PWL models of gene regulatory networks

Context

This application case studies the resolution of OCPs through a direct numerical method in piecewise linear models of gene regulatory networks^[1]. More specifically, we study state transitions between two points of the state space minimizing the nonsmooth cost function

\[J_\lambda(u) = \lambda \int_0^{t_f} |u(t)-1| \, dt + (1-\lambda) t_f\]

where $u(t) \in [u_{\min}, u_{\max}]$ is the external control, $\lambda \in (0,1)$ a fixed parameter, and $t_f$ is the free final time. In the general case, the concentration of the $i$-th gene is described by a dynamical equation of the form

\[\dot{x}_i = -\gamma_i x_i + k_i s^{\pm}(x_j,\theta_j),\]

where the positive constants $\gamma_i$, $k_i$ correspond, respectively, to the degradation and the production rates of $x_i$, and the gene expression rate $s^{\pm}$ is a piecewise constant function defined as:

\[ s^+(x, \theta) = \left\{ \begin{array}{ll} 0 \quad \textit{if } x < \theta, \\ 1 \quad \textit{if } x > \theta, \end{array} \right. \quad s^-(x, \theta) = 1 - s^+(x, \theta) = \left\{ \begin{array}{ll} 1 \quad \textit{if } x < \theta, \\ 0 \quad \textit{if } x > \theta, \end{array} \right.\]

where $s^-$ models an inhibiting effect, $s^+$ a catalyzing effect, and $\theta$ represents in both cases a threshold for transcriptional repression or activation, respectively.

In order to solve this numerical OCP, there are two difficulties:

The hybrid nature of the dynamics.
The non-smoothness of the $L^1$ Lagrangian cost.

The latter can be tackled through a regularization scheme to be detailed in next section.

Regularization strategies

Two regularization strategies are compared: through Hill functions, and through exponential functions, both depending on a parameter $k \in \mathbb{N}$. Each case is detailed in the following table.

Function	Hill	Exponential
$s^+(x, \theta)$	$\frac{x^k}{x^k + \theta^k}$	$1 - \frac{1}{1 + e^{k(x-\theta)}}$
$\mid u-1 \mid$	$(u-1) \frac{u^k - 1}{u^k + 1}$	$(u-1) \left[ 1 - \frac{2}{1 + e^{k(u-1)}} \right]$

A comparison for low values of $k$, for $s^+(x, \theta)$:

using Plots
θ = 2
k = 10
x = range(0, 4, length=100)
y1 = (x .> θ)
y2 = x.^k ./ (x.^k .+ θ^k)
y3 = 1 .- 1 ./ (1 .+ exp.(k*(x.-θ)))
plot(x, [y1, y2, y3], label=["s⁺" "Hill" "Exponential"], xlabel="x")

and for $|u-1|$:

k = 5
u = range(0, 2, length=100)
y1 = abs.(u .- 1)
y2 = (u .- 1).*(u.^k .- 1)./(u.^k .+ 1)
y3 = (u .- 1).*(1 .- 2 ./ (1 .+ exp.(k .* (u .- 1))))
plot(u, [y1, y2, y3], label=["|u-1|" "Hill" "Exponential"], xlabel="u")

Reproducibility

The documentation of this package was built using these direct dependencies,

Status `~/work/PWLdynamics.jl/PWLdynamics.jl/docs/Project.toml`
  [e30172f5] Documenter v1.13.0
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  [3a975481] PWLdynamics v0.3.3 `~/work/PWLdynamics.jl/PWLdynamics.jl`
  [91a5bcdd] Plots v1.40.14

and using this machine and Julia version.

Julia Version 1.11.5
Commit 760b2e5b739 (2025-04-14 06:53 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 4 × AMD EPYC 7763 64-Core Processor
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  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
Environment:
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A more complete overview of all dependencies and their versions is also provided.

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Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

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1Agustín G. Yabo, Nicolas Augier. On L¹ and time-optimal state transitions in piecewise linear models of gene-regulatory networks. Preprint. 2024. https://hal.science/hal-04820387.