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3.2 Delayed degradation

3.2.1 Simple Case

Delayed degradation

We will start by studying simple stochastic birth and death processes that include delay in the degradation step. A process of this type was proposed in [Bratsun et al., 2005] as a model for protein level dynamics with a complex degradation pathway.


Simple Case

We consider first the simplest possible process including delayed degradation:



τ ∅, (3.1)

that is, a particleXis created at a rateCand disappears (“dies" or “degrades") a timeτafter created. We allow the delay timeτto be randomly distributed i.e. the lifetimesτof the created particles are random variables, that for simplicity we consider independent and identically distributed, with probability density f(τ). Although not considered here, the case of non-identically distributed delay times, in particular a probability density that depends on the time from birth, can also be treated. However, as commented above, the case of non-independent delay times does not seem to be tractable with the methods we present below.

We note first that distributed delay is completely equivalent to degradation at a rate that depends on the “age"a(time form creation) of the particle, i.e. processes


τ Y, and X


−→Y, (3.2)

1Actually, the effective Markovian reduction indeed allows to consider situations in which the completion times are not statistically independent.



are equivalent if the rateγ(a) and the probability density of the delay f(τ) are related by:

γ(a)= f(a) F(a)ˆ

⇒ f(τ)=γ(τ)eR0τdaγ(a), (3.3) with ˆF(t) = 1−F(t) being F(t) = Prob(τ < t) = Rτ

0 dτf(τ) the cumulative distribution of the delay-time. This is so becauseγ(a)dais the probability of dying at the time interval (a,a+da), if the particle is still present ata, and so it is nothing but the probabilityf(a)dathat the delay timeτ belongs to that same interval conditioned to the particle still being alive at timea, an event with probability ˆF(a). In the notation of [Papoulis and Pillai, 2011],γ(a) is nothing but theconditional failure rate. Moreover, a multi-step reactions, with all steps being first-order reactions i.e. their rates do not depend on the state of the system, is also equivalent to a single reaction with some delay distribution, i.e. processes

are equivalent if the probability density of the delay, f(τ), has a Laplace transform given by:


In the particular case that all the rates are equal, this corresponds to a gamma distribution of the form f(τ) = γ(mmτm1)!1eγτ. Note that this "multi-step” procedure is not completely general, since the distributions of delay that can be obtained are always given by (3.5) (which corresponds to a superposition of exponentials). This implies that the variance is bounded respect the mean value, since

Mimicking a delay distribution with a multi-step process is convenient because the process is Markovian and usual methods can be employed [Morelli and Jülicher, 2007]. However we see that the multi-step procedure is not completely general. It is the goal of this chapter to develop methods to analyze this kind of non-Markovian processes.

We taket=0 as the time origin, so the number of alive particles at timetisn(t)=0 fort≤0.

LetP(n,t) the probability ofnparticles being alive at timet. In the remaining of this subsection we assume that there is no feedback, in the sense that the creation rateCis independent on the number of particlesn, but, for the sake of generality, we do allow it to be a function of time C(t). The non-feedback assumption allows us to obtain a full analytical solution. As shown in Appendix 1, independently of the form of the delay distribution, P(n,t) follows a Poisson distribution


n! , (3.7)

with averagehn(t)i=Rt

0 dt0C(t0) ˆF(t−t0). If the creation rate,C(t), is independent of time, a steady state is reached, in which the average number of particles ishnist =Chτi, again independently of the form of the delay distribution.

We will now compute the two-times joint probability distribution, which in particular, allows to derive the time correlation function. We shall see that the analytical expression of the correlation


function does depend on the form of the delay distribution (and not only on the average delay, as we have seen is the case for the average number of particles), which opens the possibility to determine the presence and the form of the delay distribution from macroscopic data (by macroscopic we mean based onnand not on the lifetime of individual particles). We start from the relation:

n(t+T)=nnew(t+T)+nold(t+T), (3.8) withnnew(nold) particles created after(before)t(so, in the followingT>0).nnewcan be computed exactly as before (now takingtas the time origin), so we have:

P(nnew=m,t+T|n(t))=hnnew(t+T)|n(t)im The evolution of the number of particles already present at tdepends on the age aof these particles. Their survival probability until timet+Tcan be written as:

P(alive att+T|alive att)= where we used P(a|b) = P(a;b)P(b) . Since the different particles are independent, nold follows a binomial distribution: statistically independent, we obtain the expression for the two-times probability:

P(n,t+T;n0,t)= Xn


Pold(m,t+T|n0,t)Pnew(n−m,t+T|n0,t)P(n0,t), (3.13)

withP(n0,t) given by (3.7). A more explicit formula is found for the generating function:



If C(t) = C, independent of time, a steady-state can be reached with correlation function Kst[n](T) = limt→∞K[n](t,T). For a constant rate γ, which would be equivalent to an expo-nential delay distribution f(τ)=γeγτ, it has the usual exponential decayKst[n](T)=(C/γ)eγT. For a fixed delay timeτ0, corresponding to f(τ)=δ(τ−τ0), the correlation function is a straight lineKst[n](T)=C(τ0−T) forT< τ0andKst[n](T)=0 forT≥τ0. For other distributions of delay time, the correlation function adopts different forms, but it is always monotonically decreasing.

In figure (3.1) we plot the correlation function for two different types of distribution of delay, for different values of the variance of the delay. We see that the distribution with fatter tail displays a slower asymptotic decay, and that the decay is slower as the variance of the delay increases. Numerical simulations, performed with a conveniently modified version of the Gille-spie algorithm [Cai, 2007] (see Appendix 3 for details about the numerical simulations), are in perfect agreement with this exact result, providing a check of its correctness. We remark that the functional form of the decay of the correlation function depends on the delay distributed and can differ from the exponential decay found in systems without delay.

0 1 2 3 4 5

Figure 3.1: Steady state correlation function, Eq.(3.15), as a function of time, plotted in logarithmic scale, for two different types of delay distribution, gamma and lognormal, for two values of the variance of the delay:σ2τ=0.2 (left panel) andσ2τ=5 (right panel); in both cases the average delay ishτi=1 and the creation rate isC=1. We also plot a exponential decay with exponent one (dot-dashed line), for comparison. Note that delay distributions

with larger variance and fatter tayls display slower asymptotic decay.