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The canonical equation of adaptive dynamics a new role for the effective population sizes of population genetics Hans (=JAJ *) Metz QuickTime™ en een TIFF (ongecomprimeerd)-decompressor zijn vereist om deze afbeelding weer te geven. VEOLIAEcole Polytechnique & Mathematical Institute, Leiden University QuickTime™ en een TIFF ( ongecompri meerd) -decompr essor zijn vereist om deze afbeelding weer te geven. (formerly ADN) IIASA QuickTime™ and a decompressor are needed to see this picture. Preamble Real evolution mostly takes place in higher (even infinite) dimensional trait spaces. In higher dimensions the main ideas introduced by means of the graphical constructions still apply, but the analysis of the monomorphic dynamics hinges the ordering properties of the real line. To study the dynamics in the higher dimensional case we ususally fall back on a differential equation, grandly called the Canonical Equation of adaptive dynamics. Goal: This talk is given to its derivation, and to an unexpected recently discovered link with the theory of random genetic drift. overview the ecological theatre and the evolutionary play Long term adaptive evolution proceeds through the continual filtering of new mutations by selection. The supply of new phenotypes by mutation depends on the genotypic architecture and the genotype to phenotype map. Selection is a population dynamical process. Convenient idealisation: The time scales of the production of novel variation and of gene substitutions are separated. the adaptive play Given a population dynamics one can graft onto it an adaptive dynamics: Just assume that individuals are characterised by traits that may change through mutation, and that affect their demographic parameters. The speed of adaptive evolution is proportional to the population size n . a second, silent, play: random genetic drift Given a population dynamical model one can also graft onto it a random genetic drift. Just imagine that each individual harbours two alleles without consequential phenotypic effect, which in the reproductive process are reassorted according to Mendel’s laws. The speed at which variation is lost over time is inversely proportional to n. a formal connection between the two plays Result: The constants that appear in front of n in the formulas for the speeds of adaptive evolution and random genetic drift are the same. The product of this constant and n is known as the effective population size ne. Part I The ecological theatre the population dynamical side of adaptive evolution Adaptive evolution occurs by the repeated substitution of mutants in largish populations. Mutants emerge single in an environment set by a resident population. How many mutants occur per time unit depends on the size of the resident birth stream. The mutant invasion process depends on their phenotype and on the environment in which they invade. The assumption that the resident population is large makes that both the birth stream and the environment can be calculated from a deterministic community model. modelling the ecological theatre The easiest way of accomodating all sort of life history detail in a single overarching formalism is to do the population dynamical bookkeeping from births to births. (C.f. Lotka’s integral equation.) By allowing multiple birth states it is moreover possible to accommodate i.a. - parentally mediated differences in offspring size, social status, etc. spatially distributed populations (use location as birth state component) cyclic environments (use phase of cycle as birth state component) genetic sex determination, including haplo-diploid genetics, etc genetic polymorphisms. I will proceed as if the number of possible birth states is finite. modelling the ecological theatre Odo Diekmann Mats Gyllenberg Equilibria of structured populations generally satisfy: B = L(X|EX) B, EX = F(I), I = G(X|EX) B Diekmann, Gyllenberg & Metz (2003) TPB 63: 309-338 trait vector (affects the two operators that describe individual behaviour) modelling the ecological theatre Odo Diekmann Mats Gyllenberg Equilibria of structured populations generally satisfy: B = L(X|EX) B, equilibrium Birth rate vector per unit of area EX = F(I), environmental input, i.e., the Environment as perceived by the individuals next generation operator (i.e., lij is the Lifetime number of births in state i expected from a newborn in state j) I = G(X|EX) B per capita lifetime impingement on the environment population output, i.e., Impingement on the environment modelling the ecological theatre Odo Diekmann Mats Gyllenberg Equilibria of structured populations generally satisfy: B = L(X|EX) B, equilibrium Birth rate vector per unit of area EX = F(I), environmental input, i.e., the Environment as perceived by the individuals next generation operator (i.e., lij is the Lifetime number of births in state i expected from a newborn in state j) I = G(X|EX) B result of the dynamic equilibrium of the surrounding community Part II The adaptive play two subsequent limits (concentrate on traits, and rescale time appropriately) individual-based stochastic process canonical equation limit branching t trait value limit type: system size ∞ successful mutations/time 0 mutational step size 0 the canonical equation The “canonical equation of adaptive dynamics” was derived to describe the evolution of quantitative traits in realistic ecologies. Assumptions: large population sizes, mutation limitation, small mutational steps*. dX 12 n C dt * s Y X Y Y X T and, initially, simple ODE population models the canonical equation dX 12 n C dt s Y X Y Y X T X: value of trait vector predominant in the population n: population size, : mutation probability per birth event C: mutational covariance matrix, s: invasion fitness, i.e., initial relative growth rate of a potential Y mutant population. history basic ideas and first derivation (1996) hard proofs (2003) QuickTime™ en een TIF F (ongecomprimeerd)-decompressor zijn v ereist om deze af beelding weer te geven. Ulf Dieckmann & Richard Law Mendelian discrete diploids generations extensions (2008) QuickTi me™ en een -decompressor zi jn vereist om deze afbeel ding weer t e geve n. Nicolas Champagnat & Sylvie Méléard dX 122 ne,A C dt hard proof s Y X Y Y X Michel Durinx & me T general with Poisson life histories # offspring only rigorousassumptions for Underlying rather unbiological: individuals reproduce for pure age far only ODE model dependence clonally, have exponentially distributed lifetimes andsogive birthfor at communitymodel). equilibria Tran ageneral constant rate from birth Chi onwards (i.e., an ODE population case (2006) non-rigorous not yet published an unexpected connection Part III will argue, based on conjectured extensions of theorems that have been proven for simple special cases, that ne,A = ne,D the effective population size for random genetic Drift. derivation of the canonical equation For general equilibrium population dynamics the canonical equation appears first in another form: dX 2 b 2 dt e births per unit of time B=bU, 1TU=1 C probability of mutation per birth event evolution is mutation limited R0 Y E X Y Y X T mean of [mutational step approximation for the probability that a Y-mutant invades] mutational steps are small demographic ingredients 1 2 2e R0 Y E X Y Y X * R0(Y | EX) average life-time offspring number of a mutant allele producing Y when singly substituted in the resident genotype ( calculated as dominant eigenvalue of a next generation operator L(Y | EX) ). For the resident: R0(X | EX) = 1. U stable birth state distribution of resident( allele)s , normalised dominating right eigenvector of TU =1, L(X | E ) , 1 X rate based) reproductive values of newborn resident( allele)s, V (birth co-normalised dominating left eigenvector of L(X | EX), V TU = 1. demographic ingredients 2 2e * 2 2e R0 Y E X Y Y X : Var vi mi j u j i j with mi the lifetime number of offspring born in state i, begotten by a resident allele born in state j. For the resident: R0(X | EX) = 1. U stable birth state distribution of resident( allele)s , normalised dominating right eigenvector of TU =1, L(X | E ) , 1 X rate based) reproductive values of newborn resident( allele)s, V (birth co-normalised dominating left eigenvector of L(X | EX), V TU = 1. from the theory of branching processes 1000 # individuals 100 10 1 time 0 10 20 In an ergodic environment EX a population starting from a single individual: either goes extinct, with probability Q(Y |EX), or "grows exponentially" at a relative rate s(Y | X). QuickTi me™ en een T IFF (LZW )-decompressor zi jn vereist om deze afbee ldi ng weer t e geven. J.B.S. Haldane For constant EX and small s : 2 P 1 Q 2 ln R0 e Ilan Eshel dX 2 b 2 dt e the average effective step Z = Y-X approximation for the probability that a Y-mutant invades T 2 Z Z 2 e R0 Y EX Y Y X T C probability density of mutational steps g(Z )dZ Brook Taylor Smooth on thegenotype stronger to phenotype on the assumption maps assumption thatlead the to locally thatadditive mutations genetics. are mutations distribution In diploids the mutational isunbiased symmetric effect doubles over a full around the resident substitution 1 2 C 2 2e R0 Y E X Y Y X R0 Y E X Y Y X T mutational covariance matrix T aside on genotype to phenotype maps Some evo-devo: Genotype to phenotype maps are treated as smooth maps from some vector space of gene expressions to some vector space of phenotypes. Rationale: Most phenotypic evolution is probably regulatory, and hence quantitative on the level of gene expressions. regulatory regions coding region DNA reading direction the genotype to phenotype map: Consider a representative locus, to be called A, affecting the phenotypic trait, with resident allele a and potential mutant A. Alleles are supposed to affect the phenotype in a quantitative manner, expressible through an allelic trait Xa resp. XA . The corresponding phenotypes will be written as Xaa, XaA, XAA: Xaa = ( ;Xa,Xa; ), XaA= ( ;Xa,XA; ), XAA= ( ;XA,XA; ), with subject to the restriction ( ;Xa,XA; ) = ( ;XA,Xa; ) ( no parental effects). local approximate additivity QuickTi me™ en een T IFF (ongecom pri meerd)-decompressor zi jn vereist om deze afbeel ding weer t e geven. Andrea Pugliese Tom van Dooren We assume to be sufficiently smooth, and express the smallness of the allelic effect as XA = Xa + Z. The symmetry restriction implies ∂( ;Xa,XA; ) ∂( ;Xa,XA; ) = : = A´( ;Xa,Xa; ) ∂Xa XA=Xa ∂XA XA=Xa Hence XaA = ( ;Xa,Xa; ) + A´( ;Xa,Xa; ) Z + O(2) XAA= (;Xa,Xa; )+ A´(;Xa,Xa; )Z+ A´(;Xa,Xa ;) Z + O(2) XAA = ( ;Xa,Xa; ) + 2 A´( ;Xa,Xa; )Z + O(2). the mutational covariance matrix Let, given that a mutation happens, the probability that this mutation happens at the A-locus be pA, and let the distribution of the mutational step, Y := XA-Xa, be fA . Assume that the total mutational change is not only small but also unbiased, i.e, p ( A L ;Xa ,Xa Y;L ) fA (Y )dY 0 A then C pA ( L ;Xa , Xa Y;L ) ( L ;Xa , Xa Y;L ) fA (Y )dY T A pA A ( L ;Xa , Xa ;L ) YY T fA (Y )dY A T ( L ;Xa , Xa ;L ) A mutational covariances need not be constant! In the Mendelian case the derivation from the basic ingredients, genotype to phenotype map and genotypic mutation structure, shows that the canonical equation represents but the lowest level in a moment expansion. The next level is a differential equation for change of the mutational covariance matrix, depending on mutational 3rd moments, etc. recovering population size and fitness dX 2 b 2 dt e C R0 Y E X Y Y X T Tr : average age at giving birth, Ts : average survival time ne,A Tr 2 n 2 Ts e C s Y X Y Y X of the residents. n bTs b n Ts ln R0 Y E X s Y X Tr T demographic ingredients 3 dX T 2 r2 n dt Ts e C s Y X Y Y X Ts a F T (a) U da * 0 with fi(a) the probability density of the age at death of an individual born in state i. Tr a V T (a) U da * 0 with (a) composed of the average pro capita birth rates at age a. 0 V (a) U da V L X | E X U 1. T T T aside: robustness of the CE a dna ™e mi Tk ciuQ ro s serp m oce d .erut cip sih t ee s ot ded een era Géza Meszéna For small mutational steps the influence of the mutants on the environment at higher densities comes in only as a term with a higher order in the mutational step size than the terms accounted for by the canonical equation. Whether or not some previous mutants have not yet gone to fixation has little influence on the invasion of new mutants. The applicability of the canonical equation extends well beyond the case of strict mutation limitation. Part III The new result and its ‘proof’ random genetic Drift Operational definition: The effective population size for random genetic Drift, ne,D , is most easily defined as the parameter occurring in the usual diffusion approximation for the temporal development of the probability density of the frequency p of a neutral gene. 2 p 1 p 1 1 2 2 ne,D t p 2 Interpretation: The size of a population with non-overlapping generations and multinomial pro capita off-spring numbers that produces the same (asymptotic) decay of genetic variability as the focal population. Proposition The effective population sizes ne,A for Adaptive evolution and ne,D for random genetic Drift are equal whatever the life history or ecological embedding. idea of the proof Connect the approximation formula for the invasion probability derived from branching process theory PB (s) (Eshel’s formula) with the probability for take-over for a diffusion approximation including selection, PD p0 , s%: P s,n Tr PB s lim ns, 2 2 lim lim s0 n s0 ne,D n constant s e s s0 initial frequency of mutant allele Ts 1 PD 2n T , Tr s ne,D r 2Ts n s selection coefficient Why this particular proxy for combination system size of limits ? Why this formula for the initial mutant allele frequency p0? Why this formula for the selection coefficient ~s ? idea of the proof Tr P s 2 2 lim B s0 e s lim ns, s0 ne,D n constant Ts 1 PD 2n T , Tr s ne,D r 2Ts n s Why this formula for the selection coefficient s~ ? In the diffusion approximation time is measured in generations. Hence s = Tr s~ . idea of the proof Tr P s 2 2 lim B s0 e s lim ns, s0 ne,D n constant Ts 1 PD 2n T , Tr s ne,D r 2Ts n s Why this particular combination of limits ? ne,D n reflects the behavioural laws of resident individuals. We want to determine the first order term for small positive s in the Taylor expansion of the invasion probability. We want to remove the effect of the finiteness of n. idea of the proof Tr P s 2 2 lim B s0 e s p0 lim ns, s0 ne,D n constant Ts 1 PD 2n T , Tr s ne,D r 2Ts n s Why this particular combination of limits ? Left and right we take subsequent limits in different ways: (1) from full population dynamics to branching process, followed by an approximation for the invasion probability, (2) from full population dynamics to diffusion process, followed by a two-step approximation for the invasion probability. To get equality we want the corresponding paths in parameter space to approach the eventual limit point from the same direction. idea of the proof Tr P s 2 2 lim B s0 e s lim ns, s0 ne,D n constant Ts 1 PD 2n T , Tr s ne,D r 2Ts n s Why this formula for the initial mutant allele frequency p0 ? belief Any relevant life history process can be uniformly approximated by finite state processes. In continuous time at population dynamical equilibrium (E = EX ) : B: individual level state transition generator A: average birth rate operator Order the states such that the birth states come first. Then the next generation operator L(X, EX ) can be expressed as 1 L(X, EX ) K AB K T with KT 1 0 M 0 0 L O O O O L 0 0 M 0 1 0 L M M 0 L L L 0 M M 0 idea of the proof Tr PB s 2 2 lim s0 e s lim ns, s0 ne,D n constant Ts 1 PD 2n T , Tr s ne,D r 2Ts n s Why this formula for the initial mutant allele frequency p0 ? The consists N0 = diffusion N0 = for N astructured n the fastpopulation For genetic process results in of a slow diffusion of the gene frequency 1p, preceded by a fast process T % p0 V N0 bringing the population state 2n V T : co-normalised to +B N p02of , 2 pA0 (1 p0 ), (1 p0 )2 N0, N0 , N0 left eigenvector N nU%the stationary population state without genetic differentiation, with U the normalised eigenvector of A + B. idea of the proof Tr PB s 2 2 lim s0 e s lim ns, s0 ne,D n constant Ts 1 PD 2n T , Tr s ne,D r 2Ts n s Why this formula for the initial mutant allele frequency p0 ? For N0 = N0 = N n the fast process results in 1 %T p0 2n V N0 V T : co-normalised left eigenvector of A + B For a just appeared mutant: Lemma: N0 KU Ts T V K V Tr T p0 1 Ts 2 n Tr Part IV Finale a corollary: individual-based calculation of ne,D Tr PB s 2 2 lim s0 e s lim ns, s0 ne,D n constant ne,D Ts 1 PD 2n T , Tr s ne,D r 2Ts n s Tr n , 2 Ts e a result already reached by different means by William G Hill (1972) for the simple age-dependent case, William G Hill Edward Pollak and by Edward Pollak (1979, ...) for the age dependent case with multiple birth states. a practical consequence At equal genetic and developmental architectures and population dynamical regimes the speeds of adaptive evolution and random genetic drift are inversely proportional. Already for moderately large effective population sizes adaptive processes dominate, and neutral substitutions will be largely caused by genetic draft. The End a tricky point The calculation linking the initial condition for the structured mutant population to the initial condition of the diffusion of the gene frequency assumes that there is no need to account for extinctions before the reaching of the slow manifold, in blatant disagreement with the fact that mutants arrive singly. The justification lies in a relation linking the invasion probabilities of single invaders to those of mutants arriving in groups that are - so small that their influence on the environment can be neglected, yet - so large that the probability of their extinction before their i-state distribution has stabilised can be neglected. a tricky point, cont’d Consider a mutant population starting from an initial cohort of newborns =(1, … , k), i the number in birth state i, drawn from some law . Let P(,s) be the correspondig probability of invasion. Ansatz: P(,s) = ()s + o(s), (i) =: i. Then (under some conditions on the “tail” of ) k i P( , s) E 1 1 P( i , s) o E o(s) i 1 Hence k i k 1 1 P( i , s) i 1 ( ) lim E E i i TE s0 s i i More about the robustness of the canonical equation slowing down by within population variation 1 Away from full mutation limitation: the speed of evolution tends in the case of clonal reproduction to be considerabley smaller than predicted by the canonical equation, thanks to so-called clonal interference: Without mutation limitation the effect of different mutants in no longer additive as different mutants coming from the same parent type compete (with ususally the furthest one winning the race). This effect largely disappears with Mendelian inheritance: Substitutions occur in parallel, and the local additivity of the genotype to phenotype map makes that in the deterministic realm the different loci have little influence on each other’s speed of substitution. slowing down by within population variation 2 In the initial stochastic realm evolution is slowed down a bit by the additional variability in the offspring number of newly introduced alleles coming from the variability in the partners with whom they team up to make a body. If this variability would be fixed T dR dR 2e 2e 12 0 Cp 0 dX dX Cp the covariance matrix of the variation of X in the population. If 2e would be calculated based on empirically determined components the additional term is automatically incorporated. However, the above expression only gives a very partial picture since Cp is bound to fluctuate in time. dissecting the limit lim ns, s0 ne,D n constant Ts 1 PD 2n T , Tr s r s why this particular limit? First hint: For Karlin models the chosen limit procedure naturally recovers the result from their branching process limits. Karlin models (without any population structure) In Karlin’s conditioned direct product branching process models n/ne,D = 2 = variance of offspring number 2 1 exp 2 s 1 PD n , s 1 exp 2ns 2 lim ns, s0 PD n 1, s 2s 1 , 2 which equals the limit result from branching process theory. why this particular limit? 1 exp 2ne,D s%n 1 exp 4ne,D s% First hint: For Karlin models the chosen limit procedure naturally recovers the result from their branching process limits. General heuristics: - The effect of finite population size should disappear. (This effect occurs primarily in the integrals making up the denominator of the diffusion formula.) Let ne,D ~s ∞. The goal is to obtain the coefficient of the linear term in an ~ ~ expansion in s. Divide by s and let s ∞. Eventual “justification”: A study of the paths in parameter space corresponding to the two ways of taking subsequent limit shows that in both paths the limit point is approached in a similar manner. diffusion approximation versus limit Population geneticists usually express the diffusion approximation in the variables relative gene frequency, p, and generations, t. Mathematicians express the diffusion limit in rescaled time = t/n, leading to (for the equation with selection) 2 p 1 p p 1 p 1 1 ns% 2 2 2 ne,D n p p where the expressions ne,D n ns should be interpreted as single symbols instead of as formulas: ne,D n : lim ne,D n n lim ns% ns% : n s ns n 1 the paths in parameter space ~ p0 Bold arrows: branching process limit followed by s~ 0. Thin arrows: each arrow corresponds to a separate diffusion limit; ~, their angles correspond to the values of [ns] their endpoints to the values of x0. The sequence of arrows corresponds to the limits ns~ , s~0. proof of the lemma Ts T V K V Tr T Ts T V K V Tr proof of the lemma T S : sij = expected residence time in state i of an individual born in state j U : normed stationary distribution of h-states (right eigenvector of A+ B) V : reproductive values of h-states (left eigenvector of A+ B, V TU% 1 ) The upper part of V is proportional to V L : lij = expected future # kids in state i from parent who is now in state j The leftmost square block of L equals L. Ts 1T SU U Ts1SU V T cV T L% since vi is the expected long term birth rate if the mutant population is started with a single individual in state i and vi is the expected long term population size when the mutant population is started with a single individual in state i. V U% c V T L%Ts1 SU 1 T c Ts % = Tr ? V T LSU Ts c T % = Tr ? V LSU proof of the lemma Tr is defined as Tr T T a V (a) U da V ()d da U 0 0 a with (a) containing the average pro capita birth rates at age a. For a finite state process (a) K T Ae Ba K Tr V T K T AB2 KU. so that Similarly S e Ba da B1 K 0 Therefore indeed and L T Ba T 1 K Ae da K AB . 0 % Tr . V T LSU proof of the lemma, discrete time case Tr is defined as Tr aV (a)U V T a1 T ()U a1 a with (a) containing the average pro capita births at age a. For a finite state process (a) K T ABa K so that Tr V K ABI B KU. T T 2 Similarly S B K B I B K a a1 Therefore indeed 1 and L K T ABa K T AI B . a 0 % Tr . V T LSU 1 serendipitous pay off: quicker and more insightful ways for calculating ne,D example: micro- and macrogametes Assumption: everybody can be considered to be born (stochastically) equal. R0 Y E X * 1 f Y E X m Y E X 2 with m Y EX f Y EX the average lifetime number of successful For the resident: macrogametes microgametes f X EX m X EX 1. of the mutant heterozygote example: micro- and macrogametes contribution of Mendelian sampling 2e 14 2f 2(cf,m 1) 2m * with 2f 2 m macrogametes the variance of the lifetime number of successful microgametes of the resident, and cf,m the covariance of these two lifetime numbers. example: micro- and macrogametes Denote the average pro capita rate of parenting at age a as (a). (a) f (a) m (a) (a)da f X E 1 0 0 f X m (a)da m X E X 1 * Tr 1 a 2 (a) da 0 1 2 T r,f Tr,m example: micro- and macrogametes * Ts a f(a) da 0 with f(a) the probability density of the age at death of an individual. special case: separate sexes Call the relative frequencies at which the two sexes are born into the population qf and qm and their average life-lenghts Ts,f and Ts,m , then Ts qfTs,f qmTs,m . example: micro- and macrogametes Let a subscript + indicate that the quantity in question refers to the number of offspring of a female or male, then f Y E X qf Y E X f Y E X , m Y E X qm Y E X m Y E X . Hence R0 Y E X 1 qf Y E X f Y E X qm Y E X m Y E X . 2 For the resident f =1 and m =1. Hence 1 1 f X E X , m Y E X . qf qm Moreover, the production of macro- and microgametes are mutually exclusive. Hence cfm = -1. on calculating variances Consider a discrete nonnegative random variable h such that with probability p: with probability q = (1-p): h = 0 h = k with k another nonnegative random variable, then Eh = qEk and 2 2 2 Var(h) Eh 2 Eh q Var(k) Ek qEk . When moreover Eh = 1 so that Ek = q-1: 2k Var(k) Ek 1 q Var(h) Eh 1 q Var(k) q 1 q 2 2 2 example: separate sexes 2 e 1 4 2(cf,m 2 f 2 2 1 q 1 qm f 1) f+ m+ 4qf 4qm 2 m the coefficient of variation (standard deviation/mean). Tr n Tr ne 2 Ts e Ts With the definitions ne,f 1 2f+ 1 qf 2m+ 1 qm 4 Ts Ts,f nf 4 Ts Ts,m nm Tr,m nm Tr,f nf , , ne,m 2 2 Ts,m m+ 1 qm Ts,f f+ 1 qf this gives Tr,f f , Tr Tr,m m , Tr 4ne,f ne,m ne . f ne,f m ne,m Generalises the textbook formula for the case Tr,f =Tr,m =1( f =m =1). example 2: simplest cyclic environment bj = j-1 nj-1 nj = pj bj pi probability that a newborn in phase i-1 survives till phase i bi number of births at phase i i pro capita number of births at phase i 0 p 1 1 L(X | EX ) 0 M 0 V L 0 O O L bi i k L 0 L L O O O 0 k 1 pk 1 1 b1 L k pk 0 M M 0 0 b b 2 1 0 M 0 1 bk L 0 O O L L 0 b1 bk L L 0 O M O O M 0 bk bk 1 0 b1 U 1 M bi i bk example 2: simplest cyclic environment 2 bi 2 bj 2e Var j vi mij u j i j i k b bi j j j1 i bi 1 1 2 bi 1 2j m j i i b j j k k j b 2j1 k k j b j1 Next use m j b j1 b j j p j and write ne, j : to rewrite this as b j1 mj 2 j jnj mj 2 j 1 1 1 bi . k j k j ne, j 2 e The end result is the textbook harmonic mean expression: pb b Tr n 1 1 1 ne 2 ne, j . 2 2 Ts e pb b e e nj p j 2j The end