arXiv subjects

Michael C. Mackey

Publications and source records attributed to Michael C. Mackey.

At least 19 recordsLinked to original sources

arXiv2024

Dynamics of a state-dependent delay-differential equation

We present a detailed study of a scalar differential equation with threshold state-dependent delayed feedback. This equation arises as a simplification of a gene regulatory model. There are two monotone nonlinearities in the model: one describes the dependence of delay on state, and the other is the feedback nonlinearity. Both increasing and decreasing nonlinearities are considered. Our analysis is exhaustive both analytically and numerically as we examine the bifurcations of the system for various combinations of increasing and decreasing nonlinearities. We identify rich bifurcation patterns including Bautin, Bogdanov-Takens, cusp, fold, homoclinic, and Hopf bifurcations whose existence depend on the derivative signs of nonlinearities. Our analysis confirms many of these patterns in the limit where the nonlinearities are switch-like and change their value abruptly at a threshold. Perhaps one of the most surprising findings is the existence of a Hopf bifurcation to a periodic solution when the nonlinearity is monotone increasing and the time delay is a decreasing function of the state variable.

math.DS↗

arXiv2024

Temporal Entropy Evolution in Stochastic and Delayed Dynamics

We review the behaviour of the Gibbs' and conditional entropies in deterministic and stochastic systems and continue to a formulation appropriate for a stochastically perturbed system with delayed dynamics. The underlying question driving these investigations: ``Is the origin of the universally observed unidirectionality of time in our universe connected to the behaviour of entropy?'' We focus on temporal entropic behaviour with a review of previous results in deterministic and stochastic systems. Our emphasis is on the temporal behaviour of the Gibbs' and conditional entropies as these entropies give equilibrium results in concordance with experimental findings. In invertible deterministic systems both entropies are temporally constant as has been well known for decades. The addition of stochastic perturbations (a Wiener process) leads to an indeterminate (either increasing or decreasing) behaviour of the Gibbs' entropy, but the conditional entropy monotonically approaches equilibrium with increasing time. The presence of delays in the dynamics, whether stochastically perturbed or not, leads to situations in which the Gibbs' and conditional entropies evolution can be oscillatory and not monotone, and may not approach equilibrium.

math-ph↗

arXiv2021

Asymptotic (statistical) periodicity in two-dimensional maps

In this paper we give a new sufficient condition for asymptotic periodicity of Frobenius-Perron operator corresponding to two--dimensional maps. The result of the asymptotic periodicity for strictly expanding systems, that is, all eigenvalues of the system are greater than one, in a high-dimensional dynamical systems was already known. Our new theorem enables to apply for the system having an eigenvalue smaller than one. The key idea for the proof is a function of bounded variation defined by line integration. Finally, we introduce a new two-dimensional dynamical system exhibiting the asymptotic periodicity with different periods depending on parameter values, and discuss to apply our theorem to the model.

math.DS↗

arXiv2021

Operon Dynamics with State Dependent Transcription and/or Translation Delays

Transcription and translation retrieve and operationalize gene encoded information in cells. These processes are not instantaneous and incur significant delays. In this paper we study Goodwin models of both inducible and repressible operons with state-dependent delays. The paper provides justification and derivation of the model, detailed analysis of the appropriate setting of the corresponding dynamical system, and extensive numerical analysis of its dynamics. Comparison with constant delay models shows significant differences in dynamics that include existence of stable periodic orbits in inducible systems and multistability in repressible systems. A combination of parameter space exploration, numerics, analysis of steady state linearization and bifurcation theory indicates the likely presence of Shilnikov-type homoclinic bifurcations in the repressible operon model.

math.DS↗

arXiv2021

How Can We Describe Density Evolution Under Delayed Dynamics?

Although the theory of density evolution in maps and ordinary differential equations is well developed, the situation is far from satisfactory in continuous time systems with delay. This paper reviews some of the work that has been done numerically and the interesting dynamics that have emerged, and the largely unsuccessful attempts that have been made to analytically treat the evolution of densities in differential delay equations. We also present a new approach to the problem and illustrate it with a simple example.

math.DS↗

arXiv2020

Randomly switching evolution equations

We present an investigation of stochastic evolution in which a family of evolution equations in $L^1$ are driven by continuous-time Markov processes. These are examples of so-called piecewise deterministic Markov processes (PDMP's) on the space of integrable functions. We derive equations for the first moment and correlations (of any order) of such processes. We also introduce the mean of the process at large time and describe its behaviour. The results are illustrated by some simple, yet generic, biological examples characterized by different one-parameter types of bifurcations.

math.PR↗

arXiv2018

Response of an oscillatory delay differential equation to a periodic stimulus

Periodic hematological diseases such as cyclical neutropenia or cyclical thrombocytopenia, with their characteristic oscillations of circulating neutrophils or platelets, may pose grave problems for patients. Likewise, periodically administered chemotherapy has the unintended side effect of establishing periodic fluctuations in circulating white cells, red cell precursors and/or platelets. These fluctuations, either spontaneous or induced, often have serious consequences for the patient (e.g. neutropenia, anemia, or thrombocytopenia respectively) which exogenously administered cytokines can partially correct. The question of when and how to administer these drugs is a difficult one for clinicians and not easily answered. In this paper we use a simple model consisting of a delay differential equation with a piecewise linear nonlinearity, that has a periodic solution, to model the effect of a periodic disease or periodic chemotherapy. We then examine the response of this toy model to both single and periodic perturbations, meant to mimic the drug administration, as a function of the drug dose and the duration and frequency of its administration to best determine how to avoid side effects.

math.DS↗

arXiv2018

The combined effects of Feller diffusion and transcriptional/translational bursting in simple gene networks

We study a stochastic model of biosynthesis of proteins in generic bacterial operons. The stochasticity arises from two different processes, namely from `bursting' production of either mRNA and/or protein (in the transcription/translation process) and from standard diffusive fluctuations. The amount of protein follows the Feller diffusion, while the bursting introduces random jumps between trajectories of the diffusion process. The combined effect leads to a process commonly known as a diffusion process with jumps. We study existence of invariant densities and the long time behavior of distributions of the corresponding Markov process, proving asymptotic stability in the evolution of the density.

q-bio.MN↗

arXiv2017

Normal and pathological dynamics of platelets in humans

We develop a comprehensive mathematical model of platelet, megakaryocyte, and thrombopoietin dynamics in humans. We show that there is a single stationary solution that can undergo a Hopf bifurcation, and use this information to investigate both normal and pathological platelet production, specifically cyclic thrombocytopenia. Carefully estimating model parameters from laboratory and clinical data, we then argue that a subset of parameters are involved in the genesis of cyclic thrombocytopenia based on clinical information. We provide excellent model fits to the existing data for both platelet counts and thrombopoietin levels by changing six parameters that have physiological correlates. Our results indicate that the primary change in cyclic thrombocytopenia is a major interference with or destruction of the thrombopoietin receptor with secondary changes in other processes, including immune-mediated destruction of platelets and megakaryocyte deficiency and failure in platelet production. This study makes a major contribution to the understanding of the origin of cyclic thrombopoietin as well as significantly extending the modeling of thrombopoiesis.

q-bio.TO↗

arXiv2016

Response of an oscillatory differential delay equation to a single stimulus

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

math.DS↗

arXiv2016

Mathematical model of galactose regulation and metabolic consumption in yeast

The galactose network is a complex system responsible for galactose metabolism. It has been extensively studied experimentally and mathematically at the unicellular level to broaden our understanding of its regulatory mechanisms at higher order species. Although the key molecular players involved in the metabolic and regulatory processes underlying this system have been known for decades, their interactions and chemical kinetics remain incompletely understood. Mathematical models can provide an alternative method to study the dynamics of this network from a quantitative and a qualitative perspective. Here, we employ such approaches to unravel the main properties of the galactose network, including equilibrium binary and temporal responses, as a way to decipher its adaptation to actively-changing inputs. We combine the two main components of the network; namely, the genetic branch which allows for bistable responses, and a metabolic branch, encompassing the relevant metabolic processes and glucose repressive reactions. We use both computational tools to estimate model parameters based on published experimental data, as well as bifurcation analysis to decipher the properties of the system in various parameter regimes. Our model analysis reveals that the interplay between the inducer (galactose) and the repressor (glucose) creates the bistability regime which dictates the temporal responses of the system. Based on the same bifurcation techniques, we can also explain why the system is robust to genetic mutations and molecular instabilities. These findings may provide experimentalists with a theoretical framework upon which they can determine how the galactose network functions under various conditions.

q-bio.MN↗

arXiv2015

Random Numbers from a Delay Equation

Delay differential equations (DDE) can have "chaotic" solutions that can be used to mimic Brownian motion. Since a Brownian motion is random in its velocity, it is reasonable to think that a random number generator (RNG) might be constructed from such a model. In this preliminary study, we consider one specific example of this and show that it satisfies criteria commonly employed in the testing of random number generators (from TestU01's very stringent "Big Crush" battery of tests). A technique termed digit discarding, commonly used in both this generator and physical RNG's using laser feedback systems, is discussed with regard to the maximal Lyapunov exponent. Also, we benchmark the generator to a contemporary common method: the multiple recursive generator, MRG32k3a. Although our method is about 7 times slower than MRG32k3a, there is in principle no apparent limit on the number of possible values that can be generated from the scheme we present here.

nlin.CD↗

arXiv2015

The Limiting Dynamics of a Bistable Molecular Switch With and Without Noise

We consider the dynamics of a population of organisms containing two mutually inhibitory gene regulatory networks, that can result in a bistable switch-like behaviour. We completely characterize their local and global dynamics in the absence of any noise, and then go on to consider the effects of either noise coming from bursting (transcription or translation), or Gaussian noise in molecular degradation rates when there is a dominant slow variable in the system. We show analytically how the steady state distribution in the population can range from a single unimodal distribution through a bimodal distribution and give the explicit analytic form for the invariant stationary density which is globally asymptotically stable. Rather remarkably, the behaviour of the stationary density with respect to the parameters characterizing the molecular behaviour of the bistable switch is qualitatively identical in the presence of noise coming from bursting as well as in the presence of Gaussian noise in the degradation rate. This implies that one cannot distinguish between either the dominant source or nature of noise based on the stationary molecular distribution in a population of cells. We finally show that the switch model with bursting but two dominant slow genes has an asymptotically stable stationary density.

q-bio.MN↗

arXiv2013

Adiabatic reduction of a model of stochastic gene expression with jump Markov process

This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription considered as a jump Markov process. In this model, the process of gene expression with auto-regulation is described by fast/slow dynamics. The production of mRNA is assumed to follow a compound Poisson process occurring at a rate depending on protein levels (the phenomena called bursting in molecular biology) and the production of protein is a linear function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast process (due to faster mRNA degradation than that of protein) we prove that, with appropriate scalings in the burst rate, jump size or translational rate, the bursting phenomena can be transmitted to the slow variable. We show that, depending on the scaling, the reduced equation is either a stochastic differential equation with a jump Poisson process or a deterministic ordinary differential equation. These results are significant because adiabatic reduction techniques seem to have not been rigorously justified for a stochastic differential system containing a jump Markov process. We expect that the results can be generalized to adiabatic methods in more general stochastic hybrid systems.

math.PR↗

arXiv2011

Deterministic Brownian motion generated from differential delay equations

This paper addresses the question of how Brownian-like motion can arise from the solution of a deterministic differential delay equation. To study this we analytically study the bifurcation properties of an apparently simple differential delay equation and then numerically investigate the probabilistic properties of chaotic solutions of the same equation. Our results show that solutions of the deterministic equation with randomly selected initial conditions display a Gaussian-like density for long time, but the densities are supported on an interval of finite measure. Using these chaotic solutions as velocities, we are able to produce Brownian-like motions, which show statistical properties akin to those of a classical Brownian motion over both short and long time scales. Several conjectures are formulated for the probabilistic properties of the solution of the differential delay equation. Numerical studies suggest that these conjectures could be "universal" for similar types of "chaotic" dynamics, but we have been unable to prove this.

nlin.CD↗

arXiv2010

Molecular Distributions in Gene Regulatory Dynamics

We show how one may analytically compute the stationary density of the distribution of molecular constituents in populations of cells in the presence of noise arising from either bursting transcription or translation, or noise in degradation rates arising from low numbers of molecules. We have compared our results with an analysis of the same model systems (either inducible or repressible operons) in the absence of any stochastic effects, and shown the correspondence between behaviour in the deterministic system and the stochastic analogs. We have identified key dimensionless parameters that control the appearance of one or two steady states in the deterministic case, or unimodal and bimodal densities in the stochastic systems, and detailed the analytic requirements for the occurrence of different behaviours. This approach provides, in some situations, an alternative to computationally intensive stochastic simulations. Our results indicate that, within the context of the simple models we have examined, bursting and degradation noise cannot be distinguished analytically when present alone.

q-bio.MN↗

arXiv2008

Dynamics and density evolution in piecewise deterministic growth processes

A new sufficient condition is proved for the existence of stochastic semigroups generated by the sum of two unbounded operators. It is applied to one-dimensional piecewise deterministic Markov processes, where we also discuss the existence of a unique stationary density and give sufficient conditions for asymptotic stability.

math.AP↗

arXiv2008

Central Limit Theorems for Non-Invertible Measure Preserving Maps

We establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator. We apply the result to asymptotically periodic transformations and give an extensive specific example of asymptotically periodic transformations by using the tent map.

math.PR↗