Modeling delayed processes in biological systems (original) (raw)
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A Report on the Use of Delay Differential Equations In Numerical Modelling In the Biosciences
Numerical Analysis Report, 1999
We review the application of numerical techniques to investigate mathematical models of phenomena in the biosciences using delay di erential equations. We show that there are prima facie reasons for using such models: (i) they have a richer mathematical framework (compared with ordinary di erential equations) for the analysis of biosystems dynamics, (ii) they display better consistency with the nature of the underlying processes and predictive results. We now have suitable computational techniques to treat numerically the emerging models for the biosciences.
Approximation of delays in biochemical systems
Mathematical Biosciences, 2005
In the past metabolic pathway analyses have mostly ignored the effects of time delays that may be due to processes that are slower than biochemical reactions, such as transcription, translation, translocation, and transport. We show within the framework of biochemical systems theory (BST) that delay processes can be approximated accurately by augmenting the original variables and non-linear differential equations with auxiliary variables that are defined through a system of linear ordinary differential equations. These equations are naturally embedded in the structure of S-systems and generalized mass action systems within BST and can be interpreted as linear signaling pathways or cascades. We demonstrate the approximation method with the simplest generic modules, namely single delayed steps with and without feedback inhibition. These steps are representative though, because they are easily incorporated into larger systems. We show that the dynamics of the approximated systems reflects that of the original delay systems well, as long as the systems do not operate in very close vicinity of threshold values where the systems lose stability. The accuracy of approximation furthermore depends on the selected number of auxiliary variables. In the most relevant situations where the systems operate at states away from their critical thresholds, even a few auxiliary variables lead to satisfactory approximations.
Impact of Delay on Predator-Prey Models
Cornell University - arXiv, 2022
Mathematical modeling based on time-delay differential equations is an important tool to study the role of delay in biological systems and to evaluate its impact on the asymptotic behavior of their dynamics. Delays are indeed found in many biological, physical, and engineering systems and are a consequence of the limited speed at which physiological, chemical, or biological processes are transmitted from one place to another. Since real biological systems are always subject to perturbations that are not fully understood or cannot be explicitly modeled, stochastic delay differential systems (SD-DEs) provide a more realistic approximation to these models. In this work, we study the predator-prey system considering three time delay models: one deterministic and two types of stochastic models. Our numerical results show relevant differences in their respective asymptotic behaviors.
A model of regulatory dynamics with threshold-type state-dependent delay
Mathematical Biosciences & Engineering
We model intracellular regulatory dynamics with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time. A general model which is an extension of the classical differential equation models with constant or zero time delays is developed to study the stability of steady state, the occurrence and stability of periodic oscillations in regulatory dynamics. Using the method of multiple time scales, we compute the normal form of the general model and show that the state-dependent diffusion time may lead to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of Hes1 regulatory dynamics are given to illustrate the general results.
Impact of Delay on Stochastic Predator–Prey Models
Symmetry
Ordinary differential equations (ODE) have long been an important tool for modelling and understanding the dynamics of many real systems. However, mathematical modelling in several areas of the life sciences requires the use of time-delayed differential models (DDEs). The time delays in these models refer to the time required for the manifestation of certain hidden processes, such as the time between the onset of cell infection and the production of new viruses (incubation periods), the infection period, or the immune period. Since real biological systems are always subject to perturbations that are not fully understood or cannot be explicitly modeled, stochastic delay differential systems (SDDEs) provide a more realistic approximation to these models. In this work, we study the predator–prey system considering three time-delay models: one deterministic and two types of stochastic models. Our numerical results allow us to distinguish between different asymptotic behaviours depending...
Modelling biochemical networks with intrinsic time delays: a hybrid semi-parametric approach
BMC Systems Biology, 2010
Background: This paper presents a method for modelling dynamical biochemical networks with intrinsic time delays. Since the fundamental mechanisms leading to such delays are many times unknown, non conventional modelling approaches become necessary. Herein, a hybrid semi-parametric identification methodology is proposed in which discrete time series are incorporated into fundamental material balance models. This integration results in hybrid delay differential equations which can be applied to identify unknown cellular dynamics. Results: The proposed hybrid modelling methodology was evaluated using two case studies. The first of these deals with dynamic modelling of transcriptional factor A in mammalian cells. The protein transport from the cytosol to the nucleus introduced a delay that was accounted for by discrete time series formulation. The second case study focused on a simple network with distributed time delays that demonstrated that the discrete time delay formalism has broad applicability to both discrete and distributed delay problems.
Chaos, Solitons & Fractals, 2012
Quasi-stationary approximations are commonly used in order to simplify and reduce the number of equations of genetic circuit models. Protein/protein and protein/DNA binding reactions are considered to occur on much shorter time scale than protein production and degradation processes and often tacitly assumed at a quasi-equilibrium. Taking a biologically inspired, typical, small, abstract, negative feedback, genetic circuit model as study case, we investigate in this paper how different quasi-stationary approximations change the system behaviour both in deterministic and stochastic frameworks. We investigate the consistence between the deterministic and stochastic behaviours of our time-delayed negative feedback genetic circuit model with different implementations of quasi-stationary approximations. Quantitative and qualitative differences are observed among the various reduction schemes and with the underlying microscopic model, for biologically reasonable ranges and combinations of the microscopic model kinetic rates. The different reductions do not behave in the same way: correlations and amplitudes of the stochastic oscillations are not equally captured and the population behaviour is not always in consistence with the deterministic curves.
Delayed differential equations in description of biochemical reactions channels
It is well known that the time evolution of spatially homogeneous mixture composition consisting of molecules that can react through chemical channels can be deterministically described by some set of ordinary differential equations. In [8] the method of generating stochastic simulations of such systems was developed. There is high correspondence between quantitative results obtained by these two methods.
Delay Differential Equations in the Dynamics of Gene Copying
2007
We analyze a model of gene transcription and protein syn- thesis which has been previously presented in the biological lit- erature. The model takes the form of an ODE (ordinary differ- ential equation) coupled to a DDE (delay differential equation), the state variables being concentrations of messenger RNA and protein. The delay is assumed to depend on the concentration of mRNA and is therefore state-dependent. Linear analysis gives a critical time delay beyond which a periodic motion is born in a Hopf bifurcation. Lindstedt's method is applied to the nonlinear system, resulting in closed form approximate expressions for the amplitude and frequency of oscillation.