Gustave Malécot, 1911-1998: Population Genetics Founding Father

IN November 1998 we lost one of the most important scientists of this century, Gustave Malécot. He was, in the best sense of the words, a great man. His contributions to science and mathematics began 60 years ago and continued until his death. His contributions to mathematical population genetics were arguably the most profound of any. If we will admit a fourth member to the “fathers” of population genetics, the others being Sewall Wright, R. A. Fisher, and J. B. S. Haldane, it would surely have to be Gustave Malécot. It seems safe to say that it was Malécot who first formulated models of population genetics as stochastic processes, in the sense of the term today, for example, as Markov chains. Moreover, Malécot’s mathematics were elegant and exact. They were also often compact and abstract. Recognition of Malécot’s achievements has at most times been slow, its spread a branching trickle that continues today. His achievements deserve fuller recognition. The purpose of this article is to give a personal perspective of the person of Gustave Malécot and a conceptual account of his contributions to the field of population genetics. Some essential biographical information is given, but more important are some insights he himself gave into his early career. I was privileged to have him share these insights with me over the past 5 years. There are also some mathematical formulations necessary to put his work into its conceptual as well as historical context. The structure is mostly chronological. At times this account borrows heavily from an important Perspectives article written by Nagylaki (1989), who presented many details of Malécot’s mathematics, especially the diffusion approximation approaches and other key conceptual issues. Other important accounts of Malécot’s work and life include a recent interview with Bocquet-Appel (1996) and a series of articles by Gillois (1996a,b,c, and unpublished manuscript). As was discussed in detail by Nagylaki, and as Malécot related directly to me, Malécot’s doctoral dissertation, guided by George Darmois and completed in 1939, focused on Fisher’s pioneering 1918 article on the phenotypic covariance of relatives. Prior to this, in 1935, Malécot completed his mathematics degree at the École Normale Superieure in Paris (Nagylaki 1989). Malécot told me how he had spent 2 years reading and mastering (no doubt in rigorous mathematical detail) Fisher’s article, at the Institut Henri Poincaré, also in Paris. Nagylaki (1989) describes how Malécot’s dissertation work made Fisher’s connection of biometry with Mendelian inheritance more rigorous and general. Specifically, Malécot adopted a conditional expectation approach that foreshadowed his stochastic process approach to other problems. Although he respected the fundamental connections that Fisher had made, Malécot also related to me how he had “found some problems” with the mathematics of Fisher’s theory. What followed from Malécot’s dissertation is, to my thinking, probably the first of several reasons that Malécot’s work never received the rapid and widespread recognition it deserved. Although Malécot was not at all bitter about his career, he related to me on numerous occasions various aspects of the historical context to his earliest works. He explained how the leading Darwinians at the Université de Paris and the Sorbonne dominated the intellectual scene in Paris and in France generally at that time. Malécot said that many of the French Darwinians were also communists or even Stalinists. This is despite the fact that today it may seem that Darwinism and Stalinism are mutually exclusive, given the history of the role of Lamarckianism and social construction in the Soviet Union and China. As a result, the French Darwinists would invite from Russia only sympathetic scientists, not people such as N. I. Vavilov. More to the point, many of the Parisian Darwinians viewed Malécot’s models as “anti-Darwinian” because the models were stochastic processes. The Parisian Darwinians considered Fisher’s work to be the complete and unarguable truth in reconciling Mendelian genetics and Darwin’s theory. Since Malécot found in Fisher’s (1918) article and later in his 1930 book, The Genetical Theory of Natural Selection, much that was mathematically wrong, he incurred the displeasure of these evolutionists. Malécot related how in some steps of Fisher’s work there was confusion of statistical sampling theory with stochastic process theory. Malécot benefitted from a longer tradition in France of the measure-theoretic approach to probability theory, the standard that is widely accepted today. Publicly questioning Fisher did not help the acceptance of Malécot’s work in France. Malécot related that on occasion Fisher visited Parisian universities and gave talks; Malécot asked him some critical questions, and Fisher answered politely that he simply disagreed, instead of the more usual cutting answers Fisher reputedly gave to other questioners. Malécot told me that Fisher always treated him and his work with high respect, and vice versa. In addition, Malécot published in French, and as a result the English-speaking were late in recognizing the importance of his work. It was also in the late 1930s that Malécot became intensely interested in Wright’s work, which also dealt with covariances and variances among relatives. Malécot freely and fully acknowledged that most of his work was inspired by Wright’s work and creative genius. Malécot made these early ideas of Wright mathematically rigorous, again applying what are today widely accepted interpretations of probability theory. Malécot did improve on Wright’s work. For example, Wright developed the inbreeding coefficient in terms of path coefficients and partial regression (or correlation) coefficients. Path coefficients are still used today but are limited as statistical measures, in part because they assume linearity of genetic effects (Nagylaki 1989). Malécot, in contrast, considered the probabilities that two genes are descended from various ancestral genes, using Mendel’s laws of inheritance. Such probabilities Malécot originally termed “Les coefficients de parenté” (Malécot 1941, 1942). Later, when he studied models that included mutation and focused on the population level, some but not all of his results were stated in terms of what became known as probabilities of identity by descent (Gillois 1996a). Malécot used other French names for these probabilities, and Crow first used the English term identity by descent (Crow 1954). Coefficients of consanguinity allowed what has become the preferred textbook method for teaching the calculation of inbreeding coefficients, through the chain-counting method, for example. Chains of genetic descent, often at the gametic rather than zygotic ploidy level, and probabilities of identity by descent became the center of much of Malécot’s work that followed over the next six decades. Their immediate importance was that they allowed application to general pedigrees, rather than the special cases that Wright used to illustrate his method (see Nagylaki 1989). Wright certainly knew of Malécot and his work, but it is less clear how well Wright understood the mathematical subtleties of Malécot’s early work. More importantly, Wright apparently did not recognize much of the biological importance of Malécot’s work. Perhaps it is fair to say that Wright was most interested in treating biological variables in terms of sample statistics, whereas Malécot was most interested in treating them as entities in stochastic processes. Malécot described himself as more of a mathematician than a geneticist. Wright’s (e.g., 1978) later compilations contain numerous citations of Malécot. Malécot always spoke glowingly about Wright. Malécot thought that, in contrast, the pre-World War II Parisian Darwinians had a “very low opinion” of Wright. Malécot had also met Haldane more than once and knew his work, but because most of Haldane’s models were deterministic, they generally did not spark Malécot’s interest in the same way as did those of Wright. After receiving his Doctorat d’État in 1939 for what must be viewed as a brilliant dissertation on Fisher’s work, Malécot taught mathematics from 1940 to 1942 at the Lycée (secondary school) de Saint-Étienne. Then an important mentor,Émile Borel appointed Malécot to a position as maître de conférence (similar to a university lecturer) at the Université de Montpellier (1942 to 1944; Nagylaki 1989). In 1945 he joined the faculty of the Université de Lyon, where he became a professor of applied mathematics (1946), a position he held until his retirement in 1981. Lyon being in fairly close proximity to Paris, Malécot also frequently “lectured on population genetics at the Institute of Statistics...” at the Université de Paris (Nagylaki 1989). During the time immediately following his dissertation, Malécot developed many seminal models, often focusing on extending his probabilistic genealogical framework to the population level. It is fair to say that Malécot had been branded for his critical work on Fisher. All of his published papers during this time were written in French. In addition to his work related to Fisher’s 1918 article, during this period Malécot published works on coefficients of consanguinity (Malécot 1941, 1942, 1946), inbreeding (Malécot 1941, 1942), random mating in finite populations (Malécot 1946), and “drift” of gene frequencies (Malécot 1937, 1944, 1945; Nagylaki 1989; Gillois, unpublished manuscript). During this period Malécot developed what he called “Les chaînes des kinship zygotique” (Malécot 1942, 1973a), or zygotic kinship chains, which measured the probability that a of This work was a complete and mathematically rigorous probability theory for on the of among in a and probabilities of in the This is a important point, and it allowed Malécot to results when he his interest to the population level. these coefficients, he for example, how to Wright’s inbreeding which he was the of the probabilities of the various zygotic kinship by the coefficient of in the This was the for the chain-counting method used today. It is as discussed Malécot also developed a method that much of the probability theory by to an of “Les chaînes des kinship This was important because at this time Malécot focused on population genetic processes, for many of which the is not or but the frequencies of various of mating frequencies of on the be and the be by of gametic genes rather than The method of gametic kinship was well developed in one of Malécot’s papers in a fairly French in (Malécot 1973a), to theory (e.g., Malécot also continued of his works on several other This is in of his articles from to for example, and Gillois manuscript). In Malécot often his of who was for the of diffusion theory in the the for Malécot also that this which is on the first two did not generally of the probability (Malécot and he the by using a approach (Malécot as is described in detail by Nagylaki It be that the the process is Malécot the for population genetics, as it is frequencies are or and he the importance of He also Fisher’s approach using the Malécot did not diffusion theory much in his later work. in his models, he the and The of Malécot’s science did not improve during the period from 1939 to but the important work he did was into his book, de published in The was a and became a It is to much less than his and it was and a and work that has much of the population genetics it did over the following 5 in through various leading to Recognition of the fundamental contributions in his was not immediate Malécot’s was made to English in a by by to its I with Nagylaki that the numerous I of the in my population genetics and I also found a of that not in Malécot’s Malécot early on much of his to genetics, which a great of sense in terms of genealogical approaches to population genetics. He had considered a of population processes, such as the effects of on probabilities of identity by these were among his first from the to the population level. His work on the of probabilities of in was to of which be considered simply as that share of among populations are to pedigrees, whereas the of and the of gene among populations on He on and population and gene of to because they are the same population or because they are on a Malécot freely he was inspired and by the seminal works of Wright (e.g., the still the population models and the general of to the work of and and and Malécot had much to say about this he certainly was not he was he had developed the population in a series of articles in the and (Malécot and the was well developed by works were written in French and published in not to English-speaking In Malécot to the on and gave of his articles to He also said that, of those he only with him about in apparently had only of the of populations as published in his which or and did not to put it the was (Malécot interview with Bocquet-Appel a also in Malécot’s published on presented at the (Malécot was of importance in with what Malécot had and it was not until the articles of and and and that had the of by among Malécot said that him Malécot that he had not used his work because he did not French, but Malécot not to Although and did Malécot’s book, have more to recognition of the more fundamental contributions Malécot of Malécot’s work on and on where were the same for a they were general. The articles were always mathematically The of the is that it the to be used to and Malécot used it with in time. He used diffusion a method put to by many most and Nagylaki (e.g., Nagylaki Malécot often said that his formulations of for the probability theory were and this certainly seems to be he used the and its a of the and in the the such in the Malécot’s were always clear in the steps where were He studied and models, of by for example, that in or from the models were in as to what I term the of which only among Malécot the probabilistic for and population genetics, there have been many important by others in more recent of these are discussed by Nagylaki 1989). models and and were always in terms of the or covariance in gene It is a fact that these models the that the gene frequencies in the are those in the population from which they This may be as as there are of to the population genetic that but it is not where the in many This be by stochastic effects that were not included in models until my work on the time series models known as or There is a that Malécot’s genetics models from in that Malécot did not or covariances in gene In he with all mathematical models of the of covariances in articles as early as (Malécot He often used of variables for the of genes in to from probabilities of identity by descent to covariances in frequencies (e.g., Malécot He used the and as in the models in terms of probabilities of identity by models were following and from the there is a with and for that with all that that and and my work on Specifically, the the probability of consanguinity among genes from the same and this a to the covariance for example, the term in 2 Malécot that was one for not It is clear that this is over time but less clear what it may have on the the of models, or them that more than for the approximation on the being Moreover, and other statistical are much more with the Malécot two of covariances of gene frequencies and also considered coefficients of or on his probability theory. were the a and the conditional or a as used in He also used of variables to the a and conditional covariances for the same of models as in This confusion in to the covariances for of genetic in for As discussed in more detail Malécot was interested in and this probably did not help his work become more widely It is also that Malécot’s models were for were often of interest to and genetics. In in most of his articles Malécot first developed models on the genealogical and results for but later in article he would the which the some The coefficient mutation or from the some of or of these important in the when Malécot was by J. to give a at the on and The gave important to his work in the he published his in English (Malécot Malécot’s was and he told me he used this to As of his Malécot was to fair For example, he told me that in the with two was and He that is a critical important in Malécot’s and in the of the field of genetics, at a on the genetic structure of populations held at the of in to which Malécot was by Wright was of the The of the papers published in the (see to Malécot a Malécot also his with Wright in a account in a recent interview with Bocquet-Appel It must have been a and The include not only the papers but also of and many in the of genetics. Malécot had a and important with Malécot was interested in and other aspects of statistical of of even he had a interest in and his were in a of statistical and on the by Malécot’s stochastic Malécot’s work and As and were for at the center of of genetic for this a great to spread recognition of Malécot’s work on and stochastic processes. (e.g., focused on Malécot’s models that were in terms of the conditional kinship or conditional covariance of gene The connection of to stochastic process is in processes. The and probabilities must be made, and much of this to of the of a about these and often probably did not in what have been or of Malécot’s models by the and who Moreover, Malécot did not always with all of and again this is given the it seems that these probably the of other statistical that were less with stochastic and that mostly did not Malécot’s work. it be that inbreeding coefficients populations and be to Malécot’s Malécot’s theory of and genetic had fully by He had a series of articles (Malécot in French. The first two are important and on the coefficients de and covariances in gene The article (Malécot is of even more only did it it also some of his most elegant mathematical using various of the and and He for example, elegant formulations of by in terms of probabilities of identity by descent, for general models in with one or two The article is written at a level, in French. He also developed an the probabilities of identity by descent to the probabilities of chaînes de kinship years the same was developed in the (e.g., it is how Malécot’s work is in this Malécot’s on some not the importance of the work in this The theory developed by had aspects that were and much of with of statistical of once again important by Malécot are in of not receiving widespread recognition. The of Malécot’s work on populations is his article in (Malécot which is written in It is a and of and stochastic models of It much that is in but also some Malécot, as often to his theory and for a of models into with The article mostly on probabilities of identity by descent, but also and probabilities of gametic kinship chains, or what be called It is also and to Malécot did not a great he became of the Université de After there was only one a an unpublished to in Malécot and Nagylaki, personal It with one of the genes is at a in a or in and this approach from The two models is Malécot’s apparently would only one gene whereas it is well known that the genes are from a has become a it a mutation the mutation probabilities of The work that developed is of it is important to that to to rather where all of sample genes be treated as because of the of of the most models studied is the in which two and of genes are Some have used theory, for example, to gene for but the fact that among of genes on the of the of populations most of his Malécot published He developed his and a of work. He fully knew the importance of his work, but he was also His interest in was to the important and fundamental results that from his He was interested only in contributions that of how stochastic and explained Malécot was not interested in There is a that by Malécot was among the first to models in which are always to this is the mutation for the or many Crow 1989; Gillois, unpublished manuscript). This is because years ago would it was a in important mutation is the which information in for example. Malécot’s work on probabilities be many models or to the may there is the is not for some of Malécot’s theory in terms of probabilities of identity by descent and the in which they probabilities of at In addition, it seems that models of all information on because they are simply or they are various and must be that these be but they not be to be it is how often there is information using the rather than the I of only two results on this found the which the may have more information than the for a population of genetic and that for most these would not be and that for the by for most of the information in the was who had recent It is that these by one and certainly not by more than one and probabilities of identity by descent most of the Malécot’s included Gillois, and (Nagylaki 1989). They have made various important in and other the of French population had the to Malécot, when he gave a at the 1998 de des in France. He received a Perhaps the best I give into Gustave Malécot’s from my I had studied Malécot’s work for many years and viewed him as a great mathematician and who always Although his articles are to they are always not only but also of and more than 5 years I to him and him some of my I had been to him in the of his by Gillois, a of Malécot, I met in The work of that I to him was approaches that I had developed and papers on of population genetic of which his met for the first time the next at his in a in France. The were I knew I was a of his but still not I and asked some where Malécot’s They that there was professor in the After several the Malécot, that is the who the His did not even they had an mathematician in This was of his personal I his he was in the much to my I that he had not been there during the of that when I he to my the that by this time also as Malécot was He was a and all that mostly in I was that, population genetics he immediately would such and such that be in one of his papers from years in on his It was a to have known and I to have with The work we were was to be the of my He was a great in the and an After this we ideas and the next Malécot me to a as well as some The of was in the of that we frequently over the following years. The were mostly mathematical were the when he I will the on my with him as Malécot’s and my or are He me, are in such a the have in of I to France to time with Gustave and to He was a and I than to him during my with him He was as as and I told he was of to Gustave Malécot was and in a in the de the of a des In he who in he in to He and which he continued his this to his In the years I knew he and His intellectual are He had an interest in and and he developed an early in He did not in and his personal was Gustave Malécot and it is to that he probably as as His told me that he had been the of as he was to He to his work doubt on population genetics theory. time later found In addition to Gustave Malécot is by and and Malécot’s work never received a of its recognition. he has received a of de des des de and de Nagylaki all of which his If we not admit Gustave Malécot as a fourth of population genetics, he must be its I Malécot and Malécot for personal and and for on an of this