Compartmental discretization and simulation of gene regulation networks (GRNs)

Abstract: Gene expression is a fundamental biological process of actually realizing DNA information in the form of proteins in living organisms. It is widely accepted that the stochastic nature of gene expression has to be taken into consideration during quantitative modeling due to uncertainties coming from e.g., both translational and transcriptional bursting. In (Friedman, 2006) a model in partial integro-differential equation (PIDE) form was proposed for the description of the temporal evolution of protein distribution. This model was generalized to multiple dimensions in (Pájaro, 2017). To make numerical solution and analysis easier, a finite volume discretization scheme in the space of molecule numbers was proposed in (Vághy, 2024) which gives a non-homogeneous master equation which is essentially a linear time-varying compartmental model. In this presentation, I will show recent results on the simulation and control of GRNs using the discretized compartmental model form. Existence and uniqueness of equilibria (i.e., stationary distributions) and stability of solutions will be studied using the theory of non-autonomous master equations (see, e.g. Diblík, 2024).

Coauthors: Irene Otero-Muras, Mihály A. Vághy, Manuel Pájaro, Christian Fernandez Perez, Mihály Pituk, Josef Diblík

References:

Friedman, N., Cai, L., & Xie, X. S. (2006). Linking stochastic dynamics to population Distribution: an analytical framework of gene expression. Physical Review Letters, 97(16), 168302.

Pájaro, M., Alonso, A. A., Otero-Muras, I., & Vázquez, C. (2017). Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting. Journal of Theoretical Biology, 421, 51-70.

Vághy, M. A., Otero-Muras, I., Pájaro, M., & Szederkényi, G. (2024). A kinetic finite volume discretization of the multidimensional PIDE model for gene regulatory networks. Bulletin of Mathematical Biology, 86(2), 22.

Diblík, J., Pituk, M., & Szederkényi, G. (2024). Large time behavior of nonautonomous linear differential equations with Kirchhoff coefficients. Automatica, 161, 111473.

chemical biologychemical kineticsalgebraic geometrydynamical systemsprobability

Audience: researchers in the topic

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Seminar on the Mathematics of Reaction Networks

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This seminar series focuses on progress in mathematical theory for the study of reaction networks, mainly in biology and chemistry. The scope is broad and accommodates works arising from dynamical systems, stochastics, algebra, topology and beyond.

We aim at providing a common forum for sharing knowledge and encouraging discussion across subfields. In particular we aim at facilitating interactions between junior and established researchers. These considerations will be represented in the choice of invited speakers and we will strive to create an excellent, exciting and diverse schedule.

The seminar runs approximately every other week on Thursdays, at 17:00 Brussels time (observe that this webpage shows the schedule in your current time zone). Each session consists of two 25-minute talks followed by 5-minute questions. After the two talks, longer discussions will take place for those interested. To this end, we will use breakout rooms. For this to work well, you need to have the latest version of Zoom installed (version 5.3.0 or higher), and use the desktop client or mobile app (not supported on ChromeOS).

We look forward hearing about new work and meeting many of you over zoom! Many of the talks are recorded; to see the recording, from Past Talks, open details of the listed talk for a video link.

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