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In this paper, we make use of semi-Markov processes, where the absolute waiting time distribution from a given state to another is a function of the time since the system entered the first state. The mean time, obtained by stochastic simulations, to produce V virions combining a mixture distribution for G, as defined above (budding events), and a log-normal one for recovery events (instead of Exp(μ)) is given by the filled black circles in Fig. The bimodal distribution is a mixture of two Gaussian distributions centered at times \(\tau _1\) and \(\tau _2\), inspired by the experimental observations discussed in ref.

4b The probability distributions considered for the absolute waiting time are shown in panel (b). (a) From any given state \(i\in \mathscrS\), the absolute waiting times to any state \(j\in A(i)=\i_1,i_2,\ldots ,i_n_i-1\\) are exponentially distributed, \(T_i\to j\sim Exp(\lambda _ij)\), except for at most one state \(i_n_i\). In particular, if the process is Markovian (more precisely, a continuous time Markov chain), then all the absolute inter-event times, \(\T_i\to j,\,j\in A(i)\\), for any event leaving state i, are exponentially distributed.

To show that considering (e.g., biological) processes with the inappropriate assumption that all inter-event times are exponentially distributed can lead to incorrect predictions (e.g., when analysing first passage times). (b) Markovian version of the process \(\mathscrX\), where all events \(i\to j\), with i, \(j\in \mathscrS\), occur with exponentially distributed times \(T_i\to j\sim Exp(\lambda _ij)\). Another biological process where it has been suggested that the Markovian assumption might be too strong is cell proliferation 5 , where Erlang distributions for modelling the inter-event times within the cell cycle might be more appropriate than the exponential distribution 5 It is also important to keep in mind that, when more than one transition is possible out of a given state, experimentally observed distributions of times for a particular transition to occur are always conditioned (also referred to as censored 6 ) on the chosen transition actually taking place.

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A Night at the Theatre Sunday 21st October 2018 – Doors 18:30- Show 19:00 A Night at the Theatre brings together all of Royal Holloway’s Creative Arts societies in the biggest, brightest event of the year. A Night at the Theatre – Saturday 20th October 2018 – Doors 18:00 – Show 19:00 A Night at the Theatre brings together all of Royal Holloway’s Creative Arts societies in the biggest, brightest event of the year. Bel, G., Munsky, B. & Nemenman, I. The simplicity of completion time distributions for common complex biochemical processes.

Computer codes to generate the results derived in this paper are available online at Castro M, López-García M, Lythe G, Molina-París C. First passage events in biological systems with non-exponential inter-event times (computer codes). As a final caveat, it is worth placing our approach in the context of the standard framework provided by ODE models, that are accurate as long as the probability of extinction is small, the population is large and the waiting time distributions of the underlying stochastic process are exponential. 9 , where M = 3. Although the resulting A(z) might be sparse, so that efficient computational techniques could be in principle applied, computational limitations might still arise for significantly large values of N, when approximating more than one general distribution by a phase-type one for many events, or when high-dimensional phase-type approximations were needed (instead of the three-dimensional one used in Fig.

We have shown the relationship between the absolute waiting times to conditioned waiting times and the transition probabilities of the process. (b) Black circles: simulation of case study 2 with a mixture of Gaussian distributions for budding events and a log-normal one for the recovery events. The prediction of the theory captures accurately the simulations and shows the large overestimations arising from different approximations that make use of exponentially distributed waiting times.

10b , the black solid line is the theoretical prediction arising when the log-normal recovery time distribution is substituted by this PH distribution (see Supplementary Information for further details), while keeping the general bimodal distribution G for the budding events. (a) Relative difference in the mean time to produce V = 1 virions between the Markovian hypothesis \((G\equiv Exp(\frac1\tau ))\) and the choice of G as a mixture of Gaussian distributions with different shifts, \(\rm\Delta \), and variances, σ2. We note that values of \(\tau _1\) and \(\tau _2\) are adjusted according to Eq. ( 7 ), so that \(ET_G=\tau \). (b) Plot of the distribution G for q = 0.26. Percentage difference between the mean FPT \(ET_(H,0)(V)\) under the Markovian hypothesis and under general distributions G for each budding event (see Fig.

In the latter, the cell releases one virion at a time after generally distributed times, TG, with common general distribution G, and Laplace-Stieltjes transform \( \mathcal L _G(z)=Ee^-zT_G\). 4a ) and gamma distributions with shape \(K=2,\ldots ,5\) and a deterministic distribution with time δ−1 (see densities in Fig. 5 ), we set the mean time for a phosphorylation event to be the same, \(ET_G_k=\delta ^-1\) independently of k and we set the distribution G (\(G_k\equiv G\) for all k) and consider the case given by Fig.

However, the mean of each transition event is not enough when analysing the mean FPT to state \(\rm\Omega \) if general distributions are considered for scenarios more complex than Fig. In our framework, these four scenarios can be summarised by providing the matrix A(z) in Eq. ( 1 ). We recall that if an absolute waiting time is exponentially distributed, \(T_i\to j\sim Exp(\lambda _ij)\), its Laplace-Stieltjes transform is given by.

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