Counting generations in birth and death processes with competing Erlang and exponential waiting times
Abstract Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Nature Portfolio
2022-07-01
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| Series: | Scientific Reports |
| Online Access: | https://doi.org/10.1038/s41598-022-14202-0 |
| _version_ | 1811324358655737856 |
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| author | Giulia Belluccini Martín López-García Grant Lythe Carmen Molina-París |
| author_facet | Giulia Belluccini Martín López-García Grant Lythe Carmen Molina-París |
| author_sort | Giulia Belluccini |
| collection | DOAJ |
| description | Abstract Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, then each cell can be imagined as sampling from a probability density of times to division and death. The exponential density is the most mathematically and computationally convenient choice. It has the advantage of satisfying the memoryless property, consistent with a Markov process, but it overestimates the probability of short division times. With the aim of preserving the advantages of a Markovian framework while improving the representation of experimentally-observed division times, we consider a multi-stage model of cellular division and death. We use Erlang-distributed (or, more generally, phase-type distributed) times to division, and exponentially distributed times to death. We classify cells into generations, using the rule that the daughters of cells in generation n are in generation $$n+1$$ n + 1 . In some circumstances, our representation is equivalent to established models of lymphocyte dynamics. We find the growth rate of the cell population by calculating the proportions of cells by stage and generation. The exponent describing the late-time cell population growth, and the criterion for extinction of the population, differs from what would be expected if N steps with rate $$\lambda$$ λ were equivalent to a single step of rate $$\lambda /N$$ λ / N . We link with a published experimental dataset, where cell counts were reported after T cells were transferred to lymphopenic mice, using Approximate Bayesian Computation. In the comparison, the death rate is assumed to be proportional to the generation and the Erlang time to division for generation 0 is allowed to differ from that of subsequent generations. The multi-stage representation is preferred to a simple exponential in posterior distributions, and the mean time to first division is estimated to be longer than the mean time to subsequent divisions. |
| first_indexed | 2024-04-13T14:14:00Z |
| format | Article |
| id | doaj.art-27fd6d7efb8547b89c1ca1f5f9aa87f5 |
| institution | Directory Open Access Journal |
| issn | 2045-2322 |
| language | English |
| last_indexed | 2024-04-13T14:14:00Z |
| publishDate | 2022-07-01 |
| publisher | Nature Portfolio |
| record_format | Article |
| series | Scientific Reports |
| spelling | doaj.art-27fd6d7efb8547b89c1ca1f5f9aa87f52022-12-22T02:43:43ZengNature PortfolioScientific Reports2045-23222022-07-0112112010.1038/s41598-022-14202-0Counting generations in birth and death processes with competing Erlang and exponential waiting timesGiulia Belluccini0Martín López-García1Grant Lythe2Carmen Molina-París3School of Mathematics, University of LeedsSchool of Mathematics, University of LeedsSchool of Mathematics, University of LeedsSchool of Mathematics, University of LeedsAbstract Lymphocyte populations, stimulated in vitro or in vivo, grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, then each cell can be imagined as sampling from a probability density of times to division and death. The exponential density is the most mathematically and computationally convenient choice. It has the advantage of satisfying the memoryless property, consistent with a Markov process, but it overestimates the probability of short division times. With the aim of preserving the advantages of a Markovian framework while improving the representation of experimentally-observed division times, we consider a multi-stage model of cellular division and death. We use Erlang-distributed (or, more generally, phase-type distributed) times to division, and exponentially distributed times to death. We classify cells into generations, using the rule that the daughters of cells in generation n are in generation $$n+1$$ n + 1 . In some circumstances, our representation is equivalent to established models of lymphocyte dynamics. We find the growth rate of the cell population by calculating the proportions of cells by stage and generation. The exponent describing the late-time cell population growth, and the criterion for extinction of the population, differs from what would be expected if N steps with rate $$\lambda$$ λ were equivalent to a single step of rate $$\lambda /N$$ λ / N . We link with a published experimental dataset, where cell counts were reported after T cells were transferred to lymphopenic mice, using Approximate Bayesian Computation. In the comparison, the death rate is assumed to be proportional to the generation and the Erlang time to division for generation 0 is allowed to differ from that of subsequent generations. The multi-stage representation is preferred to a simple exponential in posterior distributions, and the mean time to first division is estimated to be longer than the mean time to subsequent divisions.https://doi.org/10.1038/s41598-022-14202-0 |
| spellingShingle | Giulia Belluccini Martín López-García Grant Lythe Carmen Molina-París Counting generations in birth and death processes with competing Erlang and exponential waiting times Scientific Reports |
| title | Counting generations in birth and death processes with competing Erlang and exponential waiting times |
| title_full | Counting generations in birth and death processes with competing Erlang and exponential waiting times |
| title_fullStr | Counting generations in birth and death processes with competing Erlang and exponential waiting times |
| title_full_unstemmed | Counting generations in birth and death processes with competing Erlang and exponential waiting times |
| title_short | Counting generations in birth and death processes with competing Erlang and exponential waiting times |
| title_sort | counting generations in birth and death processes with competing erlang and exponential waiting times |
| url | https://doi.org/10.1038/s41598-022-14202-0 |
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