| 1. |
- Baker, J., et al.
(author)
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On the establishment of a mutant
- 2020
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 80, s. 1733-1757
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Journal article (peer-reviewed)abstract
- How long does it take for an initially advantageous mutant to establish itself in a resident population, and what does the population composition look like then? We approach these questions in the framework of the so called Bare Bones evolution model (Klebaner et al. in J Biol Dyn 5(2):147-162, 2011. https://doi.org/ 10.1080/ 17513758.2010.506041) that provides a simplified approach to the adaptive population dynamics of binary splitting cells. As the mutant population grows, cell division becomes less probable, and it may in fact turn less likely than that of residents. Our analysis rests on the assumption of the process starting from resident populations, with sizes proportional to a large carrying capacity K. Actually, we assume carrying capacities to be a(1)K and a(2)K for the resident and the mutant populations, respectively, and study the dynamics for K -> infinity. We find conditions for the mutant to be successful in establishing itself alongside the resident. The time it takes turns out to be proportional to log K. We introduce the time of establishment through the asymptotic behaviour of the stochastic nonlinear dynamics describing the evolution, and show that it is indeed 1/rho log K, where rho is twice the probability of successful division of the mutant at its appearance. Looking at the composition of the population, at times 1/rho log K + n, n is an element of Z(+), we find that the densities (i.e. sizes relative to carrying capacities) of both populations follow closely the corresponding two dimensional nonlinear deterministic dynamics that starts at a random point. We characterise this random initial condition in terms of the scaling limit of the corresponding dynamics, and the limit of the properly scaled initial binary splitting process of the mutant. The deterministic approximation with random initial condition is in fact valid asymptotically at all times 1/rho log K + n with n is an element of Z.
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| 2. |
- Chigansky, P., et al.
(author)
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What can be observed in real time PCR and when does it show?
- 2018
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 76:3, s. 679-695
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Journal article (peer-reviewed)abstract
- Real time, or quantitative, PCR typically starts from a very low concentration of initial DNA strands. During iterations the numbers increase, first essentially by doubling, later predominantly in a linear way. Observation of the number of DNA molecules in the experiment becomes possible only when it is substantially larger than initial numbers, and then possibly affected by the randomness in individual replication. Can the initial copy number still be determined? This is a classical problem and, indeed, a concrete special case of the general problem of determining the number of ancestors, mutants or invaders, of a population observed only later. We approach it through a generalised version of the branching process model introduced in Jagers and Klebaner (J Theor Biol 224(3):299-304, 2003. doi: 10.1016/S0022-5193(03) 001668), and based on Michaelis-Menten type enzyme kinetical considerations from Schnell and Mendoza (J Theor Biol 184(4):433-440, 1997). A crucial role is played by the Michaelis-Menten constant being large, as compared to initial copy numbers. In a strange way, determination of the initial number turns out to be completely possible if the initial rate v is one, i.e all DNA strands replicate, but only partly so when v < 1, and thus the initial rate or probability of succesful replication is lower than one. Then, the starting molecule number becomes hidden behind a "veil of uncertainty". This is a special case, of a hitherto unobserved general phenomenon in population growth processes, which will be adressed elsewhere.
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| 3. |
- Fan, J. Y., et al.
(author)
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Convergence of the age structure of general schemes of population processes
- 2020
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In: Bernoulli. - : Bernoulli Society for Mathematical Statistics and Probability. - 1350-7265. ; 26:2, s. 893-926
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Journal article (peer-reviewed)abstract
- We consider a family of general branching processes with reproduction parameters depending on the age of the individual as well as the population age structure and a parameter K, which may represent the carrying capacity. These processes are Markovian in the age structure. In a previous paper (Proc. Steklov Inst. Math. 282 (2013) 90-105), the Law of Large Numbers as K -> infinity was derived. Here we prove the central limit theorem, namely the weak convergence of the fluctuation processes in an appropriate Skorokhod space. We also show that the limit is driven by a stochastic partial differential equation.
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| 4. |
- Hamza, K., et al.
(author)
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On the establishment, persistence, and inevitable extinction of populations
- 2016
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In: Journal of Mathematical Biology. - : Springer Science and Business Media LLC. - 0303-6812 .- 1432-1416. ; 72:4, s. 797-820
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Journal article (peer-reviewed)abstract
- Comprehensive models of stochastic, clonally reproducing populations are defined in terms of general branching processes, allowing birth during maternal life, as for higher organisms, or by splitting, as in cell division. The populations are assumed to start small, by mutation or immigration, reproduce supercritically while smaller than the habitat carrying capacity but subcritically above it. Such populations establish themselves with a probability wellknown from branching process theory. Once established, they grow up to a band around the carrying capacity in a time that is logarithmic in the latter, assumed large. There they prevail during a time period whose duration is exponential in the carrying capacity. Even populations whose life style is sustainble in the sense that the habitat carrying capacity is not eroded but remains the same, ultimately enter an extinction phase, which again lasts for a time logarithmic in the carrying capacity. However, if the habitat can carry a population which is large, say millions of individuals, and it manages to avoid early extinction, time in generations to extinction will be exorbitantly long, and during it, population composition over ages, types, lineage etc. will have time to stabilise. This paper aims at an exhaustive description of the life cycle of such populations, from inception to extinction, extending and overviewing earlier results. We shall also say some words on persistence times of populations with smaller carrying capacities and short life cycles, where the population may indeed be in danger in spite of not eroding its environment.
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| 5. |
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| 6. |
- Jagers, Peter, 1941, et al.
(author)
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Random variation and concentration effects in PCR
- 2003
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In: J. Theoret. Biol. 224, 299-304 (2003). - 0022-5193 .- 1095-8541. ; 224, s. 299-304
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Journal article (peer-reviewed)abstract
- Even though the efficiency of the PCR reaction decreases, analyses are made in terms of Galton-Watson processes, or simple deterministic models with constant replication probability (efficiency).Recently Schnell and Mendoza have suggested that the form of the efficiency can be derived from enzyme kinetics. This results in the sequence of molecules numbers forming a stochastic process with the properties of a branching process with population size dependence, which is supercritical, but has a mean reproductionnumber that approaches one. Such processes display ultimate linear growth, after an initial exponential phase, as is the case in PCR. It is also shown that the resulting stochastic process for a large Michaelis Menten constant behaves like the deterministic sequence x_n arising by iterations of the function f(x) = x+x/(1+x).
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