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Applied Mathematics, 2011, 2, 11-22 doi:10.4236/am.2011.21002 Published Online January 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Discrete Evolutionary Genetics: Multiplicative Fitnesses and the Mutation-Fitness Balance Thierry Huillet1, Servet Martinez2 1Laboratoire de Ph ysique Théorique et Modélisation, CNRS-UMR 8089, Université de Cergy-Pontoise, Cergy-Pontoise, France 2Departamento de Ingeniería Matemática, Centro Modelamiento Ma temático, UMI 2807, UCHILE-CN RS, Santiag o, Chile E-mail: Thierry.Huillet@u-cergy.fr, smartine@dim.uchile.cl Received June 18, 2010; revi sed October 16, 2010; accepted October 20, 2010 Abstract We revisit the multi-allelic mutation-fitness balance problem especially when fitnesses are multiplicative. Using ideas arising from quasi-stationary distributions, we analyze the qualitative differences between the fitness-first and mutation-first models, under various schemes of the mutation pattern. We give some sto- chastic domination relations between the equilibrium states resulting from these models. Keywords: Evolutionary Genetics, Fitness Landscape, Selection, Mutation, Stochastic Models, Quasi-Stationarity 1. Introduction and Outline Population genetics aims at elucidating the fate of the allelic population composition when various dr iving ‘for- ces’ such as selection or mutation are at stake in the gene pool. This requires to identify first the updatin g mechan- isms responsible of the gene frequency-distributions evo- lution over time. In this note, we shall briefly revisit the basics of the deterministic dynamics arising in discrete- time asexual evolutionary genetics when the origin of motion is either the fitness or the mutations or both. We start with the multi-allelic haploid case before dealing with the diploid case. First, we consider general fitness mechanisms, then general mutation mechanisms and then we shall combine the two. The general purpose of the Sections 2-3 is to introduce separately the marginal allelic dyna mics driv en by fitness and then the one driven by mutations. These issues are of course part of the standard models discussed for exa- mple in [1-4]. In Section 4, we stress that there are two different ways to combine the fitness and the mutation effects. One (fitness-first), which is classical, consists in app- lying first the fitness mapping and then let mutation act on the result. The other (mutation-first) consists in rev- ersing the order. Stochastic models pertaining to the mutation/selection combination are numerous. See [5,6] (and the References therein) for the relation of a muta- tion/selection model with ancestral branching processes. A recent discussion on a Markov chain evolution to study the probability that a new mutant beco mes fixed in a Moran type model can be found in [7]. A work descri- bing phenotypic variation and natural selection by mod- eling population as a Mark ov point process can be found in [8]. In Section 5, we focus on a model with multiplicative fitnesses and general mutation pattern and we analyze both the fitness-first and mutation-first dynamics. Starting with the fitness-first dynamics, we observe that it has the structure of a discrete-time nonlinear master equation of some Markov process whose construction we give. In this stochastic interpretation, the polymorphic equilibr- ium state interprets as a quasi-stationary distribution of the Markov process conditioned to be currently alive. It is the left eigenvector of some sub-stochastic matrix A associated to its spectral radius. The corresponding right survival eigenvector makes sense in this in terpretation. A similar interpretation can be given when dealing with the mutation-first dynamics driven now by some sub-stocha- stic matrix B with its own left and right Perron-Frobe- nius eigenvectors. The matrices A and B are diago- nally similar. Using these stochastic tools, we observe that the mean fitness at equilibrium of the model B is larger than the on e of model A , together with some sto- T. HUILLET ET AL. Copyright © 2011 SciRes. AM 12 chastic domination properties between both the left and right Perron-Frobenius eigenvectors of the models A and B. If we specify the structure of the mutation ma- trix to be reversible, then the right and left Perron- Frobenius eigenvectors of each model can be related to one another by using an appropriate Schur product. Some simplifications also occur if we deal with symmetric mutations because the right (left) eigenvector of A coincides with the left (right) eigenvector of B. Section 6 particularizes the study of Section 5 when a house of cards condition holds for the mutation matrix. Because this mutation model is quite restrictive, some simplifications occur and the shapes of the polymorphic equilibrium states can be made more explicit. The interpretation of the fitness-first and mutation-first dynamics in terms of a stochastic process conditioned on not being currently absorbed in some coffin state sugg- ests that related conditional models for the evolutionary dynamics involving multiplicative fitness and mutations could also be relevant. In Section 7 we suggest to con- dition the process on its non-extinction either locally (stepwise) or to condition it globally on not getting ex- tinct in the remote future. Models A and B lead to different conditional dynamics. 2. Evolution under Fitness: The Deterministic Point of View We briefly describe the frequency distribution dynamics when fitness only drives the process. We start with the haploid case before moving to the diploid case. 2.1. Single Locus: Haploid Population with K Alleles Consider K alleles k A , =1, ,kK attached to a single locus. Suppose the current time-t allelic fre- quency distribution is given by the column vector :k x x, =1, ,kK1. We therefore have =1 =::= =1 K Kk k Sx xx0x the K simplex. Let := >0, k ww =1,,kK denote the absolute fitnesses of the alleles. Let * := = ll l wwx xwx (1) be the mean fitness of the population at time t. The variance in absolute fitness 2 x and the variance in relative fitness 2 x are given respectively by 2 2 =1 22 2 2 =1 =; =1=. K kk k Kk k k xw w w xww xx x xxx (2) 2.1.1. Dynamics The discrete-time update of the allele frequency distri- bution on th e simplex K S is given by2: =:=, =1,,. 'kk kk xw x pkK w xx (3) As required, the vector := k ppxx , =1, ,kK, maps K S into K S. In vector form, with :=Dx diag , =1,, k x kK, the nonlinear deterministic dynamics reads3: 11 == , 'DD ww wx xpxx= w xx or, with := ' xxx , the increment of x 1 =DI w w xx. x Without loss of generality, we can assume that 1 0< =1. K ww Thus that allele K A has largest fitness. Let ,: kl KK SS be the involution exchanging the coordinates k and l. When = kl ww, we have that ,, =() ' ' kl kl xx, and so the evolution is symmetric under ,kl . In that case, the alleles k and l can be merged into a single one. For K S x, let support =:>0, k kxx. Let =:=1 k kw be the set of alleles with maximal fitness. Any K S x such that support x is called an equilibrium state. A vector ==0,,0,1,0, ,0 k xe with k is called a pure (or monomorphic) equilibrium state. 2.1.2. Mean Fitness Increase According to the dynamical system (3), unless its equili- brium state is attained, the absolute mean fitness wx increases. Indeed, with =' wwwxxx : 2 == 1 =0, k kk kk kk kk k w wwxwx w wx w w xx x x (4) 1In the sequel, a boldface variable, say x, will represent a col- umn-vector so that its transpose, say * x, will be a line-vector. Simi- larly, * A will stand for the transpose of some matrix A . 2The symbol ' is a common and useful notation to denote the up- dated frequency. 3 D xw clearly is the Schur product of x and w. See [3] page 238 for a sim ilar notational convenience. T. HUILLET ET AL. Copyright © 2011 SciRes. AM 13 and it is >0 except when support x. The mean fitness is maximal at equilibrium. The rate of increase of wx is: 22 2 =1==. k k k kk k x ww x ww x xx xx (5) These last two facts are sometimes termed the 1930s Fisher fundamental theorem of natural selection (FTNS). Then, if there is an allele whose fitness is strictly larger than the ones of the others starting from any initial state of K S which is not an extremal point, the haploid traje- ctories will converge to this fittest state. 2.2. Single Locus: Diploid Population with K Alleles We now run into similar considerations but with diploid populations. 2.2.1. Joint Evolutionary Dynamics Let ,0 kl w, ,=1, ,kl K stand for the absolute fit- ness of the genotypes kl A A attached to a single locus. Assume ,, = kl lk ww (,kl w being proportional to the pro- bability of an kl A A surviving to maturity, it is natural to take ,, = kl lk ww ). Let then W be the symmetric fitness matrix with ,klentry ,kl w. Assume the current frequency distribution at time t of the genotypes kl A A is given by ,kl x . Let X be the frequencies array with ,klentry ,kl x . The joint evolu- tionary dynamics in the diploid case is given by the updating: , ,, ,, , = where ()=. () kl ' klklkl kl kl w x xXxw X (6) The relative fitness of the genotype kl A A is ,kl wX . The joint dynamics takes the matrix form: 11 == ' X XW WX XX where stands for the (commutative) Hadamard pro- duct of matrices. Let J be the KK matrix whose entries are all 1 (the identity for ). T h en 1 := = 1 =. ' X XX XJW X WXJ X Let 2 2,, ,=1 22 , 2,2 ,=1 ()= (); () =1= () () K kl kl kl Kkl kl kl XxwX X Xx X X (7) stand respectively the genotypic variance in absolute fit- ness and the diploid variance in relative fitness. The in- crease of the mean fitness is given by 2, ,, ,, ,, 2 == () =()()0, kl kl klklkl kl kl w Xxwxw X XX (8) which vanishes only at the equilibrium states maximizing lk w,, with a relative rate of increase: 2 =wX wXX . This is the diploid version of the FTNS. 2.2.2. Marginal Allelic Dynamics Assuming a Hardy-Weinberg equilibrium, the frequency distribution at time t, say ,kl x , of the genotypes kl A A is given by: ,= klkl x xx where , = kkl l x x is the mar- ginal frequency of allele k A in the whole genotypic population. The frequency information is = X x1 (1 is the unit K-vector) and the mean fitness is given by the quadratic form: * , , := = kl kl kl x xw W xxx . Let 2 2, ,=1 22 , 22 ,=1 =; =1= K kl kl kl Kkl kl kl xx w w xx xx x xxx (9) be respectively the genotypic varian ce in ab solute fitness and the diploid variance in relative fitness. If we first define the frequency-dependent marginal fitness of k A by , =:= kkll l k wW wx xx , the mar- ginal dynamics is given as in (3) by: 1 == =:,=1,,. k ' kkk k k w x xxWpkK xxx xx (10) In vector form (10) reads 11 ===:, 'W DW D xx xxxpx xx where p maps K S into K S. Iterating, the time-t fre- quency distribution tx is the ttimes composition of p applied to some 0x. In the diploid case, assuming fitnesses to be multi- plicative, say with ,= klk l www , then selection acts on the gametes rather than on the genotypes. Observing T. HUILLET ET AL. Copyright © 2011 SciRes. AM 14 *= kk ll l ww wx W x xx , the dynamics (10) boils down to (3). However, the mean fitness in this case is 2 =ll lwx x and not =ll lwx x as in the haploid case. 2.2.3. Increase of Mean Fitness Again, the mean fitness x, as a Lyapunov function, increases as time passes by. We indeed have ,, 2,, = 1 =0, ' kk klllkkll kl kl xwwxwxwx xxx xx x and vanishes only when the process has reached equili- brium. Its partial rate of increase due to frequency shifts only is := kk kxw xx. It satisfies 22 2 1 =1==, 2 k k kA kk k x w xx xx x xx (11) where 2 A x is the allelic variance in relative fitness 2 2 =1 :=21 . Kk Ak k w x x xx (12) 2.2.4. An Alternative Representation of the Allelic Dynamics There is an alternative vectorial representation of the dynamics (10) emphasizing its gradient-like character. Define the matrix * =GD x xxx. It is symmetric, positive semi-positive whose quadratic form vanishes only for the constants. Gx is partially invertible on the space orthogonal to the constants with left-inv e rse 11 1 =.GIJD K x x Note 1 GGxxδδ. Looking for a left-inverse in the weaker sense of the quadratic form, that is satisfying 1 ** =GG I δxxδδδ for all δ with =0δ, every 11 =GIJD K x x would do the job for any R . In particular =0 . Introduce the quan ti ty 1 =log 2 W V xx. Then, (10) may be recast as the gradient-like dynamics: 1 ==, W GWG V xxxxx x (13) with * ==0x1x as a result of ** =G1x0. Note ** =0. WWW VVGV xxx xx Based on [9,10], the dynamics (13) is of gradient-type with respect to th e Shashahani-Svirezhev distance metric given by 1/2 1/2 2 1 *1 =1 ,== . K ' Gkk k dG xx xxx xx Its trajectories are perpendicular to the level surfaces of W V with respect to this metric. From (11) and (12), ' G dx, x, which is the length of x, is also the square- root of half the allelic variance (the standard deviation) in relative fitness. 3. The Mutation Mapping We now briefly describe the frequency distribution dyna- mics when mutation is the only driving source of motion. Assume alleles mutate according to the scheme: kl A Awith probability ,0,1 kl satisfying ,=0 kk and , 0< 1 kl lk for all k. Let , := kl M be the mutation pattern matrix; we shall assume that the non-negative matrix M is irreducible. We first con- sider the deterministic diploid model involving muta- tions. 3.1. Only Mutations Considering first an updating mechanism of the freq- uencies where only mutations operate, we get ,, =,=1,,. ' kk lklkkl lk lk x xxxkK (14) In matrix form, with * M the transpose of M * ==:=:, 'MM MD 1 xx xxMxpx (15) and the update of the frequencies with mutations is given by the affine transformation * := . M I DM 1 M We have 0M and * = M M if and only if M is stochastic, = M 11. Also ** =1M1 and then M maps K S into K S because if *=11x, then * *** * == ==1 ' 1x1MxM1 x1x. The matrix * M is stochastic and irreducible and so, by Perron-Frobenius theorem, it has a unique strictly positive probability left- eigenvector associated to the real dominant eigenvalue 1. Let * eq x be this line-vector, then *** = eq eq xxM, or T. HUILLET ET AL. Copyright © 2011 SciRes. AM 15 = eq eq xMx. Under the irreducibility assumption on M , the frequencies d ynamics involving only mutations has a unique polymorphic equilibrium fixed point > eq x0. When M is primitive then * = lim t teq Mx1. This shows that * :==00 =, teq eq t t xx Mxx1xx regardless of the initial condition 0x belonging to K S. Note finally that from (15): ==:,IV M xM xx (16) where * 1 =2 VI MxxMx is the quadratic mutation potential. The probability right-eigenvector eq x of M uniquely solves =0VMx with =0 eq VMx maxi- mal. 3.2. Remarks and Special Cases 1) Reversible mutations: Let eq x solve *** = eq eq xxM. Define *1 =. eq eq DD xx MM We have *1 == eq eq D x M1Mx 1 , so * M is the sto- chastic matrix of the time-reversed process at equili- brium with invariant measure * eq x. If ** =MM , then the mutation pattern is said to be time-reversible. In this case , ,, , =. eq k kllk eql x x 2) If * = M M, then * =MM and M is doubly sto - chastic. In that case, * 1 =1,,1=: eq b K xx. A model with symmetric mutations by assuming for instance mul- tiplicative mutations: ,= klkl . In this case, with μ the column vector of the k s, * =. I Dμ Mμμ μ Alternatively, assuming ,1 =0, 1 kl K for all kl leads to =1 J KI M which is also symmetric. It is not necessary that * = M M in order to have M doubly stochastic. It suffices to impose * = M M11 . In that case although * MM , the overall input-output mutation probabilities attached to any state coincide and the equilibrium state again matches with the barycenter b x of K S. 3) (Kingman house of cards, [4]). Assume the muta- tion probabilities only depend on the terminal state, that is: ,= kll for all kl , still with ,=0 kk . Let *1 =,..., K μ. Then, * = M Dμ 1μ, =M 1μ1μ where * min<:=< 1max kk μμ1, * =1 I Mμ1μ and * ===1 . 'MDM x xMxxx1μμx (17) The equilibrium state is = eq xμμ . Note that 1 11 K μ. This model is reversible. In this model the coordinates are decoupled: =1 ' kk k x x μ, depends only on k x . 4) Assume the mutation probabilities only depend on the initial state, that is: ,= kl k for all lk. Then * =.IKD μ M1μ This mutation model is also reversible and the equi- librium state is ,1/ =. 1/k eq kl l x 5) (Random walk). In this case, M is tri-diagonal with ,,1 = kl kk u and eq x is a truncated geometric distribution with common ratio =uu : 1 ,1 =1 =,=1,,. k eq kKl l x kK This model is reversible. 6) (Cyclic mutation pattern). Here, ,,1 = klk lk , =1, ,1kK and ,,1 = K lKl . This model is not reversible and ,1 =. 1k eq kl l x 4. Combining Fitness and Mutations Let us now consider the dyna mics driven both by fitness and mutation. There are two ways to combine the fitness and mutation effects. One (fitness-first), which is clas- sical, consists in applying first the fitness mapping and then let mutation act on the result. The other (mutation- first), which seems to be less popular, consists in rever- sing the order. 4.1. Fitness-First Dynamics It is typically obtained by applying first the fitness ope- rator and then the mutation one to give the ‘fitness-first’ T. HUILLET ET AL. Copyright © 2011 SciRes. AM 16 dynamics [11]: ** 11 ==, 'W DDW WW xx xMxMx xxxx F (18) defining a new nonlinear transformation. Alternatively, = ' xpx where * 1 =DW Wx pxM x xx is the new mapping from K S to K S to consider. Component-wise, this is also as required ,, 1 =, =1 ,,. ' kkklkllkkkl lk lk xxw wxxw kK xxx x (19) We have: * ,1 ,1,,1 , ()= ,,,1,,, kkkkkl kkkKK lk S pe and so the extremal states k e are not invariant under p and from the fixed-point theorem, there exists some equilibrium state in K S. Using the representation (13) and (15): = =. W W IGV VGV M xM xMxx xMx x (20) This is not a gradient-like dynamics in general b ecau se there is a competition between the mutation and fitness potentials VM and W V. When = I M (no mutatio n) (20) boils down into (13) and when JW= (no selection), (20) boils down into (16). When both = I M (no mutation) and =WJ (no selection), =0xwith corresponding neutral =pxx . 4.2. Mutation-First Because W was assumed symmetric * =WW, there is another way to combine the mutation-selection effects. It is obtained by applying first the mutation operator and then the fitness operator to give the `mutation-first’ dyn- amics: '1 =: =, W D Mx xpxMx Mx (21) where * := W MxMxMx . We have: * *==1W 1p xMxMxMx if and only if * =WW and under this condition, this n ew ()px again maps the K simplex K S onto itself. The dynamics of :=yMx is 1 =, 'W D y yMy y which is of the form (18) and 1 = xMy may be reco- vered as an output from y only if M is invertible. Component-wise, each component k px may be read from ,, =1, =1,,. 'k kkkllkl lk lk W x xxkK Mx Mx (22) Equation (21) may also be recast as: 1 = =, W IGW VG V M xM xMxMx Mx xMx Mx (23) where, as before, =log 2 W V xx. The mean fit- ness function appearing in (23) is *** := ,WW MxMxMx=x MMx or else, the new fitness matrix to consider is * := ,WW MMMwhich is itself symmetric because * =WW. Just like (20), (23) neither is a gradient-like dynamics. Although natural, this alternative `mutation-first' way to combine mutations and fitness effects seems to have been less studied in the literature. 5. Multiplicative Fitness We now focus on the multiplicative fitness model. Fol- lowing the previous observations, we shall distinguish two cases. 5.1. Fitness-First In the haploid case or in the diploid case when fitnesses are multiplicative, ,= klk l www, with =ll l wwx x ,, ==, =1,,, ' llkk klk llk kkl llk lk wwxwxxwwxxw kK xx or 11 ==, 'DD ww xw xMwMx xx (24) where w is the constant colu mn-vector of the k ws and * =wxwx. When dealing with multiplicative fitn esses models, we shall assume >0 minkk w, =1 maxkk w and the second largest <1 k w. The image of the extremal states =l xe by the trans- formation ' xx reduces to l Me which belongs to the interior of K S. In that case, there exists a unique, globally stable polymorphic equilibrium state which is the fixed-point of (24). This follows from the Perron- Frobenius theorem commented in the forthcoming para- graph. Recall that in the absence of mutations, the multi- T. HUILLET ET AL. Copyright © 2011 SciRes. AM 17 plicative fitness model cannot have a polymorphic equili- brium state. 5.1.1. Polymorphic Equilibrium and Steady Mean Fitness Let * =AD wM, the latter ‘selection-first’ recurrence may be recast as ** * 1 1= .ttA tA xx x1 (25) Under our assumptions on w, =A1w and therefore A is sub-stochastic. By iteration ** * 1 =(0). 0 t t tA A xx x1 When M is primitive, so is 0A which has Per- ron-Frobenius left and right probability eigenvectors *>0 A x and >0 A y associated to its largest eigenvalue 1> >0 A . Then, * ,, 11 = lim t AA tAAkAk k Axy yx showing that, 0:=> , lim A tt xxx0 which is the required limiting polymorphic state. The value of A (respectively 2 A ) is the limiting haploid (diploid) mean fitness because: ** :==== AA AAA ww A xwxx1 . When looking at the equivalent reformulation (25) of (24), A x can be inter- preted as a quasi-stationary distribution as developed now. For the precise definition, see [12]. 5.1.2. A Stochastic Interpretation of the Deterministic Dynamics (25) A vector x of K S can be thought of as a probability vector. The dynamical Equation (24), as a nonlinear update mapping from K S to K S may be viewed as the discrete-time nonlinear master equation of some Markov process whose construction we now give. We shall need to introduce an extra state, say =0 which will be absorbing for the process we shall now construct. It will be useful to extend the matrix A to in the follow- ing way: ,0, 0,,0 =1 =1, =. K kklll l AAA Let then Lt be the random labels distribution of an individual at ti m e t, with enlarged state-space 0,1, , K . Let :=1,2 t Ut be an i.i.d. driving sequence of uniformly distributed random variables on [0,1]. Consider the random evolution equation 1, 1= ,==<=, 't' Lt l LtlLt lUALt l 111 where ,0,1,, ' ll K. From this construction :Ls s t is measurable with respect to : s Ust and we ge t , 1== . '' L tl LtlLt A (26) Let be the first time that Lt hits the absorbing state =0. Using the extinction time (26) may be recast as , 1=, > 1=>. Ltk LtktLtAt 1 Putting :== ,> k ztLt kt , we get an un -nor- malized version of (25): * , 1=>=,{1,, }. kLtkk ztAttA kK 1z We clearly have , ,, >= , lim Ak t Ak tAk Ak k y txy (27) and so 1> >0 t A t geometrically fast. From the last expression, the right-hand-side may be inter- preted as the propensity of a type-k allele to survive to its fate: the eventual extinction. If ,, > Ak Al yy indeed, the extinction time of the process started at k is larger than the one started at l (has larger survival asymptotic tails). We shall call A y the survival probability vector. Defining the normalized conditional probabilities =1 ===>, k kK k k zt x tLtkt zt we obtain the normalized haploid dynamics (25) * * =1 =,1,,. 'k kK k k A x kK A x x It may now be viewed as the nonlinear master equa- tion of some stochastic Markovian process. In view of this construction, the vector A x is the quasi-stationary distribution of Lt given>t . We note that the appeal to the coffin state was a necessary step to understand the normalization kk zx, and the stochastic interpretation of (25) allows to give sense to the right eigenvector A y of A . Clearly the above construction can be done for N particles, in particular =2N in the diploid case. 5.2. Mutation-First If instead of (24) the dynamics is of the type ‘mutation- first’ T. HUILLET ET AL. Copyright © 2011 SciRes. AM 18 * 1 =, 'Dw xMx wMx (28) because the mutation operator was applied first in the composition of the fitnesses and mutation effects, the latter recurrence may be recast as: ** * 1 1= (),ttB tB xx x1 (29) where now * =BD w M. By iteration ** * 1 =(0). 0 t t tB B xx x1 5.2.1. Equilibrium and Mean Fitness at Equilibrium Let > B x0 and > B y0 be now the left and right probability Perron-Frobenius eigenvector of B asso- ciated to its largest eigenvalue >0 B . We have * ,, 11 = lim t B B tBBkBk k Bxy yx showing that 0:=> . lim B tt xxx0 We have 1 =BDAD ww and so B is diagonally similar to A . Therefore = B A and = B AA Dw w xx toge- ther with 11 = B AA DD ww yyy. The limiting equilibrium mean fitness is now ** :== BBB wwxx w (or 2 B w in the diploid case). Re- calling = B AA Dw w xx, we have 2, , =. kAk k BkAk k wx wwx Since , == AkAkA k wwx , under the multiplicative fitness hypothesis we obtain: Proposition 1. =<<1. AAB ww Remark: The quantity 2 2, =1 k AAk kA w x w x is the variance in relative fitness at equilibrium for the model A We therefore have 22 =1or=1 . B ABAA A www w xx The equilibrium fitness of the second model is larger than the one of the first. Without mutations, only the fittest state, say K under our hypotheses, will survive, leading to an equilibrium mean fitness equal to * 0,0, ,0,1==1 K ww. Therefore, both mutation mo- dels lead to a decrease of the equilibrium mean fitness, when compared to the one without mutations. However, the first model involves mutations which are more dele- terious than the ones relative to the second one where mutations appear more advantageously. Note finally that **1* *1* ===== B BB BBBBA DB D ww wMx wxwx1x and, since * =BD w M then 1* =BD wM so that 1=B w1 if * 111 1 :=,, K ww w is the reciprocal fit- ness vector. As a result, *1*1 ==1 BBB B xwxw so that , *1 1 :== . Bk B B kk x w xw 5.2.2. A Stochastic Interpretation of the Deterministic Dynamics (29) We can repeat the above construction substituting B for A and we are done. 5.2.3. The Stochastic Dominations xx B st A and yy AstB For two K dimensional probability vectors a and b, we put st ab if for each l =1=1 . ll kk kk ab Proposition 2. We have B st A xx. Proof: = B AA Dw w xx and therefore , , , =1 =. kAk Bk K kAk k wx x wx With , =1 := l lAk k x , we have ,, , =1=1= 1 ,, =1= 1 =1 . Kl K Ak Ak kAkl klk lK kk kl Ak Ak kkl xx wx ww xx Since ,1 =1 , =1 , lAk kl l k Ak k x www x and ,1 =1 , =1 ,, KAk klK K kl Ak kl x www x , , =1=1 , =1 , Kl Ak kAkk l kk Ak k x wx w x and therefore , =1 ,, =1 =1 , =1 =, l kAk ll k Ak Bk K kk kAk k wx x x wx which means B st A xx. T. HUILLET ET AL. Copyright © 2011 SciRes. AM 19 We point out that we used the order 1 0< =1 K ww on .wWould we have consi- dered the reverse order, we would get the opposite dominat ion relati onship. We also have the following stochastic domination pro- perty between the two survival probability vectors: Corollary 3. AstB yy. Proof: Because B and A are diagonally similar, we also have = ABB DD ww yyy and the same argu- ment applies substituting , AB yy for , B A xx in the previous pro of . 5.3. Symmetric Mutations When * =MM, mutations are symmetric and * =BA. Therefore = B A xy and = B A yx. The left (right) pro- bability eigenvector of B matches with the right (left) probability eigenvector of A .s In this case, there is a stochastic domination property between the left and right eigenvectors of both models, namely Proposition 4. If mutations are symmetric AstA yx and B st B xy. 5.4. Reversible Mutations When dealing with reversible mutations with equilib rium distribution eq x, we show now that the r ight eigenvector of A (or B) can be computed from the left eigen- vector by using an appropriate Schur product involving w and eq x Let A be an irreducible non-negative matrix. Let , AA xy be the left and right probability eigenvectors of A , associated to the spectral radius A of A . If there exists a positive vector η such that ,, = kkl llk A A for all , kl , A is said to be reversible with respect to η. Consider the stochastic matrix 11 A A A A DAD yy (30) Its left probability invariant measure is easily seen to be ,, A y AAkAk Ak Dxy xx , which is the normalized Schur product of A x and A y We have Lemma 5. If A is reversible with respect to η, then A is reversible with respect to 2 A y Dη the Schur product of A y, A y and η Proof: 2 ,,, 11 ,,,, 2 ,, , . AkAk Ak kk lkA lkAklkl AllAl lAl yyy AAA A yy y As an illustration, we shall consider the fitness-first dynamics for which * =AD wM. This A will be rever- sible with respect to η if and only if * M itself is rev- ersible. Indeed, η must satisfy ,, =, kkkl lllk ww (31) and if this is the case * M must be reversible with respect to eq x, leading to , =. keqkk x w In case * M is reversible, A is reversible with respect to 2 A Dyη with entries proportional to 2 ,,eqkAkk x yw. But this must be the invariant measure of A which, up to a normalizing constant, is A A Dyx with un-normalized entries ,,AkAk x y. For this point, see also [13]. We conclude: Proposition 6. If the mutation matrix is reversible with respect to eq x, then A is reversible with respect to 21 12 = weq weq AA DD DD yy xx and 1 1 =. wA eq A wA eq DD DD x x y x y Example: If * M is symmetric, it is reversible with respect to the uniform measure ,=1 . eq k x K Therefore 11 =. AwAwA DD xyy Up to a normalization constant, we have 22 2 ,,,, ,, ===, eqk AkkAkkkAkAk Ak x ywywwxxy the Schur product of A x and A y. Substituting B for A and , B B xy for , AA xy , the same holds true for * =BD w M and w e ge t If the mutation matrix is reversible with respect to eq x, then 11 B B B BDBD yy is reversible with respect to 22 = weq weq BB DD DD yy xx and ,, ,,, = or =. wB eqkeq kB k BBk keqkBk wB eq k DD wx y xwx y DD x x y x y 6. Multiplicative Fitness and the House of Cards Condition We shall again distinguish two cases. 6.1. Fitness-First Assume the house of cards condition holds, leading to: * =AD wM, with ** =1 I M1μμ. In this case, the computations become more explicit. Since for all k, ,==1 kll k lk lk μ, we have 1 and 1. 11 K K K μμ T. HUILLET ET AL. Copyright © 2011 SciRes. AM 20 Under the multiplicative fitness and th e house of cards conditions, Equation (19) reads * 1 = 1 =. ' kkkkllkkl lk lk kkk xxwwxxw w wx x μ xw The equilibrium frequency distribution is therefo re the solution to the equ a tion * 1 =, AA A D w μ xxμ xw which, s i n ce * = AA xw, is exactly seen to be: ,=,=1,,, 11 / k Ak kA x kK w μ (32) where 0,1 A is such that ,=1 Ak kx . See [4] where these results appear first. Alleles k A with largest frequencies are those for which both , kk w are large. The equilibrium mean fitness is *=, 11 / kk AA kkA w w xw= μ (33) the spectral radius of A and * =AD wM satisfying =A1w. Because > A x0, we have: =>1 . max AAk k ww μ (34) If =1,μ,=, Ak k x =1, ,kK and = Akk kw . 6.2. Mutation-First When fitnesses are multiplicative: * =Www is symme- tric, (21) is also * 1 =: =. 'Dw xpx Mx wMx With * 1 =,, , k www this simplifies to give: ,, * =1 ,=1,,. 'k kkkllkl lk lk w x xxkK wMx (35) When the house of cards condition holds, * =BD w M, with ** =1 I M1μμ. Equation (35) further simpli- fies to: ** =1,=1,,. 1 'k kkk w x xkK μ wμμwx (36) From Equation (36), the equilibrium frequency distri- bution is the solu tion to the equations ,, ** =1,=1,,, 1 k Bkk Bk B w x xkK μ wμμxw which is exactly seen to be: ,* =,=1,,, 1 kk Bk Bk w x kK ww wμμ (37) where * =0,1 BB wxw is such that ,=1 Bk kx . Alleles k A with largest frequencies are those for which the product kk w is largest. Because A and B are diagonally similar, we have = B AA Dw w xx where == AAB w . From the expression (32) of ,Ak x , we get the alternative expression ,=,=1,,. 1 kk Bk Bk w x kK w μ (38) We also have 2 *=, 1 kk B B kBk ww w xw= μ (39) the equilibrium mean fitness under * =BD w M . Comparing the tw o expressions (3 7) and (38) o f kB x,, this suggests that * =1 B B w wμμ. Thus * === 1 AAB B ww wμμ and * =>0. BA B ww wμwμ (40) As a result * 1 > B wwμ μ (41) gives a lower bound for B w in terms of the average of w with respect to the mutation equilibrium probability measure =. eq xμμ Because > B x0, from (37), we also have: * 1<1,. kB wwk μwμμ If 1>0μ, this means * >1 maxk Bk wwwμ/μ whereas if 1<0μ this means * <1. mink Bk ww wμ/μ When =1μ, 2 = B kk kk kk www is explicit, together with ,=,=1,,. kk Bk kk k w x kK w Recalling that when =1:μ== AA kk k ww , we can check in this particular case that: > B A ww. 7. Alternative Conditional Models for the Evolutionary Dynamics The interpretation of (24) and of (29) in terms of a sto- chastic process conditioned on not being currently ab- T. HUILLET ET AL. Copyright © 2011 SciRes. AM 21 sorbed in suggests that other conditional models for the evolutionary dynamics involving multiplicative fit- ness and mutations could also be worth investigating. Consider first the fitness-first model (24) driven by * =AD wM. Let 1 1A A DA and consider the updating dynamics on the simplex ** 1ttAxx (42) Because =A1w, we have * A M which is the pure mutation stochastic matrix. Using the terms of the stochastic inter preta t i o n of ( 2 4) , we have: ,011 kl Lk APL l which is the transition ma- trix of a one-step conditioned process. Therefore condi- tioning locally Lt on non-extinction brings one back to the pure underlying mutation model with eq A xx. Let us consider a similar conditioning but for the mutation-first dynamics. With now* =,BD w Mlet 1 1 =B BDB and consider the dynamics ** 1ttBxx (43) We have * 11* 1 =B BDBDD w Mw M which is stochastic but cannot be reduced to mutation effects in general. This is an additional illustration of the differences bet- ween the two models based on A or B. Note that B and A are not diagonally similar. Let now 11 yA A Ay A DAD and consider the dyna- mics on the simplex ** 1ttAxx (44) We have: ,0 lim 1 kl sLk A PLls which is the transition matrix of a process conditioned on not getting extinct in the remote future (see [14]). Therefore conditioning globally ()Lt on non-extinction in the far future brings one back to a standard (linear) Chapman- Kolmogorov evolution equation. This conditioning being more stringent than the one involved in (24), one expects its limiting frequ enc y distribution (which is ,, AAAkAk Ak Dxy x xy ) to stay away more signifi- cantly from the origin {0}. For this model, the mean fitness at equilibrium will be: ,, * ,, . A kAkAk k A AAk Ak k wxy wD xy x wy When mutations are reversible, 12 ,, 12 ,,, . keqkAk Ak keqkAk k wx x xwxx Similar conclusions can be drawn if we define 11 : B B B BDAD yy . The main interest is that in both conditioning (either local or global), the deterministic updating mechanisms are now linear in sharp contrast with (24) and (29) in- volving rational updating mechanisms. 7. Acknowledgments The authors are indebted for support of the Basal Conicyt project and S. M. to the Guggenheim fellowship. 8. References [1] W. J. Ewens, “Mathematical Population Genetics. I. Theoretical Introduction,” 2nd Edition, Interdisciplinary Applied Mathematics, Springer-Verlag, New York, Vol. 27, 2004. [2] R. Bürger, “The Mathematical Theory of Selection, Re- combination, and Mutation,” Wiley Series in Mathemati- cal and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000. [3] S. Karlin, “Mathematical Models, Problems, and Contro- versies of Evolutionary Theory,” Bulletin of the American Mathematical Society (N.S.), Vol. 10, No. 2, 1984, pp. 221-273. [4] J. F. C. 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