Dimension reduction for stochastic dynamical systems forced onto a manifold by large drift: a constructive approach with examples from theoretical biology

Parsons, Todd L.; Rogers, Tim

doi:10.1088/1751-8121/aa86c7

Cited by 36 publications

(41 citation statements)

References 68 publications

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“…The full calculation is too long to reproduce here, however a clear and coherent explanation is given in Ref. [51]. There it is shown that if a system has M variables and a one-dimensional slow-subspace, then the equation for the reduced/effective dynamics can be expressed by…”

Section: Revealing Noise-induced Selection Via Fast Variable Eliminationmentioning

confidence: 91%

Exploiting Fast-Variables to Understand Population Dynamics and Evolution

Constable

McKane

2017

J Stat Phys

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We describe a continuous-time modelling framework for biological population dynamics that accounts for demographic noise. In the spirit of the methodology used by statistical physicists, transitions between the states of the system are caused by individual events while the dynamics are described in terms of the time-evolution of a probability density function. In general, the application of the diffusion approximation still leaves a description that is quite complex. However, in many biological applications one or more of the processes happen slowly relative to the system's other processes, and the dynamics can be approximated as occurring within a slow low-dimensional subspace. We review these time-scale separation arguments and analyse the more simple stochastic dynamics that result in a number of cases. We stress that it is important to retain the demographic noise derived in this way, and emphasise this point by showing that it can alter the direction of selection compared to the prediction made from an analysis of the corresponding deterministic model.

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Section: Revealing Noise-induced Selection Via Fast Variable Eliminationmentioning

confidence: 91%

Exploiting Fast-Variables to Understand Population Dynamics and Evolution

Constable

McKane

2017

J Stat Phys

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“…Then using methods outlined in recent work (Constable et al. ; Parsons and Rogers ), which built off of standard techniques (Katzenberg ; Berglund and Gentz ), the slow timescale dynamics of p are governed by the stochastic differential equation (SDE)

\begin{matrix} true \\ right d p & = & left \underset{a false(p false)}{\underset{︸}{(ν false(1 - 2 p false) - ε β false(x_{p} false) p false(1 - p false))}} d t \\ left + {(\underset{σ^{2} (p)}{\underset{︸}{\frac{p (1 - p)}{normalΩ} R false(x_{p} false)}})}^{1 / 2} d W_{t}, \end{matrix}

…”

Section: Modelmentioning

confidence: 99%

“…Therefore, we wish to reduce it to a more tractable form. To do so, we observe that since selection is weak and mutations are rare, system (1) admits what is commonly referred to as a "slow manifold" (Berglund and Gentz 2006;Parsons and Rogers 2017), which in this case is a curve in (x, y 1 , y 2 )-space satisfying β(x) = μ(x). The reason that this curve is referred to as a slow manifold is that in the vicinity of this curve, the per capita growth rate of the different pathogen strains is very small (since mutations are rare and selection is weak), and so changes in the composition of the pathogen population occur slowly.…”

Section: Modelmentioning

confidence: 99%

“…Let p ≡ y 1 /(y 1 + y 2 ) be the proportion of infected hosts that are infected with strain 1. Then using methods outlined in recent work (Constable et al 2016;Parsons and Rogers 2017), which built off of standard techniques (Katzenberg 1991;Berglund and Gentz 2006), the slow timescale dynamics of p are governed by the stochastic differential equation…”

Section: Modelmentioning

confidence: 99%

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Why is sterility virulence most common in sexually transmitted infections? Examining the role of epidemiology

McLeod

Day

2019

Evolution

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Sterility virulence, or the reduction in host fecundity due to infection, occurs in many host–pathogen systems. Notably, sterility virulence is more common for sexually transmitted infections (STIs) than for directly transmitted pathogens, while other forms of virulence tend to be limited in STIs. This has led to the suggestion that sterility virulence may have an adaptive explanation. By focusing upon finite population models, we show that the observed patterns of sterility virulence can be explained by consideration of the epidemiological differences between STIs and directly transmitted pathogens. In particular, when pathogen transmission is predominantly density invariant (as for STIs), and mortality is density dependent, sterility virulence can be favored by demographic stochasticity, whereas if pathogen transmission is predominantly density dependent, as is common for most directly transmitted pathogens, sterility virulence is disfavored. We show these conclusions can hold even if there is a weak selective advantage to sterilizing.

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“…An SDE for heteroplasmy forced onto the steady state line in the high-churn limit It has been demonstrated that SDE descriptions of stochastic systems which possess a globally-attracting line of steady states may be formed in the long-time limit by forcing the state variables onto the steady state line (Constable et al, 2016;Parsons and Rogers, 2017). Such descriptions may be formed in terms of a parameter which traces out the position on the steady state line, hence reducing a high-dimensional problem into a single dimension (Constable et al, 2016;Parsons and Rogers, 2017). In our case, heteroplasmy is a suitable parameter to trace out the position on the steady state line.…”

Section: Fokker-planck Equation For Chemical Reaction Networkmentioning

confidence: 99%

Mitochondrial network state scales mtDNA genetic dynamics

Aryaman

Bowles

Jones

et al. 2018

Preprint

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Mitochondrial DNA (mtDNA) mutations cause severe congenital diseases but may also be associated with healthy aging. MtDNA is stochastically replicated and degraded, and exists within organelles which undergo dynamic fusion and fission. The role of the resulting mitochondrial networks in the time evolution of the cellular proportion of mutated mtDNA molecules (heteroplasmy), and cell-to-cell variability in heteroplasmy (heteroplasmy variance), remains incompletely understood. Heteroplasmy variance is particularly important since it modulates the number of pathological cells in a tissue. Here, we provide the first wide-reaching theoretical framework which bridges mitochondrial network and genetic states. We show that, under a range of conditions, the (genetic) rate of increase in heteroplasmy variance and de novo mutation are proportionally modulated by the (physical) fraction of unfused mitochondria, independently of the absolute fission-fusion rate. In the context of selective fusion, we show that intermediate fusion/fission ratios are optimal for the clearance of mtDNA mutants. Our findings imply that modulating network state, mitophagy rate and copy number to slow down heteroplasmy dynamics when mean heteroplasmy is low could have therapeutic advantages for mitochondrial disease and healthy aging.

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Dimension reduction for stochastic dynamical systems forced onto a manifold by large drift: a constructive approach with examples from theoretical biology

Cited by 36 publications

References 68 publications

Exploiting Fast-Variables to Understand Population Dynamics and Evolution

Exploiting Fast-Variables to Understand Population Dynamics and Evolution

Why is sterility virulence most common in sexually transmitted infections? Examining the role of epidemiology

Mitochondrial network state scales mtDNA genetic dynamics

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