namely Hill's function and piecewise multi-affine functions. The main advantage of GODEL is that building a model of a genetic network only amounts to introduce a GODEL block for each gene and connections among regulated genes. The resulting schemes are very similar to networks drawn by hand and this considerably ease the debugging of complex models. As an example, the following genetic network

Gene regulatory network: genes 1,2 and 3 form a repressilator network [5]

corresponds to the GODEL model

GODEL representation of the network above

One can notice that to each gene corresponds a gene block that is available from the GODEL library. Furthermore, GODEL offers the possibility of specifying arbitrary regulation functions b_i and γ_i (as it as been done for gene 6, that depends upon three regulators) as well as of using pre-defined blocks corresponding to the basic Boolean functions AND, OR and NOT (see, e.g., genes 4,5 and 1). Simulation of gene-product concentrations starting from suitable initial concentrations can be done in one click so obtaining the following concentration profiles

Simulation of the network (arbitrary units)

Bibliographic references

[1] E. Plahte, T. Mestl and S. Omholt. (1998). A methodological basis for description and analysis of systems with complex switch-like interactions. J. Math. Biol., 36, 321-348.

[2] R. Porreca, E. Cinquemani, J. Lygeros, and G. Ferrari-Trecate (2010). Identification of genetic network dynamics with unate structure. Bioinformatics, 26(9), 1239-1245.

[3] R. Porreca, S. Drulhe, H. de Jong, and G. Ferrari-Trecate (2008). Structural identification of piecewise-linear models of genetic regulatory networks. J. of Comp. Biol. , 15(10), 1365-1380.

[4] H. de Jong (2002). Modeling and simulation of genetic regulatory systems: A literature review. J. Comput. Biol., 9(1), 69-105.

[5] M. Elowitz and S. Leibler (2000). A synthetic oscillatory network of transcriptional regulators. Nature, 403(6767), 335-338.