arXiv subjects

Tomislav Plesa

Publications and source records attributed to Tomislav Plesa.

12 recordsLinked to original sources

arXiv2026

Recurrent neural chemical reaction networks trained to switch dynamical behaviours through learned bifurcations

Both natural and synthetic chemical systems not only exhibit a range of non-trivial dynamics, but also transition between qualitatively different dynamical behaviours as environmental parameters change. Such transitions are called bifurcations. Here, we show that recurrent neural chemical reaction networks (RNCRNs), a class of chemical reaction networks based on recurrent artificial neural networks that can be trained to reproduce a given dynamical behaviour, can also be trained to exhibit bifurcations. First, we show that RNCRNs can inherit some bifurcations defined by smooth ordinary differential equations (ODEs). Second, we demonstrate that the RNCRN can be trained to infer bifurcations that allow it to approximate different target behaviours within different regions of parameter space, without explicitly providing the bifurcation itself in the training. These behaviours can be specified using target ODEs that are discontinuous with respect to the parameters, or even simply by specifying certain desired dynamical features in certain regions of the parameter space. To achieve the latter, we introduce an ODE-free algorithm for training the RNCRN to display designer oscillations, such as a heart-shaped limit cycle or two coexisting limit cycles.

q-bio.MN↗

arXiv2026

Simple chemical systems with chaos

A number of simple chaotic three-dimensional dynamical systems (DSs) with quadratic polynomials on the right-hand sides are reported in the literature, containing exactly 5 or 6 monomials of which only 1 or 2 are quadratic. However, none of these simple systems are chemical dynamical systems (CDSs) - a special subset of polynomial DSs that model the dynamics of mass-action chemical reaction networks (CRNs). In particular, only a small number of three-dimensional quadratic CDSs with chaos are reported, all of which have at least 9 monomials and at least 3 quadratics, with CRNs containing at least 7 reactions and at least 3 quadratic ones. To bridge this gap, in this paper we prove some basic properties of chaotic CDSs, including that those in three dimensions have at least 6 monomials, at least one of which is negative and quadratic. We then use these results to computationally find 20 chaotic three-dimensional CDSs with 6 monomials and as few as 4 quadratics, or 7 monomials and as few as 2 quadratics. At the CRN level, some of these systems have 4 reactions of which only 3 are quadratic, or 5 reactions with only 2 being quadratic. These results quantify structural complexity of chaotic CDSs, and indicate that they are ubiquitous.

nlin.CD↗

arXiv2025

Chemical systems with chaos

Three-dimensional polynomial dynamical systems (DSs) can display chaos with various properties already in the quadratic case with only one or two quadratic monomials. In particular, one-wing chaos is reported in quadratic DSs with only one quadratic monomial, while two-wing and hidden chaos in quadratic DSs with only two quadratic monomials. However, none of the reported DSs can be realized with chemical reactions. To bridge this gap, in this paper, we investigate chaos in chemical dynamical systems (CDSs) - a subset of polynomial DSs that can model the dynamics of mass-action chemical reaction networks. To this end, we develop a fundamental theory for mapping polynomial DSs into CDSs of the same dimension and with a reduced number of non-linear terms. Applying this theory, we show that, under suitable robustness assumptions, quadratic CDSs, and cubic CDSs with only one cubic, can display a rich set of chaotic solutions already in three dimensions. Furthermore, we construct some relatively simple three-dimensional examples, including a quadratic CDS with one-wing chaos and three quadratics, a cubic CDS with two-wing chaos and one cubic, and a quadratic CDS with hidden chaos and five quadratics.

math.DS↗

arXiv2024

Mapping dynamical systems into chemical reactions

Polynomial dynamical systems (DSs) can model a wide range of physical processes. A special subset of these DSs that can model chemical reactions under mass-action kinetics is called chemical dynamical systems (CDSs). A fundamental problem, central to synthetic biology, is to map polynomial DSs into dynamically similar CDSs. In this paper, we introduce the quasi-chemical map (QCM) that can systematically solve this problem. The QCM introduces suitable state-dependent perturbations into any given polynomial DS which then becomes a CDS under sufficiently large translations of variables. This map preserves robust features, such as generic equilibria and limit cycles, and generic bifurcations, as well as some temporal properties, such as periods of oscillations. Furthermore, the resulting CDSs are at most one degree higher than the original DSs. We showcase the QCM by designing relatively simple CDSs with oscillations, chaos and bifurcations, and addressing Hilbert's 16th problem in chemistry.

q-bio.MN↗

arXiv2024

Recurrent neural chemical reaction networks that approximate arbitrary dynamics

Many important phenomena in biochemistry and biology exploit dynamical features such as multi-stability, oscillations, and chaos. Construction of novel chemical systems with such rich dynamics is a challenging problem central to the fields of synthetic biology and molecular nanotechnology. In this paper, we address this problem by putting forward a molecular version of a recurrent artificial neural network, which we call recurrent neural chemical reaction network (RNCRN). The RNCRN uses a modular architecture - a network of chemical neurons - to approximate arbitrary dynamics. We first prove that with sufficiently many chemical neurons and suitably fast reactions, the RNCRN can be systematically trained to achieve any dynamics. RNCRNs with relatively small number of chemical neurons and a moderate range of reaction rates are then trained to display a variety of biologically-important dynamical features. We also demonstrate that such RNCRNs are experimentally implementable with DNA-strand-displacement technologies.

q-bio.MN↗

arXiv2021

Integral feedback in synthetic biology: Negative-equilibrium catastrophe

A central goal of synthetic biology is the design of molecular controllers that can manipulate the dynamics of intracellular networks in a stable and accurate manner. To address the fact that detailed knowledge about intracellular networks is unavailable, integral-feedback controllers (IFCs) have been put forward for controlling molecular abundances. These controllers can maintain accuracy in spite of the uncertainties in the controlled networks. However, this desirable feature is achieved only if stability is also maintained. In this paper, we show that molecular IFCs can suffer from a hazardous instability called negative-equilibrium catastrophe (NEC), whereby all nonnegative equilibria vanish under the action of the controllers, and some of the molecular abundances blow up. We show that unimolecular IFCs do not exist due to a NEC. We then derive a family of bimolecular IFCs that are safeguarded against NECs when uncertain unimolecular networks, with any number of molecular species, are controlled. However, when IFCs are applied on uncertain bimolecular (and hence most intracellular) networks, we show that preventing NECs generally becomes an intractable problem as the number of interacting molecular species increases.

q-bio.MN↗

arXiv2021

Stochastic approximations of higher-molecular by bi-molecular reactions

Biochemical reactions involving three or more reactants, called higher-molecular reactions, play an important role in theoretical systems and synthetic biology. In particular, such reactions underpin a variety of important bio-dynamical phenomena, such as multi-stability/multi-modality, oscillations, bifurcations, and noise-induced effects. However, only reactions with at most two reactants, called bi-molecular reactions, are experimentally feasible. To bridge the gap, in this paper we put forward an algorithm for systematically approximating arbitrary higher-molecular reactions with bi-molecular ones, while preserving the underlying stochastic dynamics. Properties of the algorithm and convergence are established via singular perturbation theory. The algorithm is applied to a variety of higher-molecular biochemical networks, and is shown to play an important role in nucleic-acid-based synthetic biology.

q-bio.MN↗

arXiv2019

Robust control of biochemical reaction networks via stochastic morphing

Synthetic biology is an interdisciplinary field aiming to design biochemical systems with desired behaviors. To this end, molecular controllers have been developed which, when embedded into a pre-existing ambient biochemical network, control the dynamics of the underlying target molecular species. When integrated into smaller compartments, such as biological cells in vivo, or vesicles in vitro, controllers have to be calibrated to factor in the intrinsic noise. In this context, molecular controllers put forward in the literature have focused on manipulating the mean (first moment), and reducing the variance (second moment), of the target species. However, many critical biochemical processes are realized via higher-order moments, particularly the number and configuration of the modes (maxima) of the probability distributions. To bridge the gap, a controller called stochastic morpher is put forward in this paper, inspired by gene-regulatory networks, which, under suitable time-scale separations, morphs the probability distribution of the target species into a desired predefined form. The morphing can be performed at the lower-resolution, allowing one to achieve desired multi-modality/multi-stability, and at the higher-resolution, allowing one to achieve arbitrary probability distributions. Properties of the controller, such as robust perfect adaptation and convergence, are rigorously established, and demonstrated on various examples. Also proposed is a blueprint for an experimental implementation of stochastic morpher.

q-bio.MN↗

arXiv2018

Noise-induced Mixing and Multimodality in Reaction Networks

We analyze a class of chemical reaction networks under mass-action kinetics and involving multiple time-scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks, and consist of a slow-subnetwork, describing conversions among the different gene states, and fast-subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a novel phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast-subnetworks which are `mixed' by the slow-subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

q-bio.MN↗

arXiv2017

Noise Control for DNA Computing

Synthetic biology is a growing interdisciplinary field, with far-reaching applications, which aims to design biochemical systems that behave in a desired manner. With the advancement of strand-displacement DNA computing, a large class of abstract biochemical networks may be physically realized using DNA molecules. Methods for systematic design of the abstract systems with prescribed behaviors have been predominantly developed at the (less-detailed) deterministic level. However, stochastic effects, neglected at the deterministic level, are increasingly found to play an important role in biochemistry. In such circumstances, methods for controlling the intrinsic noise in the system are necessary for a successful network design at the (more-detailed) stochastic level. To bridge the gap, the noise-control algorithm for designing biochemical networks is developed in this paper. The algorithm structurally modifies any given reaction network under mass-action kinetics, in such a way that (i) controllable state-dependent noise is introduced into the stochastic dynamics, while (ii) the deterministic dynamics are preserved. The capabilities of the algorithm are demonstrated on a production-decay reaction system, and on an exotic system displaying bistability. For the production-decay system, it is shown that the algorithm may be used to redesign the network to achieve noise-induced multistability. For the exotic system, the algorithm is used to redesign the network to control the stochastic switching, and achieve noise-induced oscillations.

q-bio.MN↗

arXiv2017

Test Models for Statistical Inference: Two-Dimensional Reaction Systems Displaying Limit Cycle Bifurcations and Bistability

Theoretical results regarding two-dimensional ordinary-differential equations (ODEs) with second-degree polynomial right-hand sides are summarized, with an emphasis on limit cycles, limit cycle bifurcations and multistability. The results are then used for construction of two reaction systems, which are at the deterministic level described by two-dimensional third-degree kinetic ODEs. The first system displays a homoclinic bifurcation, and a coexistence of a stable critical point and a stable limit cycle in the phase plane. The second system displays a multiple limit cycle bifurcation, and a coexistence of two stable limit cycles. The deterministic solutions (obtained by solving the kinetic ODEs) and stochastic solutions (noisy time-series generating by the Gillespie algorithm, and the underlying probability distributions obtained by solving the chemical master equation (CME)) of the constructed systems are compared, and the observed differences highlighted. The constructed systems are proposed as test problems for statistical methods, which are designed to detect and classify properties of given noisy time-series arising from biological applications.

q-bio.MN↗

arXiv2015

Chemical Reaction Systems with a Homoclinic Bifurcation: an Inverse Problem

An inverse problem framework for constructing reaction systems with prescribed properties is presented. Kinetic transformations are defined and analysed as a part of the framework, allowing an arbitrary polynomial ordinary differential equation to be mapped to the one that can be represented as a reaction network. The framework is used for construction of specific two- and three-dimensional bistable reaction systems undergoing a supercritical homoclinic bifurcation, and the topology of their phase spaces is discussed.

math.DS↗