Renewal Process Models for Crossover During Meiosis

Transcription

1 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 1 Renewal Process Models for Crossover During Meiosis Today we have another example of stochastic modeling in genetics. This time we will investigate models to describe the occurrence of crossover events among strands of chromosomes during the cell-division process called meiosis which is responsible for the generation of gametes from the germ cells of diploid organisms. The distance between crossover events is well-suited to modeling by a renewal process. We will consider two different interarrival densities (exponential and gamma, respectively), and, in particular, investigate what they imply for the existence (or lack thereof) of interference between crossover sites. In preview we will: Describe the biological process of crossing over Define position interference for disjoint intervals Consider crossovers occurring as a Poisson process (i.e., exponentially-distributed interarrival times) and note the lack of interference with such a model Consider gamma-distributed interarrival times Define a stationary renewal process along the chromosome Demonstrate that this model has position interference Meiosis and crossover Explanation and description follows the figures on Pages 7 and 8. Most importantly, for today, we will consider a crossover event between any two non-sister chromatids on the chromatid bundle to be an arrival in a renewal process, and we will depict the process graphically as: * * * * where this straight line depicts a bundle that had four crossover events occur upon it. If we think of time moving forward from left to right in the above diagram, we can imagine that the location of crossover events might be suitably modelled as the W n s of a renewal process. Intervals along the bundle and the notion of interference When doing genetic research, people are not able to see the exact locations of the crossovers. In fact, they typically can only observe whether or not crossover events occurred in certain intervals defined by the available genetic markers. So, the situation governing what one might be able to observe looks more like: Interval 1 Interval 2 * * * * With intervals placed on the time line as above, we have new random variables Z 1 and Z 2 which count the number of arrivals that occur in interval one and two respectively. 1 (In the above picture Z 1 1 and Z 2 2.) 1 In fact, it is a bit more complicated still, because one typically infers crossovers from observed recombinations in genetic data, but two crossovers in an interval will appear as a non-recombinant with respect to the endpoints of that interval. We pretend for this discussion that all crossovers in an interval are discernable.

2 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 2 Position interference describes the phenomenon whereby the number of arrivals in one interval affects the probability of arrivals in another interval. Position interference is often observed in data on genetic crossovers typically, a crossover in a particular interval is less probable if there was a crossover in a nearby interval. Notice that the interference is a non-independence between the Z i s for disjoint intervals. In order to demonstrate that a renewal process of a particular type would lead to position interference, one could show that under that renewal process, the conditional probability P (Z 2 Z 1 z 1 )isnot equal to the conditional probability P (Z 2 Z 1 z 1 ) when z 1 z 1. We will use that later. Stationarity Thinking about throwing intervals down brings us to think about stationary renewal processes. The reason is that, for modeling and inference purposes, one would like the expected number of crossovers in an interval of length l to be the same regardless of where on the bundle (or time line) that interval might happen to get placed. This will not be the case in a delayed renewal process. (To convince yourself of this, imagine that the expected interarrival time is very long, but with a narrow distribution, and your interval is very short; then consider the expected number of arrivals in the interval if a) you put the interval down right at time zero, or b) you put the interval down centered at the mean interarrival time.) So, to make the expected number of arrivals in an interval depend only on the interval s length l and not on its position, we must make the renewal process stationary by assigning to X 1, the time to the first occurrence, the excess life distribution. Hence if the interarrival times X 2,X 3,... are iid with density f(x), then we must assign to X 1 the density g(x) (EX) 1 (1 F (x)). (See Page 6 of Dr. Thompson s notes on renewal processes.) If we do this, it also means that if we were to choose any point at random on the time line as our starting point, the time to the next arrival will have the same density (the stationary excess life distribution). We will use this later, as well. A specific example Exponentially-distributed interarrival times The most commonly-used model for crossovers on a bundle assigns iid exponential random variables with mean, say, 1/λ, to the interarrival times. So, f(x) λe λx. This, of course, says that crossovers arrive as a Poisson process with rate λ. You already know very much about Poisson processes, but two points of review which are germane to the discussion of crossover models are: The number of events occurring in a length of time in a Poisson process is independent of the number occurring in any other disjoint interval of time. The excess lifetime distribution in the Poisson process is the same as the interarrival distribution, i.e. in this case, g(x) [1 F (x)]/ex [1 (1 e λx )]/(1/λ) λe λx f(x). Knowing these properties of the Poisson process we have already answered what we are interested in. Namely, the process requires no special tweaking to be made stationary, and there is no interference under the Poisson model (by the first point above, P (Z 2 Z 1 z 1 ) is the same for all z 1 ). Even though we know all this immediately we will go through a somewhat lengthy calculation showing the special case that P (Z 2 1 Z 1 1)P (Z 2 1 Z 1 ). (1)

3 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 3 Why are we doing this? We know this to be true already. And if we didn t, proving this special case would not bring us any closer to having proved that there is no interference in the Poisson crossover model, since the particular does not prove the general. The reason is that a particular case can disprove the general simply, we will investigate the same quantities as those in Equation 1 with the crossover model with gamma interarrival densities to show that there is interference (i.e., that there is not no interference) in that model. Hence, consider two adjacent, but disjoint, intervals of length l 1 and l 2 respectively that have been thrown down in a random position, with left endpoint at t. Imagine that there are no arrivals in Interval 1, and there is one arrival in Interval 2. This will look like: Interval 1 Interval 2 * * Since the process is stationary, t is irrelevant and we can just call the time from t to the first arrival X 1. The joint probability that Z 1 and Z 2 1 is just the probability that X 1 is between l 1 and l 2. Note that X 1 has the excess life distribution, which in this special case is the interarrival distribution. Thus, +l 2 P (Z 1,Z 2 1) λe λx dx e λl 1 e λ(l 1+l 2 ). l 1 The marginal probability that Z 1 is just the probability that X 1 is greater than l 1, and so Therefore the conditional probability is P (Z 1 )P (X 1 >l 1 )1 F (l 1 )e λl 1. P (Z 2 1 Z 1 ) P (Z 1,Z 2 1) P (Z 1 ) e λl 1 e λ(l 1+l 2 ) e λl. 1 Calculating P (Z 2 1 Z 1 1) proceeds similarly. Now the situation looks like The joint probability is Interval 1 Interval 2 * * * x 1 x 2 P (Z 1 1,Z 2 1) λe λx1 P (l 1 x 1 <X 2 l 1 + l 2 x 1 )dx 1 λe λx 1 [F (l 1 + l 2 x 1 ) F (l 1 x 1 )]dx 1 λe λx 1 [e λ(l 1 x 1 ) e λ(l 1+l 2 x 1 ) ]dx 1 λ[e λl 1 e λ(l 1+l 2 ) ]dx 1 λl 1 (e λl 1 e λ(l 1+l 2 ) ).

4 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 4 While the marginal probability, P (Z 1 1) is merely the probability that X 1 is between and l 1 and that X 2 is greater than l 1 x 1 : P (Z 1 1) And so for the conditional probability we have λe λx 1 [1 F (l 1 x 1 )]dx 1 λe λx 1 e λ(l 1 x 1 ) dx 1 λe λl 1 dx 1 λl 1 e λl 1 P (Z 2 1 Z 1 1) P (Z 1 1,Z 2 1) P (Z 1 1) λl 1(e λl 1 e λ(l 1+l 2 ) ) λl 1 e λl 1 e λl 1 e λ(l 1+l 2 ) e λl. 1 Rejoice! This is the same as P (Z 2 1 Z 1 ). If it were not, we would feel deep consternation since we knew that this had to be the case. Next we consider interarrival times with gamma densities and will set up a similar calculation to show that interference does exist between intervals thrown down on top of that renewal process. Gamma-distributed interarrival times Imagine that X 1,X 2,X 3,... are iid following the Gamma(α, λ) density. Note that different people might parametrize their gamma distributions differently. In this case we are talking about an interarrival density f(x) λα Γ(α) xα 1 e λx (2) where Γ(α) x α 1 e x dx. This distribution has mean, EX α/λ. It is very easy to visualize this process when α is taken to be an integer one can imagine watching events occuring as a Poisson process, rate λ, but you only count every α th event as an arrival. This follows from the fact that the sum of α exponential random variables with the same scale parameter λ has a Gamma(α, λ) density. Note that this renewal process is not a thinned Poisson process (i.e., a Poisson process in which each event is deleted with some probability, p, and it is, in fact, not a Poisson process at all! Making the process stationary As stated before, we want the process to be stationary because we want the expected number of arrivals in an interval to depend only on the length of the interval, not on where it starts. To do so we must change the process a little bit so that X 1 has the excess life density g(x), g(x) 1 EX (1 F (x)) λ ( x ) 1 f(s)ds λ α α x λ α Γ(α) sα 1 e λs ds λα+1 s α 1 e λs ds Γ(α +1) x (3) So long as X 1 has that density, if you decide to start waiting for arrivals at any random point on the bundle, the density to the first arrival always has the density g(x). (Subsequent to the first arrival, of course, the time until the next arrival is gamma-distributed.)

5 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 5 Interference with gamma-distributed interarrival times In order to demonstrate that this model for crossover on a bundle (with α 1) admits some position interference we need merely show that P (Z 2 1 Z 1 1) P (Z 2 1 Z 1 ) (4) for two disjoint intervals randomly placed on the bundle, for any values of l 1,l 2,α 1, and λ. We would proceed just as we did for the Poisson process, but we have to be somewhat more careful because the distribution of X 1 differs from that for X 2. Writing F (x) f(x)dx and letting f(x) be defined as in Equation 2, and g(x) be defined as in Equation 3 we would have, for example P (Z 1 1,Z 2 1) g(x 1 )[F (l 1 + l 2 x 1 ) F (l 1 x 1 )]dx 1. P (Z 1,Z 2 1) may also be found, as can the marginal probabilities P (Z 1 1) and P (Z 2 ). These integrals are fairly messy, but to demonstrate that interference exists it would suffice to show that (4) holds for just one particular set of parameters (with α 1). I leave it to the interested student to carry out the computations (perhaps computing the integrals numerically with MatLab or Mathematica) to show that (4) holds for, say, λ 2,l 1 1,l 2 1 and α 2. This should not surprise us the Poisson process (which has no interference) is a very special process. Further reading This has merely scratched the surface of a very interesting part of statistical genetics. For a fuller treatment, good summer reading on the topic is McPeek, M.S. and T.P. Speed Modeling interference in genetic recombination. Genetics 139: Addendum We can numerically evaluate the conditional probabilities of Equation 4 with the following code in Mathematica: g[x_]: (lambda^(alpha+1)/gamma[alpha+1]) * NIntegrate[s^(alpha-1) Exp[-lambda s], {s,x,infinity}]; f[x_] : ((lambda^alpha)/gamma[alpha])(x^(alpha-1)) Exp[-lambda x]; cdf[x_]: NIntegrate[f[s],{s,,x}]; jointoneone[ell1_,ell2_] : NIntegrate[g[x](cdF[ell1+ell2-x] - cdf[ell1-x]),{x,,ell1}]; jointzeroone[ell1_,ell2_] : NIntegrate[ g[x],{x,ell1,ell1+ell2}]; marginalzero[ell1_] : NIntegrate[g[x],{x,ell1,Infinity}]; marginalone[ell1_] : NIntegrate[g[x](1-cdF[ell1 - x]),{x,,ell1}]; condprobonezero[ell1_,ell2_] : jointzeroone[ell1,ell2] /marginalzero[ell1]; condproboneone[ell1_,ell2_] : jointoneone[ell1,ell2] /marginalone[ell1]; For the values λ 2,l 1 1,l 2 1 and α 2,weget P (Z 2 1 Z 1 1) P (Z 2 1 Z 1 )

6 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 6 showing us that, indeed, crossovers modeled as a renewal process with gamma-distributed interarrival times exhibit interference. Checking to make sure this gives reasonable results for the Poisson process we use the values λ 2,l 1 1,l 2 1 and α 1 (because a Gamma(1,λ) is an exponential random variable), and we find as it should be. P (Z 2 1 Z 1 1) P (Z 2 1 Z 1 ),

7 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 7 Nucleus of Germ Cell Then the chromosomes unwind Maternal Paternal Moments after a mitotic division And, Lo and Behold! Maternal Each chromosome has an Paternal Then the DNA itself back up exact copy winds again And each reproduces That is to say: and, itself Then, if this cell happens to be entering Meiosis I, the maternal and the paternal copies (and their own copies) form a "bundle" This is where the "good stuff" (i.e., recombination) occurs.

8 Stat 396 May 24, 1999 (Eric Anderson) REVISED NOTES (25 May 1999) 8 Crossover and Thinning on a Bundle "TIME" A Crossover Event When one tries to observe crossovers, there are some complications because you only get to observe one final strand. But we aren't going to worry about that. All we care about are crossovers and we will think of the above as events along a line: * * *