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In Fig. 16 we presented the age structure of different human populations and for comparison, the mortality curves In the standard Penna model, if Verhulst factor regulates the size of the population only at the level of birth, there are no random deaths later, the mortality curve is in fact of s-shape with very low dea... |
For the oldest part of populations some downward deviation of mortality in comparison with the Gompertz law is observed. This deviation, called plateau, is controversial and according to Stauffer |
it is a result of imperfect demographic data available for the oldest parts of the human populations (nowadays it concerns data from the end of nineteenth century) |
. The results of standard Penna model simulations reproduce the age distribution of the human population in its middle part after the minimum reproduction age, but not the oldest part of populations. The noisy model enables simulation and analysis of parameters which influence the mortality of the youngest individuals.... |
. There is an open question if the heterogeneity observed in the human population is Ageing and the loss of complexity When we consider the ageing processes, usually we are thinking about deterioration of physiological functions of organism which leads to the loss of its adaptive possibilities, improper responses to th... |
. Even in the simplest organisms, several genes and usually much more gene products interplay with each other . They are connected in the complicated metabolic networks with huge number of different feedback regulatory loops. In fact there is no single function, which is not connected with many other functions of organ... |
For the further extension of the Penna model we have used the version with noise described in the above section; the internal ,,personal” fluctuations can be superimposed on the fluctuations of the environment, |
the sum of both fluctuations describes the health status of organism, the fluctuations higher than the assumed limit kill the individual. |
In the new version the random Gaussian fluctuations are characteristic for the normal (wild) version of a gene. Defective genes change their random fluctuations for highly correlated signal of the same energy as the previous fluctuations. The idea is biologically justified, since in many examples the highly synchronize... |
(as in case of Huntington disease). The difference between the new proposed model and the Noisy Penna Model is in behavior of an individual state. In this version we assume that the individual state is a composition of many small fluctuations corresponding to [MATH] particular genes, thus |
[EQUATION] where [EQUATION] Defects are reflected in positive correlation between the switched on defective genes. In other words, if genes [MATH] and [MATH] are defective and switched on then |
[EQUATION] If all [MATH] are independent, then [MATH] , but the correlations between gene functions increase the variation of personal state. The most sharp effect is if the [MATH] . In this case, the variation of [MATH] totally correlated signals is equal to [MATH] |
In Fig. 17 we have compared the results [MATH] noises, it is high enough to determine the higher mortality. Why women live longer than men |
Comparing the mortality curves or age structures of men and women of many nations or ethic groups it could be clearly shown that the life expectancy of men is much lower than that of women. Especially, mortality of men at the middle age is almost 50% higher than the mortality of women at the same age. Stauffer blamed f... |
Assuming these differences in the genome structure of male and females Schneider et al., and Kurdziel et al., analysed the age structure of populations with such a sexual dimorphism. They found, that the differences in mortality of both sexes could be explained by the lack of the second copy of X chromosome in the male... |
Thus, this simple hypothesis of the role of X chromosome has been supported by the simulations. But there is another problem for biologists. Males usually have two sex chromosomes: X and Y. There are some premises that originally these two chromosomes were homologous and in the course of evolution the Y chromosome has ... |
The result of simulations are shown in Fig. 18 . After relatively short time of evolution, the Y chromosome accumulated defective genes and eventually the only genetic information it possessed was the determination of the male sex - the marker glued to the chromosome. The other effect of evolution was lower fraction of... |
Additionally, male after reproducing with one female is going back to the pool of males. Thus, one male individual can reproduce several times during the one time unit while each female can reproduce only once in the time step. This property of panmictic population has been changed - male individual after reproduction ... |
It is still not the whole story of the sex chromosome evolution. In the above versions of the model it has been assumed that the recombination between X and Y chromosomes was switched off from the beginning of the evolution. Biologists are not sure how and when the recombination between these chromosomes was switched o... |
The role of the crossover rate for the strategies of evolution In the standard diploid Penna model recombination between the bitstrings mimics the crossover and usually it is assumed that the frequency of crossover is 1 per bitstrings’ pair during the gamete production. There are some other possibilities of the impleme... |
We have compared the results of two different simulations: with one recombination event between the bitstrings (haplotypes) and without recombination. In the second version, one randomly drawn haplotype after mutation was considered as gamete. The results are shown in Fig. 19 |
The populations evolving without recombination are smaller and they have very high level of defective genes even in the part of the genomes expressed before the reproduction age. The frequency of defects reaches 0.5. The minimum reproduction age was set for 20, thus, assuming the random distribution of defects at this ... |
Acknowledgements Calculations have been carried out in Wrocław Centre for Networking and Supercomputing (), grant #102. Index allele |
§1 autosomes §8 bitstring §1 crossover §1 §8 diploid §1 dominant §1 environment §3 gamete §1 genetic death 2nd item genetic pool |
§2 §3 genome §1 haploid §1 haplotype §1 hemizygous §8 inner state §3 locus (loci) §1 mother care §4 mutation §1 §1 panmictic §8 id1.14.1 |
phenotype §1 recessive §1 recombination §8 rectangularization §3 reversion §1 sex chromosomes §1 theory of ageing §1 Verhulst factor 7th item Weismann’s trap §1 |
# Source: arxiv 0901.2227 # Title: Rare event simulation for T-cell activation # Sections: all # Downloaded: 2026-03-03T05:14:26.663941+00:00 |
Rare Event Simulation for T-cell activation (Received: date / Accepted: date) Rare event simulation for T-cell activation (Received: date / Accepted: date) |
Abstract The problem of statistical recognition is considered, as it arises in immunobiology, namely, the discrimination of foreign antigens against a background of the body’s own molecules. The precise mechanism of this foreign-self-distinction, though one of the major tasks of the immune system, continues to be a fun... |
Keywords: Immunobiology statistical recognition large deviations rare event simulation pacs: 87.16.af 87.16.dr 87.18.Tt MSC: 92-08 92C99 60F10 |
Introduction The notion of statistical recognition between randomly encountered molecules is central to many biological phenomena. This is particularly evident in biological repertoires, which contain enough molecular diversity to bind practically any randomly encountered target molecule. The receptor repertoire of the... |
This ability of the immune system to discriminate safely between foreign and self molecules is a fundamental ingredient to everyday survival of jawed vertebrates; but how this works exactly is still enigmatic. Indeed, the immune system faces an enormous challenge because it must recognise one (or a few) type(s) of (pot... |
(henceforth referred to as BRB) and further developed by Zint, Baake and den Hollander Zint:2008 . It describes (random) encounters between the two crucial types of white blood cells involved (see Figs. and ): the antigen-presenting cells (APCs), which display a mixture of self and foreign antigens at their surface (a ... |
To be biologically more precise, we consider the encounters of so-called naive T-cells with professional APCs in the secondary lymphoid tissue naive T-cell is a cell that has finished its maturation process in the thymus and has been released into the body, where it has not yet been exposed to antigen. It tends to dwel... |
identical copies on the surface of the particular T-cell. A large number (estimated at [MATH] in Arstila:1999 ) of different receptors, and hence different T-cell types, are present in an individual (every type, in turn, is present in several copies, which form a T-cell clone). However, the number of potential antigen ... |
Hunt:1992 Mason:1998 Stevanovic:1999 together with, possibly, one (or a small number of) foreign types; the T-cells therefore face a literal “needle in a haystack” problem. |
For an encounter between a pair of T-cell and APC, both chosen randomly from the diverse pool of T-cells and APCs, the probability to react must be very small (otherwise, immune reactions would occur permanently); this is a central theme in the analysis. It entails that some questions may be answered analytically with ... |
The T-cell model In this Section, we briefly motivate and introduce the model of T-cell recognition as first proposed by BRB in 2001 Berg:2001 and further developed by Zint, Baake and den Hollander Zint:2008 . More precisely, we only consider the toy version of this model, which neglects the modification of the T-cell ... |
When T-cells and APCs meet, the T-cell receptors bind to the various antigens presented by the APC Davis:2003dg . For every single receptor-antigen pair, there is an association-dissociation reaction, the rate constants for which depend on the match of the molecular structures of receptor and antigen. Assuming that ass... |
Every time a receptor unbinds from an antigen, it sends a signal to the T-cell, provided the association has lasted for at least one time unit (i.e., we rescale time so that the unit of time is this minimal association time required). The duration of a binding of a given receptor-antigen pair follows the [MATH] distrib... |
[EQUATION] i.e., the dissociation rate times the probability that the association has lasted long enough. (If the simplifying assumption of unlimited receptor abundance is dispensed with, Eq. ( ) must be modified, see |
Berg:2003a .) As shown in Fig. the function [MATH] first increases and then decreases with [MATH] with a maximum at [MATH] , which reflects the fact that, for [MATH] , the bindings tend not to last long enough, whereas for [MATH] , they tend to last so long that only few stimuli are expected per time unit. |
The T-cell sums up the signals induced by the different antigens on the APC, and if the total stimulation rate reaches a certain threshold value, the cell initiates an immune response. This model relies on several hypotheses, which are known as kinetic proofreading |
McKeithan:1995cq Rabinowitz:1996pt Lord:1999kh Hlavacek:2002xq serial triggering Valitutti:1995bc Valitutti:1997to Sousa:2000xu Borovsky:2002dz Utzny:2006hi dushek:2008 counting of stimulated TCRs Viola:1996ir Rothenberg:1996mw and the optimal dwell-time hypothesis Kalergis:2001le Gonzalez:2005bx |
Due to the huge amount of different receptor and antigen types, it is impossible (and unnecessary) to prescribe the binding durations for all pairs of receptor and antigen types individually. Therefore, BRB chose a probabilistic approach to describe the meeting of APCs and T-cells. A randomly chosen T-cell (that is, a ... |
[MATH] th type of antigen is taken to be a random variable denoted by [MATH] . The [MATH] are independent and identically distributed (i.i.d.) and are assumed to follow the [MATH] distribution, i.e., the exponential distribution with mean [MATH] where [MATH] is a free parameter. Note that there are two exponential dist... |
[MATH] , the mean duration of such a binding (where the receptor is chosen once and the times are averaged over repeated bindings with a [MATH] antigen) is itself an exponential random variable, with realisation [MATH] . Finally, its mean, |
[MATH] , is the mean binding time of a [MATH] -antigen (and, due to the i.i.d. assumption, of any antigen) when averaged over all encounters with the various receptor types. The exponential distribution of the individual binding time is an immediate consequence of the (first-order) unbinding kinetics. In contrast, the ... |
The total stimulation a T-cell receives is the sum over all stimulus rates [MATH] that emerge from antigens of the [MATH] ’th type. It is further assumed that there is at most one type of foreign antigen in [MATH] copies on an APC, whose signal must be discriminated against the signals of a huge amount of self antigens... |
[MATH] and [MATH] different types of class [MATH] and [MATH] . The indices [MATH] and [MATH] stand for constitutive and for variable, respectively; but for the purpose of this article, only the abundancies are relevant, in particular, [MATH] |
and [MATH] Over the whole APC the total number of antigens is then [MATH] if no foreign antigen is present. If [MATH] foreign molecules are also present, the self molecules are assumed to be proportionally displaced (via the factor |
[MATH] ), so that the total number of antigens remains unchanged at [EQUATION] The total stimulation rate in a random encounter of T-cell and APC can then be described as a function of [MATH] |
[EQUATION] i.e., a weighted sum of i.i.d. random variables. Alternatively, we consider the extension of the model proposed by Zint et al. Zint:2008 , which, instead of the deterministic copy numbers [MATH] , uses random variables [MATH] |
distributed according to binomial distributions with [MATH] where [MATH] denotes expectation (so the expected number of antigens per APC is still [MATH] ). The model then reads |
[EQUATION] In line with Berg:2001 Zint:2008 , we numerically specify the model parameters as follows: [MATH] [MATH] [MATH] [MATH] [MATH] (and hence [MATH] ). The distributions in the extended model are the binomials |
[MATH] and [MATH] for [MATH] and [MATH] respectively, where [MATH] The relevant quantity for us is now the probability [EQUATION] |
that the stimulation rate reaches or surpasses a threshold [MATH] . To achieve a good foreign-self discrimination, there must be a large difference in probability between the stimulation rate in the case with self antigens only ( [MATH] ), and the stimulation rate with the foreign antigen present, i.e., |
[EQUATION] for realistic values of [MATH] Note that both events must be rare events – otherwise, the immune system would “fire” all the time. Thus [MATH] must be much larger than |
[MATH] (which, due to ( ) and the identical distribution of the [MATH] , is independent of [MATH] ). Evaluating these small probabilities is a challenge. So far, two routes have been used: analytic (asymptotic) theory based on large deviations (LD) and straightforward simulation (so-called simple sampling). Both have t... |
Rare event simulation: general theory The general problem we now consider is to estimate the probability [MATH] of a (rare) event [MATH] under a probability measure [MATH] . The straightforward approach, known as simple sampling, uses the estimate |
[EQUATION] where the [MATH] are independent and identically distributed (i.i.d.) random variables with distribution [MATH] [MATH] |
denotes the indicator function, and [MATH] is the sample size; we will throughout use [MATH] for an estimate of a quantity [MATH] |
[MATH] is obviously an unbiased and consistent estimate, but, for small [MATH] , the convergence to [MATH] is slow, and large samples are required to get reliable estimates. |
Various simulation methods are available that deal with this problem and yield a better rate of convergence (see the monograph by Bucklew Bucklew:2004 for an overview). Most of them achieve this improvement by reducing the variance of the estimator. We will concentrate here on the most wide-spread class of methods, nam... |
3.1 Large deviation probabilities Consider a sequence [MATH] of random variables on the probability space [MATH] , where [MATH] is the Borel [MATH] -algebra of [MATH] . Let [MATH] be the family of probability measures induced by [MATH] , i.e., [MATH] for |
[MATH] . We assume throughout that [MATH] satisfies a large deviation principle (LDP) according to the following definition Dembo:1998 Dieker:2005 |
Definition 1 (Large deviation principle) A family of probability measures [MATH] on [MATH] satisfies the large deviation principle (LDP) with rate function |
[MATH] if [MATH] is lower semicontinuous and, for all [MATH] [EQUATION] where [MATH] and [MATH] denote the interior and the closure of |
[MATH] , respectively. [MATH] is said to be a good rate function if it has compact level sets in that [MATH] is compact for all [MATH] . ∎ |
A set [MATH] is called an [MATH] continuity set if [EQUATION] If [MATH] is such a set, the LDP means that [MATH] decays exponentially for large [MATH] with decay coefficient [MATH] A point [MATH] is called a minimum rate point of [MATH] |
if [MATH] Large deviation principles are well known for many families of random variables, like empirical means of i.i.d. random variables or empirical measures of Markov chains. For the application we have in mind, which involves sums of independent, but not identically distributed random variables, we need the fairly... |
[MATH] , be the moment-generating function of [MATH] , where [MATH] denotes the scalar product and [MATH] denotes the expectation of a random variable with respect to the probability measure [MATH] |
Theorem 3.1 (Gärtner-Ellis) Assume that (G1) [MATH] (G2) [MATH] is the effective domain of [MATH] (G3) [MATH] is lower semi-continuous on [MATH] |
(G4) [MATH] is differentiable on [MATH] (G5) Either [MATH] or [MATH] is steep at its boundary [MATH] i.e., [MATH] Then, [MATH] satisfies the LDP on [MATH] with good rate function [MATH] where [MATH] is the Legendre transform of [MATH] , i.e., |
[EQUATION] The function [MATH] in (G1) is convex. If there is a solution [MATH] of [EQUATION] one has [EQUATION] If [MATH] is strictly convex in all directions, [MATH] is unique. See Fig. for a one-dimensional example (the T-cell application, in fact). |
3.2 Simulating rare event probabilities Let now [MATH] be a rare event in the sense that [MATH] Here, the first inequality implies that |
[MATH] becomes exponentially unlikely as [MATH] whereas the second inequality serves to exclude nongeneric cases (in particular cases where the event is impossible). An important notion for the rare event simulation of [MATH] |
is that of a dominating point Bucklew:2004, , p. 83) A point [MATH] is a dominating point of the set [MATH] if it is the unique point such that |
a) [MATH] b) [MATH] a unique solution [MATH] of [MATH] , and c) [MATH] A dominating point, if it exists, is always a unique minimum rate point (see Bucklew:2004, , p. 83) ). Convexity of [MATH] implies existence of a dominating point (cf. Dieker:2005 ). |
Following Dieker:2005 we now turn to the problem of simulating [MATH] The naive simple-sampling estimate obtained from [MATH] i.i.d. copies [MATH] |
[MATH] ), drawn from [MATH] , is, as in ( ), given by [EQUATION] It is unbiased and converges (almost surely) to [MATH] in the limit [MATH] but it is inefficient since it requires that [MATH] |
increase exponentially with [MATH] to yield a meaningful estimate. Instead of [MATH] , one therefore considers an alternative family of random variables, [MATH] |
with distribution family [MATH] , again on [MATH] under which [MATH] occurs more frequently. Assuming that [MATH] and [MATH] are absolutely continuous with respect to each other, one can use the identity |
[EQUATION] where [MATH] is the Radon-Nikodym derivative of [MATH] with respect to [MATH] . The resulting importance sampling estimate then relies on i.i.d. samples |
[MATH] from [MATH] and reads [EQUATION] where [MATH] acts as a reweighting factor from the sampling distribution to the original one. It is reasonable to assume that |
[MATH] is continuous to avoid the usual problems with [MATH] -functions; this is no restriction for our intended application. An adequate optimality concept in this context is that of |
asymptotic efficiency . According to Dieker:2005 it is based on the relative error [MATH] defined via its square [EQUATION] (where [MATH] denotes the variance of a random variable with respect to the probability measure [MATH] ). The relative error is proportional to the width of the confidence interval relative to the... |
Definition 2 (Asymptotic efficiency) An importance sampling family [MATH] is called asymptotically efficient for the rare event [MATH] if |
[EQUATION] where [MATH] for some given maximal relative error [MATH] [MATH] In words, asymptotic efficiency means that the number of samples required to keep the relative error below a prescribed bound [MATH] increases only subexponentially (rather than exponentially as with simple sampling). The concrete choice of [MA... |
An obvious idea from large deviation theory would be to use, as sampling distributions, the family of measures [MATH] that are exponentially tilted with parameter [MATH] , that is, |
[EQUATION] [MATH] then takes the role of [MATH] The task remains to find a suitable [MATH] , i.e., a tilting parameter that makes [MATH] asymptotically efficient. Necessary and sufficient conditions for this are given in Dieker:2005, , Assumption 1 and Corollary 1) |
and are summarised below, in a form adapted to the present context. Theorem 3.2 (Dieker-Mandjes 2005) Assume that, for some given [MATH] |
(V1) [MATH] satisfies an LDP with good rate function [MATH] (V2) [MATH] for some [MATH] and, likewise, with [MATH] replaced by [MATH] |
(V3) The rare event [MATH] is both an [MATH] -continuity set and an [MATH] -continuity set. Then, the tilted measure [MATH] is asymptotically efficient for simulating [MATH] if and only if |
[EQUATION] We use assumption (V2) here to replace the weaker but less easy to verify condition (2) in Assumption 1 of Dieker:2005 in line with the paragraph below (2) in Dieker:2005 , or |
Dembo:1998, , Thm. 4.3.1) Note also that (V2) holds automatically if [MATH] exists for all [MATH] – but this is not mandatory here, since only a given [MATH] is considered. |
The proof of Theorem 3.2 is given in Dieker:2005 and need not be recapitulated here; but we would like to comment briefly on what happens in the central condition ( 19 ). Replacing [MATH] by [MATH] |
in ( 16 ) and ( 15 ), we can rewrite [MATH] as [EQUATION] Obviously (by (V1) and (V3)), [MATH] (i.e., the right-hand side of 19 )) is the exponential decay rate of [MATH] Inspection of the proof of Theorem 3.2 reveals that the left-hand side of ( 19 ) is the exponential decay rate of |
[MATH] It is clear from ( 20 ) that, for asymptotic efficiency to hold, [MATH] must tend to zero at least as fast as [MATH] . But it cannot decrease faster, since |
[MATH] is nonnegative, so that [MATH] for arbitrary [MATH] . Hence, the exponential decay rates must be exactly equal, as stated by ( 19 ). (A closely related argument is given in Bucklew:2004, , Ch. 5.2) .) |
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