The Effect of Gene Conversion on the Divergence Between Duplicated Genes

Corresponding author: Hideki Innan, School of Public Health, University of Texas Health Science Center, 1200 Hermann Pressler, Houston, TX 77030., hideki.innan{at}uth.tmc.edu (E-mail)

Communicating editor: J. B. WALSH

	ABSTRACT

TOP ABSTRACT MODEL AND SIMULATION RESULTS DISCUSSION LITERATURE CITED

Nonindependent evolution of duplicated genes is called concerted evolution. In this article, we study the evolutionary process of duplicated regions that involves concerted evolution. The model incorporates mutation and gene conversion: the former increases d, the divergence between two duplicated regions, while the latter decreases d. It is demonstrated that the process consists of three phases. Phase I is the time until d reaches its equilibrium value, d₀. In phase II d fluctuates around d₀, and d increases again in phase III. Our simulation results demonstrate that the length of concerted evolution (i.e., phase II) is highly variable, while the lengths of the other two phases are relatively constant. It is also demonstrated that the length of phase II approximately follows an exponential distribution with mean , which is a function of many parameters including gene conversion rate and the length of gene conversion tract. On the basis of these findings, we obtain the probability distribution of the level of divergence between a pair of duplicated regions as a function of time, mutation rate, and . Finally, we discuss potential problems in genomic data analysis of duplicated genes when it is based on the molecular clock but concerted evolution is common.

TO understand the evolutionary importance of gene duplication, it is critical to know when duplication events occurred. This is not very difficult as long as two duplicated genes have accumulated mutations independently, because we can estimate the time to the duplication event from the level of nucleotide divergence between two duplicates. The idea that nucleotide divergence has a linear correlation with time is known as the "molecular clock." However, the molecular clock hypothesis does not always hold for duplicated genes because of the phenomenon called "concerted evolution" (reviewed in OHTA 1980 , OHTA 1983 ; ARNHEIM 1983 ; LI 1997 ). Under concerted evolution, the level of divergence between two duplicated genes is maintained very low, so that the observed divergence is usually much lower than the expectation when the molecular clock is assumed.

Gene conversion has been considered as the most important mechanism for this homogenization in duplicated genes (i.e., a small multigene family with copy number of 2), although unequal crossing over could also be important for large- or middle-size multigene families. Clear evidence for gene conversion is seen when DNA polymorphism data are available for both of the duplicated genes, because gene conversion creates "shared polymorphic sites" (INNAN 2003A ), at which both of the two corresponding sites in the duplicated genes are polymorphic (Fig 1A). Although such data sets are few, examples include several pairs of duplicated genes in Drosophila (INOMATA et al. 1995 ; KING 1998 ; LAZZARO and CLARK 2001 ; BETTENCOURT and FEDER 2002 ), human (INNAN 2003B ), plants (SATO et al. 2002 ; CHARLESWORTH et al. 2003 ), and Plasmodium falciparum (NIELSEN et al. 2003 ).

View larger version (25K): [in this window] [in a new window] [Download PPT slide]   Figure 1. (A) Shared polymorphic sites created by gene conversion. (B) Real tree (history) of gene duplication and speciation (left) and observed tree under concerted evolution (right).

The action of gene conversion can also be suggested from phylogenetic studies. For example, suppose that duplicated genes I and II exist in species A and B. This means the gene duplication event predates the speciation (Fig 1B). Without gene conversion, it is expected that we observe a tree that is consistent with the real tree. However, if these genes have undergone concerted evolution, the observed tree might be inconsistent with the real tree. That is, the two duplicated genes in each species are more closely related (Fig 1B, right tree). On the basis of this idea, recently, ROZEN et al. 2003 reported that there are abundant gene conversions between several pairs of duplicated regions on the Y chromosome of human and chimpanzee.

Thus, there are many lines of evidence for gene conversion between duplicated genes. However, the effect of gene conversion on the divergence between duplicated genes has not been well understood theoretically. The purpose of this article is to investigate the behavior of the divergence between duplicated genes after gene duplication. We modify gene conversion models of WALSH 1987 . In our model, gene conversion occurs such that a certain length of DNA fragment is transferred from one gene to the other, although gene conversion tracts cover the entire gene in Walsh's models. In our general model, we investigate the effect of the mutation and gene conversion rates and the length of gene conversion tract on the divergence between duplicated genes by simulations, and we attempt to formulate the probability distribution of the divergence as a function of time. Then, we discuss potential problems in genomic data analysis of duplicated genes on the basis of the molecular clock when concerted evolution is common.

	MODEL AND SIMULATION

TOP ABSTRACT MODEL AND SIMULATION RESULTS DISCUSSION LITERATURE CITED

The evolutionary process of a pair of duplicated regions is considered. We study the behavior of d, the number of nucleotide differences between duplicated regions after their birth (i.e., duplication). The process involves mutation and gene conversion; the former increases the level of divergence while the latter decreases it. Parameters used in this section are summarized in Table 1.

Suppose a duplication event creates two identical sequences in a genome at time T = 0. In this article, we consider the evolutionary process of a pair of subregions that are within the duplicated regions as illustrated in Fig 2. Each considered region is represented by a large box and assigned to an interval of (0, 1). After the duplication event, the regions start accumulating mutations and gene conversion works to homogenize variation. Mutation occurs independently in the two regions at rate µ per region per generation, so that the number of mutations in each region follows a Poisson distribution with mean µ. For each mutation, its position is determined as a random variable between 0 and 1 (i.e., infinite-site model).

View larger version (8K): [in this window] [in a new window] [Download PPT slide]   Figure 2. Illustration of gene conversion events. The simulated regions are represented by large boxes: the shaded box represents the original region and the open box represents the duplicated region before gene conversion. Two gene conversion tracts are shown by small boxes.

Gene conversion transfers a DNA fragment from one to the other. Fig 2 shows two examples of gene conversion events. Gene conversion I transfers a DNA segment between positions 0.2 and 0.5 from the original region to the duplicated region, so that the corresponding part of the duplicated region is replaced by the original sequence. Note that shaded boxes represent the sequence of the original region before gene conversion and open boxes show the duplicated region. Gene conversion could involve regions outside of the interval (0, 1). Gene conversion II in Fig 2 consists of a region from position 0.85 to 1 and a fragment outside of position 1.

Gene conversion is simulated assuming the length of the conversion tract follows a geometric distribution in a finite length of DNA region. This assumption is from WIUF and HEIN 2000 , who studied homologous gene conversion (in this study, we consider interlocus gene conversion that occurs between two duplicated regions). Although the mechanism of interlocus gene conversion is not well understood, it might be reasonable to assume that it may be similar to that of homologous gene conversion. Following WIUF and HEIN 2000 , simulating gene conversion events involves two parameters, g and q. g represents the rate that a gene conversion event initiates and q is the geometric distribution parameter to determine the length of the gene conversion tract. The geometric distribution for the length of the converted tract in base pairs, z, is given by q(1 – q)^z–1. Suppose that each of the simulated regions is L bp long. If we assume L is very large and q is very small (i.e., infinite-sites model), the distribution of the length of a tract (see Fig 2) is given by an exponential distribution with parameter Q = qL, and the average length of a conversion tract is 1/Q (the unit of length is L bp). In this model, the rate that a particular site of one region is transferred to the other is given by

(1)

which corresponds to the gene conversion rate per site per generation defined in INNAN 2002 , INNAN 2003A . The algorithm to simulate gene conversion tracts follows WIUF and HEIN 2000 .

In this study, we consider another two probabilities. One is the probability that determines whether the two regions make pairing in meiosis, which is considered to be required for gene conversion to occur. This probability, S₁(x), should be a function of x, the divergence between the two regions. For example, S₁(x) is defined as

(2)

when the two regions make pairing when the divergence is less than s₁.

The other is the probability that a gene conversion event is successfully completed. Let S₂(y) be this probability, which is given by a function of y, the divergence in a gene conversion tract. One example is that gene conversion occurs only when y is smaller than a certain threshold, s₂. That is,

(3)

Another example might be that S₂(y) linearly decreases as y increases:

(4)

Note that S₁(x) depends on the divergence in the whole region, x, while S₂(y) is given for each gene conversion tract and y represents the level of divergence in a tract. Similar models are used in WALSH 1987 . Note that WALSH 1987 defined only one probability because all gene conversion tracts cover the whole region in his model, where the two probabilities mean the same thing. See WALSH 1987 and LI 1997 for the molecular genetic background behind this sequence-dependent gene conversion model.

In addition to nucleotide mutations, WALSH 1987 also considered mutations that block gene conversion. Such mutations are called "terminator mutations." Large indels and transposable elements might belong to terminator mutations. We assume such mutations occur at rate m per region per generation. The position of mutation is given as a random variable between (0, 1). Gene conversion is assumed to be completely suppressed if a conversion tract includes the position of the terminator mutation.

Note that the numbers of mutations and gene conversion events per generation follow Poisson distributions. When the expected numbers of these events are very small, we can obtain essentially the same simulation results by an approximate method, in which the Poisson processes are simulated every k generation with the parameters multiplied by k.

	RESULTS

TOP ABSTRACT MODEL AND SIMULATION RESULTS DISCUSSION LITERATURE CITED

The behavior of the divergence between duplicated regions, d, is investigated by computer simulation. Throughout this study µ = 10^–6/region is assumed: the simulated region corresponds to a 1-kb region if we assume that a standard mutation rate = 10^–9/site/generation. Also, k = 10⁵ is assumed. We assume 1/Q = 0.1, s₁ = 100 in (2), and s₂ = 100z' in (3), where z' is the gene conversion tract length in the interval (0, 1). The results are in Fig 3, which shows four independent realizations of the trace of d for c = {0, 1, 3, 5, 10} x 10^–8 from top to bottom. Without gene conversion (top), d increases linearly with increasing time (i.e., molecular clock). When c = 10^–8, a little delay is observed in the increasing function, and the delay gets bigger as c increases. When c = 5 x 10^–8, d fluctuates around an equilibrium value for a quite long time, and then d starts increasing linearly. The bottom indicates that c = 10^–7 is so high that the fluctuation time of d is extremely long (but the fluctuation does not continue forever).

View larger version (24K): [in this window] [in a new window] [Download PPT slide]   Figure 3. Traces of d over time under concerted evolution. We assume a gene conversion rate = {0, 1, 3, 5, 10} x 10–8 from top to bottom. Four independent traces are shown for each gene conversion rate.

Thus, the evolutionary process of duplicated regions involves a long fluctuation time of d unless the gene conversion rate is small. Fig 4 illustrates a typical behavior of d, which consists of three phases: phase I is the time until d reaches its equilibrium value, d₀, which depends mainly on the mutation rate and gene conversion rate (INNAN 2002 , INNAN 2003A ). In phase II, d keeps fluctuating around d₀. Then, the process enters phase III, in which the two regions start accumulating mutations independently. Note that there are not clear borders between the three phases. The definition of the three phases is intuitively understood such that the divergence approximately follows a molecular clock if phase II is removed. The length of phase II approximately represents the length of concerted evolution.

View larger version (9K): [in this window] [in a new window] [Download PPT slide]   Figure 4. Illustration of the behavior of d.

To study the length of concerted evolution, we consider T_d1, the waiting time for the first hit of d = d₁, d₁ >> d₀. T_d1 is divided into three parts, t₁, t₂, and t₃, according to the phases (see Fig 4). We expect that T_d1 is directly related to the length of concerted evolution, t₂, because t₂ T_d1 – d₁/(2µ). The effect of gene conversion rate on T_d1 is investigated by simulation. We assume d₁ = 200. Table 2 summarizes the results of simulations for c = {1, 2, 3, ... , 10} x 10^–8 when 1/Q = , s₁ = 100, and S₂ is given by (3) with s₂ = 100. It is demonstrated that as c increases, the average of T_d1 increases exponentially. The variance of T_d1 is huge, indicating that T_d1 is highly variable. Similar results are obtained for the case of 1/Q = 0.1 (also see Fig 5A).

View larger version (23K): [in this window] [in a new window] [Download PPT slide]   Figure 5. The relationship between c and Td1 when 1/Q = (squares) and 0.1 (stars). (A) Solid squares and stars represent results of simulations in which S1 and S2 are given by (2) and (3) with s1 = s2 = 100. Shaded squares and stars represent the case where S1 and S2 are given by (2) and (4) with s1 = s3 = 100. (B) Results with terminator mutations are shown (shaded squares and stars). m = 1.25 x 10–11 is given. Solid squares and stars indicate the same as those in A.

View this table: [in this window] [in a new window]   Table 2. Averages and variances of

We also investigate the relationship between c and T_d1 under various gene conversion models (Fig 5). First we assume 1/Q = 0.1, s₁ = 100 in (2) and S₂ is given by (3) with s₂ = 100z'. The result is presented by solid stars in Fig 5A. T_d1 is larger than that for 1/Q = (solid squares), indicating that the gene conversion tract length has a significant effect on the length of concerted evolution (see below). Next we consider T_d1 for 1/Q = and 1/Q = 0.1 in a model where s₁ = 100 in (2) and S₂ is given by (4) with s₃ = 100z'. It is shown that the increase of T_d1 against c is slower in comparison with the previous model. This is expected because the effective gene conversion rate given by (4) is smaller than that given by (3) if c is the same and s₂ = s₃.

In Fig 5B, T_d1 in a model with terminator mutation is shown. We assume m = 1.25 x 10^–11, s₁ = 100 in (2) and S₂ is given by (3) with s₂ = 100z'. When 1/Q = , as c increases, T_d1 saturates around 1/(2m) = 4 x 10¹⁰ because phase II is terminated with probability 2m. The situation is complicated when 1/Q = 0.1. T_d1 saturates somehow around 1/(2m) when c 8–9 x 10^–8, but again starts increasing for c 10 x 10^–8. This is because after a terminator mutation gene conversion is suppressed in a short region around the mutation when 1/Q is small. That is, as c increases, in most regions phase II continues for a quite long time even after a terminator mutation.

Fig 6A shows the effect of the length of gene conversion tract on T_d1 when c = 3 x 10^–8, m = 0, and s₁ = 100. As 1/Q decreases, the average of T_d1 increases dramatically. This observation could be understood as follows. As shown in Fig 6B, 1/Q has a positive correlation with the variance of d in phase II. The variance of d is a very important factor to determine the time of phase II because phase II is terminated when d happens to exceed a threshold value of d, d_t. d_t is defined as the minimum value of d that is too big for gene conversion to occur (see Fig 4). That is, once d hits d_t, there is no chance that the system returns to phase II. It is obvious that the larger the variance of d, the more chance that d hits d_t, creating the negative correlation between T_d1 and 1/Q. Note that 1/Q has no effect on the expectation of d in phase II (INNAN 2002 , INNAN 2003A ).

View larger version (14K): [in this window] [in a new window] [Download PPT slide]   Figure 6. (A) The effect of the length of the conversion tract, 1/Q, on Td1. (B) The effect of the length of the conversion tract on the variance of d in phase II.

Thus, the time of phase II, t₂, might be considered as a waiting time for an event when d first hits d_t. Let be the expectation of this waiting time. It is expected that t₂ approximately follows an exponential distribution with mean when is large, and the variance of T_d1 is approximately given by

(5)

because t₁, t₂, and t₃ are almost independent. Table 2 demonstrates that Equation 5 holds quite well when c 3 x 10^–8. This supports the hypothesis that t₂ approximately follows an exponential distribution.

Our simulation results have indicated that the relationship between d and time is complicated because many parameters (c, 1/Q, s₁, s₂, and m) affect t₂. However, the probability distribution function (pdf) of d might involve only three parameters, µ, , and T, because summarizes all parameters that affect t₂. Here, we attempt to obtain the pdf of d as a function of µ, , and T, assuming the pdf of t₂ is given by

(6)

We define t_e as the effective time that directly contributes to the linear accumulation of mutations, which is given by

(7)

We assume T >> t₁. If concerted evolution is still going on at T (i.e., phase II), t_e = t₁ and d is somewhere around d₀. The probability that the system is in phase II at T is given by

(8)

and the pdf of t_e is given by

(9)

Note that t₁ is unknown here, but the numerical calculation of (9) can be done assuming t₁ 0 because T >> t₁. Then, the pdf of d is given by convolution:

(10)

This equation works best for d >> d_t. The probability that d < d_t is almost identical with that in Equation 8.

We can also obtain the pdf of T_d1. That is,

(11)

when µ is small.

	DISCUSSION

TOP ABSTRACT MODEL AND SIMULATION RESULTS DISCUSSION LITERATURE CITED

The evolutionary process of a pair of duplicated regions is studied by simulations. The model incorporates mutation and gene conversion: the former increases d, the divergence between two duplicated regions, while the latter decreases d. It is demonstrated that the process consists of three phases. Phase I is the time until d reaches its equilibrium value, d₀. In phase II d fluctuates around d₀, and d increases again in phase III. These three phases are defined such that d has a positive linear correlation with time if phase II is deleted. Phase II approximately corresponds to the time under concerted evolution. The lengths of the three phases, t₁, t₂, and t₃, could be almost independent of each other. t₁ and t₃ are relatively constant, while t₂ is highly variable. Our simulations demonstrated that t₂ approximately follows an exponential distribution because t₂ is a waiting time for a random event that initiates phase III. The rate that such events occur determines , the expectation of t₂, which depends on the mutation rate (µ and m), gene conversion rate (c together with S₁ and S₂), and the average length of gene conversion tract (1/Q). It seems extremely difficult to obtain an equation for as a function of these parameters, but we were able to obtain the pdf of d given , µ, and T.

In this study, we considered mutations as only a mechanism to terminate phase II. However, strong selection could also be a factor to stop concerted evolution. INNAN 2003B demonstrated that the gene conversion rate is effectively reduced around the target site of selection, because selection works against gene conversion to keep the variation between two regions. Evidence for such selection is seen in exon 7 in the human RH genes (INNAN 2003B ).

Recent genomic sequence data provide great opportunities to study the evolution of gene duplication (e.g., LYNCH and CONERY 2000 ; FRIEDMAN and HUGHES 2001 ; BAILEY et al. 2002 ; GU et al. 2002 ; MCLYSAGHT et al. 2002 ). Some of these studies estimate the date of gene duplications on the basis of the molecular clock, ignoring concerted evolution. This ignorance is partly because we do not know how common concerted evolution via gene conversion is at the genome level, even though there are many reports of gene conversion for specific genes or regions (see Introduction). Here, we consider potential problems in genomic data analysis of duplicated genes on the basis of the molecular clock, if concerted evolution is common. As an example, we use the data of LYNCH and CONERY 2000 , who estimated the birth (duplication) and death (deletion or pseudogenization) rates on the basis of the molecular clock. Their estimation is based on the idea that the frequency distribution (spectrum) of d is approximately given by an exponential distribution when the birth and death rates (a and b, respectively) are constant over time. The solid bars in Fig 7B show this relationship when b = 10^–8. It is obvious that the spectrum is expected to be flat when b = 0 (solid bars in Fig 7A). The number of observed duplicated genes with very low d in a genome reflects the duplication rate, a.

View larger version (20K): [in this window] [in a new window] [Download PPT slide]   Figure 7. Expected and observed frequency distributions of d. Expected distributions are obtained by simulations. 1/Q = and s1 = 100 are assumed and S2 is given by (3) with s2 = 100z'. a is assumed to be constant, but not specified. The distributions are shown in adjusted frequencies relative to the flat distribution of the case b = c = 0. (A) Solid bars represent the distribution of d for b = c = 0, and open bars represent that for b = 0 and c = 3 x 10–8. (B) Solid bars are for b = 10–8 and c = 0, and open bars for b = 10–8 and c = 3 x 10–8. (C) Observed distributions of Ks for the D. melanogaster genome. Data are from LYNCH and CONERY 2000 .

Fig 7A shows the expected frequency distribution of d when b = 0 and c = 3 x 10^–8 (open bars), indicating that gene conversion creates 10 times more duplicated genes with low divergence (d < 50) than the case without gene conversion (solid bars) does because d spends a long time around d₀. The two distributions are identical for d > 100 because gene conversion does not occur (i.e., d_t = 100). It is indicated that gene conversion alone can also create an exponential-like distribution of d as well as the constant death process. The open bars in Fig 7B show that the frequency distribution under the joint effect of the death process and gene conversion is also similar to an exponential distribution. The peak of genes with low divergence is approximately three times higher than that of the case of the death process only, and the distribution decreases very quickly as d increases.

Thus, if concerted evolution of duplicated genes via gene conversion is common, the effect of gene conversion on the frequency distribution of d cannot be ignored. In such a case, it is indicated that the result of duplicated gene analysis based on the molecular clock should be biased. For example, suppose a and b are estimated by fitting an exponential distribution. An estimate of a depends on the number of duplicated genes with very low d: the more genes with low d, the higher the rate of gene duplication. If the number of duplicated genes of low divergence is increased by gene conversion, a should be overestimated. This excess of genes of low divergence might also contribute to an overestimation of b because an estimate of b depends on how quickly the distribution decreases as d increases.

Fig 7C shows the observed frequency distributions of the level of divergence measured by Ks, the expected number of synonymous substitutions per site in the Drosophila melanogaster genome. The data are from LYNCH and CONERY 2000 . We chose this species as an example because DNA polymorphism data for several pairs of duplicated genes exhibit clear evidence for frequent gene conversion (INNAN 2003A ). Duplicated genes are classified into two groups, pairs on the same chromosome and those on different chromosomes, because gene conversion is expected to occur more frequently between pairs of genes on the same chromosome. We find a clear difference in the distributions: an excess of pairs of low divergence is observed only in the former class. If this difference is due to the difference in the gene conversion rate, it might be suggested that the high peak of low-divergence genes on the same chromosome might be a signature of frequent gene conversion rather than a rapid birth-and-death process. LYNCH and CONERY 2000 have also argued this possibility from the positive correlation between Ks and physical distance between genes.

Our results suggest that analyzing duplicated genes on the basis of the molecular clock might be misleading if concerted evolution is common. Therefore, before analyzing data, it is very important to test the molecular clock hypothesis, which might require genomic information from other related species. Unfortunately, such data are few at this moment, but will be available in time.

	ACKNOWLEDGMENTS

J. S. Conery and M. Lynch kindly provided us information on their data in LYNCH and CONERY 2000 . H.I. is supported by a grant from the University of Texas and K.M.T. is supported by a grant to R. Chakraborty.

Manuscript received September 16, 2003; Accepted for publication December 2, 2003.

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