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Biological introduction The Galton - Watson model A model for Neandertals’ extinction

The Mathematics and Physics of HumanOrigins

Armando G. M. [email protected]

UFMG - Departamento de Matemática

30/1/2014


Outline

Biological introduction

The Galton - Watson model

A model for Neandertals’ extinction


Coautores

Carlos H. C. Moreira (UFMG -Departamento de Matemática)

Maurizio Serva (Universitàdell’Aquila - Itália)


Uma breve revisão da Biologia Matemática

• Tempo contínuo ou

tempo discreto.• Com estrutura espacial ou sem estrutura espacial.• Determinístico ou aleatório.



• Tempo contínuo ou tempo discreto.

• Com estrutura espacial ou sem estrutura espacial.• Determinístico ou aleatório.



• Tempo contínuo ou tempo discreto.• Com estrutura espacial ou

sem estrutura espacial.• Determinístico ou aleatório.



• Tempo contínuo ou tempo discreto.• Com estrutura espacial ou sem estrutura espacial.

• Determinístico ou aleatório.



• Tempo contínuo ou tempo discreto.• Com estrutura espacial ou sem estrutura espacial.• Determinístico ou

aleatório.



• Tempo contínuo ou tempo discreto.• Com estrutura espacial ou sem estrutura espacial.• Determinístico ou aleatório.


Natural history of humans

• First hominid fossils date to 7 to 6 million years ago:divergence between hominids and chimpanzees.

• All hominid fossils dated up to more or less 1 million yearsago were excavated exclusively in Africa.

• Although hominid fossils and archaeological remains beginto appear in Asia and Europe by 1 million years ago, theyare considered mostly primitive.

• Anatomically modern fossils appear in Ethiopia around 200thousand years ago.


Neandertals

• Neandertals inhabited Europeand West Asia from 350thousand to 30 thousand yearsago, when they became extinct.

• Anatomically modern humanscoexisted with Neandertals in theMiddle East for at least 130thousand years and in Europe forat least 10 thousand years. Inother parts of the world, modernhumans coexisted with otherancient human forms.


Mitochondrial DNA

• Most of the genetic information in higher animals is locatedat nuclear DNA.

• In spite of that, some DNA may be found in the subcellularorganelles called mitochondria. It is called mitochondrialDNA, abbreviated mtDNA.

• mtDNA is very short; in humans it consists in only 16,569base pairs carrying the information for 37 genes.

• mtDNA of humans and many other species has been theobject of intensive research in the last decades, both for itsavailability (because mitochondria are so numerous) andpeculiar inheritance mechanism.

• Unlike nuclear DNA, which is inherited in equal parts frommother and father, mtDNA is inherited only from themother.


mtDNA inheritance


The mitochondrial Eve

• By examining mtDNA in a sample of 147 living humans andtaking into account mutations, Cann, Stoneking and Wilsonasserted in 1987 that mtDNA of all living humans could bedescribed as mutations in the mtDNA of a single woman.

• This woman was called mitochondrial Eve.

• Using known mutation rates and geographical correlations,it could be inferred that the mitochondrial Eve lived inAfrica between 200 and 100 thousand years ago.

• Unlike the biblical Eve, the mitochondrial Eve is notsupposed to be the only woman living at her time.

• The time and the place in which the mitochondrial Evelived are considered a strong evidence for the Out of Africamodel for the origin of our own species, now accepted bythe vast majority of geneticists and paleoanthropologists.


The mitochondrial Eve

• By examining mtDNA in a sample of 147 living humans andtaking into account mutations, Cann, Stoneking and Wilsonasserted in 1987 that mtDNA of all living humans could bedescribed as mutations in the mtDNA of a single woman.

• This woman was called mitochondrial Eve.• Using known mutation rates and geographical correlations,

it could be inferred that the mitochondrial Eve lived inAfrica between 200 and 100 thousand years ago.

• Unlike the biblical Eve, the mitochondrial Eve is notsupposed to be the only woman living at her time.

• The time and the place in which the mitochondrial Evelived are considered a strong evidence for the Out of Africamodel for the origin of our own species, now accepted bythe vast majority of geneticists and paleoanthropologists.


The human mtDNA tree

Wilson, Cann (1992) Sci. Am. 266 (4): 68-73.


The distribution of pairwise differences in humanmtDNA tree

Cann, Stoneking and Wilson (1987), Nature 325: 31-36.


The Out of Africa model

• Despite the long coexistence time of modern humans andother ancient forms, according to the Out of Africa model,there was no mixing between them.

• The closeness argument:

We living humans are very closeto each other in mtDNA. If some mixing had existed, wewould have to see its traces in living humans as a differenttype of mtDNA, associated with Eurasians.


The Out of Africa model

• Despite the long coexistence time of modern humans andother ancient forms, according to the Out of Africa model,there was no mixing between them.

• The closeness argument: We living humans are very closeto each other in mtDNA. If some mixing had existed, wewould have to see its traces in living humans as a differenttype of mtDNA, associated with Eurasians.


Neandertal mtDNA

• More recently, Krings et al. Cell 90 (1997) and Nat.Genetics 26 (2000) succeeded in extracting mtDNA fromsome few Neandertal fossils.

• Whereas the number of differences between pairs of livinghumans at a certain part of mtDNA is 8.0± 3.1 base pairs,they found the difference between one of the Neandertalsand living humans is 27.0± 2.2. Similar figures were foundfor the other Neandertal specimens.

• Indeed Neandertal’s mtDNA is very different of ours.• The distance argument:

Individuals with such distantmtDNAs cannot belong to our species.


Neandertal mtDNA

• More recently, Krings et al. Cell 90 (1997) and Nat.Genetics 26 (2000) succeeded in extracting mtDNA fromsome few Neandertal fossils.

• Whereas the number of differences between pairs of livinghumans at a certain part of mtDNA is 8.0± 3.1 base pairs,they found the difference between one of the Neandertalsand living humans is 27.0± 2.2. Similar figures were foundfor the other Neandertal specimens.

• Indeed Neandertal’s mtDNA is very different of ours.• The distance argument: Individuals with such distant

mtDNAs cannot belong to our species.


Recent facts

• By the 1980’s the large majority of the scientific communityhad adopted the Out of Africa model based on its stronggenetic support.

• Sequencing mtDNA of a few Neandertal fossils (1997)gave further genetic support to the Out of Africa model.

• In 2003, fossils found in Ethiopia finally provide the “hardevidence" for the Out of Africa model.

• Among the resistants, M. Serva (works in 2004, 2005 and2006) and A. G. M. N. and C. H. C. Moreira (works in 2005and 2006).

• Sequencing of the nuclear DNA of a Neandertal fossil wasaccomplished only in 2010 (Green et al., Science 328).

• It showed that non-African living humans are geneticallycloser to Neandertals than Africans.

• The above paper is considered as an experimental proofthat the strict Out of Africa model is false.


The model

• In 1874 Galton and Watson proposed a simple stochasticmodel for explaining disappearance of family names inEngland.

• The same model may be used for studying survival ofmtDNA lineages if we exchange family names by mtDNAlineages and men by women.

• Model hypothesis:• Generations are non-overlapping.

• The number of daughters per woman is a random variable.• The probability distribution for this variable, the offspring

distribution, is the same for any woman in any generation.


The model



• Model hypothesis:• Generations are non-overlapping.• The number of daughters per woman is a random variable.

• The probability distribution for this variable, the offspringdistribution, is the same for any woman in any generation.


The model



• Model hypothesis:• Generations are non-overlapping.• The number of daughters per woman is a random variable.• The probability distribution for this variable, the offspring

distribution, is the same for any woman in any generation.


Model solution

• Let qr , r = 0,1,2, . . . be the probability that a woman has rdaughters.

• Let m = q1 + 2q2 + 3q3 + . . . be the mean number ofdaughters per woman.

• If 0 < m ≤ 1, we may prove that the process will lead topopulation extinction with probability 1.

• If m > 1, the population may survive for an infinite numberof generations with positive probability:

phase transition.• m = 1 is the critical point.


Model solution

• Let qr , r = 0,1,2, . . . be the probability that a woman has rdaughters.

• Let m = q1 + 2q2 + 3q3 + . . . be the mean number ofdaughters per woman.

• If 0 < m ≤ 1, we may prove that the process will lead topopulation extinction with probability 1.

• If m > 1, the population may survive for an infinite numberof generations with positive probability: phase transition.

• m = 1 is the critical point.


Infinite time survival probability as a function of m:phase transition

1 2 3 4m

0.2

0.4

0.6

0.8

1.0

Θ


The correlation time

• Let θn be the probability that the Galton-Watson processwill survive more than n generations, and θ = limn→∞ θn.

• We may show that if m 6= 1, then |θn − θ| ∼ e−n/ξ, where ξis defined as the correlation time.

• ξ diverges when m→ 1. We may show that if m is close to1, then ξ ≈ 1

|m−1| .


The mitochondrial Eve in the Galton–Watson model

• Let W be the number of women which lived at the time ofthe mitochondrial Eve. Some estimates relying on nuclearDNA diversity of living humans suggest W ≈ 5000.

• If each tree belonging to each woman survives withprobability θ and lineages are independent, the number ofsurviving mtDNA lineages is a binomially distributedrandom variable.

• As an example we may take a Poisson offspringdistribution, qr = e−m mr

r ! and show that a mitochondrialEve may exist with reasonable probability only in a slightlysupercritical range. Such a range is not too much alteredfor other typical offspring distributions.


Range for the mitochondrial Eve with Poisson offspring

We show, for W = 5000, as functions of m the probabilities forsurvival of zero, one or more than one mtDNA lineage.

1.0002 1.0004 1.0006 1.0008 1.0010m

0.2

0.4

0.6

0.8

1.0

probability

A G M N and C H C Moreira, Physica A (2006), 368.


Pruned trees

A pruned tree is defined as a genealogic tree from which wedelete all individuals which left no descendants among thepresent generation population.We show below a genealogic tree and the pruned tree derivedfrom it.


Supercritical pruned trees

• A G M N and C H C Moreira (2006): The number ofgenerations between branching events in a Galton-Watsontree and its application to human mitochondrial DNAevolution. In: Mondaini RP, Dilao R, editors, BIOMAT 2006.World Scientific.

• We showed that branch lengths in infinite generationsupercritical pruned trees are geometrically distributed.Moreover

• E(`) = 11−e−1/ξ

ξ→∞∼ ξ

• In a finite generation slightly supercritical pruned tree wemay have branches of all sizes up to the whole tree withalmost uniform probability.

• This is a consequence of being close to a critical point,where correlations diverge.


The human mtDNA pruned tree

Any subtree contains both long and short branches.


The dynamics of mtDNA genealogic trees

• As time goes by, thenumber of individuals atthe tips of branches vary.

• Sometimes branches areextinguished.

• When a branch linked tothe most recent commonancestor goes extinct, thisancestor changes. Themaximum genealogicdistance betweenindividuals falls suddenly.

Eve


The dynamics of mtDNA genealogic trees

• As time goes by, thenumber of individuals atthe tips of branches vary.

• Sometimes branches areextinguished.

• When a branch linked tothe most recent commonancestor goes extinct, thisancestor changes. Themaximum genealogicdistance betweenindividuals falls suddenly.

New Eve


Sudden decreases in the genealogic distances

• M. Serva (2005),J. Stat. Mech.P07011 noticedthe phenomenonin a constantpopulationmodel.

• The samephenomenonhappens in theslightlysupercriticalGalton – Watsonmodel.


Some conclusions by Serva

• The maximum genealogic distance may be much largerthan genealogic distances within a subpopulation.

• If humans and Neandertals were part of the sameinterbreeding population, then before Neandertals’extinction, the maximum distance between pairs in thewhole population might have been much larger than it isnowadays.

• Neandertal’s extinction might have caused a suddendecrease in the maximum distance. The closeness anddistance arguments do not hold.





• Neandertal’s extinction might have caused a suddendecrease in the maximum distance.

The closeness anddistance arguments do not hold.





• Neandertal’s extinction might have caused a suddendecrease in the maximum distance. The closeness anddistance arguments do not hold.


Conclusions by A G M N and C H C Moreira

• The mitochondrial Eve may exist in the Galton – Watsonmodel with a good probability only in a slightly supercriticalregime.

• So, even for an exponentially slightly expanding population,given that some mtDNA lineages survive, most probablyonly one such lineage will survive.

• Existence of a single human mtDNA lineage is thus ademographic phenomenon, not related with Neandertalsbeing or not being part of the human population.

• The sudden drops in maximum distances phenomenonoccurs in the slightly supercritical Galton – Watson model,too.


The new experimental discovery

• A first draft of a Neandertal nuclear DNA was published inMay 2010: Green et al., Science 328, 710 (2010).

• By comparison with 5 living individuals (2 Africans, 1European, 1 East Asian and 1 fom Papua New Guinea),Green et al. discovered that Neandertals are significantlyand equally closer to all non-Africans than to Africans.

• Based on that discovery, they finally admitted that mixingbetween anatomically modern humans coming from Africaand Neandertals had indeed occurred.

• They estimate that living non-Africans possess between 1and 4% of their nuclear DNA of Neandertal origin.

• Mixing probably occurred in Middle East, before Africansarrived to Europe and the rest of Asia.

• Fossils in some caves in Israel document alternateoccupation by Neandertals and modern humans for atleast 130,000 years.


Opinions change


A model for the mixing of modern humans andNeandertals (with M. Serva, PLoS ONE 7(10): e47076

(2012))

• Two subpopulations coexisting in the same geographicalregion.

• Subpopulation 1 are African ancestors of Eurasians (AAE)and subpopulation 2 are Middle Eastern Neandertals(MEN).

• Constant total population equal to N individuals during thecontact.

• Absence of biological obstacles to mixing.• Slow mixing due to “cultural” obstacles.• At each generation α individuals of each subpopulation

would migrate to the other subpopulation.• Individuals born in a subpopulation are considered as

elements of that subpopulation, even if one or both theirparents were originated in the other subpopulation.


The model

• Neutrality: the mean number of children of the individualsdoes not depend on the subpopulation label.

• Instead of measuring time in generations g, rescale it as:t = g/N.

• Let x(t) ∈ [0,1] be the fraction of AAE in the totalpopulation.

• We will eventually suppose that at some time TNeandertals were extinct: x(T ) = 1.

• Let y1(t) be the mean fraction of modern nuclear DNAamong the AAE.

• Let y2(t) be the mean fraction of modern nuclear DNAamong the MEN.


The equations

• Then y1(t + 1N ) =

(1− α

Nx(t)

)y1(t) + α

Nx(t) y2(t)

y2(t + 1N ) = α

N(1−x(t)) y1(t) +(

1− αN(1−x(t))

)y2(t)

.

• On deducing the above equations we tacitly made a meanfield hypothesis.

• Such hypothesis is justified by results in B. Derrida, S. C.Manrubia, D. H. Zanette, Phys. Rev. Lett. 82, 1987 (1999)if α� 1/ log2 N.


Solving the model

• Taking the N →∞ limit we get{y ′1(t) = − α

x(t) (y1(t)− y2(t))y ′2(t) = α

1−x(t) (y1(t)− y2(t)).

• Initial condition: y1(0) = 1, y2(0) = 0.• Defining z1(t) = y1(t)− y2(t) and z2(t) = y1(t) + y2(t) we

may solve the ODEs:•

z1(t) = exp[−∫ t

0

α

x(s)(1− x(s))ds]

and

z2(t)−1 = y2(t)−(1−y1(t)) =

∫ t

0

α(2x(s)− 1)x(s)(1− x(s))

z1(s)ds .

• What should we use as a realistic x(t)?


Solving the model


x(t) (y1(t)− y2(t))y ′2(t) = α

1−x(t) (y1(t)− y2(t)).

• Initial condition: y1(0) = 1, y2(0) = 0.

• Defining z1(t) = y1(t)− y2(t) and z2(t) = y1(t) + y2(t) wemay solve the ODEs:

•

z1(t) = exp[−∫ t

0

α

x(s)(1− x(s))ds]

and

z2(t)−1 = y2(t)−(1−y1(t)) =

∫ t

0

α(2x(s)− 1)x(s)(1− x(s))

z1(s)ds .



Solving the model


x(t) (y1(t)− y2(t))y ′2(t) = α

1−x(t) (y1(t)− y2(t)).


may solve the ODEs:

•

z1(t) = exp[−∫ t

0

α

x(s)(1− x(s))ds]

and

z2(t)−1 = y2(t)−(1−y1(t)) =

∫ t

0

α(2x(s)− 1)x(s)(1− x(s))

z1(s)ds .



Solving the model


x(t) (y1(t)− y2(t))y ′2(t) = α

1−x(t) (y1(t)− y2(t)).


may solve the ODEs:•

z1(t) = exp[−∫ t

0

α

x(s)(1− x(s))ds]

and

z2(t)−1 = y2(t)−(1−y1(t)) =

∫ t

0

α(2x(s)− 1)x(s)(1− x(s))

z1(s)ds .



The neutral Wright-Fisher process

• Suppose a fixed size population of N individuals. Each hasa genetic character determined by a gene with two alleles:either blue or red.

• Suppose that the blue allele has neither advantage ordisadvantage with respect to the red allele: neutrality.

• Let x(g) be the fraction of red individuals at generation g.


Markov chain

• The evolution of x(g) is a Markov chain with states{0, 1

N ,2N , . . . ,1}.

• States 0 and 1 are absorbing and all other ones aretransient.

• So, with probability 1 we will eventually have either x = 0or x = 1 and the chain stops.

• The probability of being absorbed in 1 is x(0).• The mean absorption time (either in 0 or in 1) is

g ≈ −2N[x(0) ln x(0) + (1− x(0)) ln(1− x(0))] .


Some realizations of the neutral Wright-Fisher process

N = 100 x(0) = 0.4

50 100 150

0.2

0.4

0.6

0.8

1.0


Estimating the interbreeding parameter α

• Experimental constraints: Neandertals were extinguished,i.e. x(T ) = 1 for some T .

• Non-African living humans have 1 to 4% of their DNA ofNeandertal origin, i.e. y1(T ) lies between 96 and 99%.

• For which values of α are these constraints met?• We may simulate several x(t) by taking random x(0) andα, drop histories in which the AAE were extinguished,calculate y1(T ) by numerically solving the ODEs and retainhistories in which y1(T ) lies in the experimental interval.





• For which values of α are these constraints met?

• We may simulate several x(t) by taking random x(0) andα, drop histories in which the AAE were extinguished,calculate y1(T ) by numerically solving the ODEs and retainhistories in which y1(T ) lies in the experimental interval.





• For which values of α are these constraints met?• We may simulate several x(t) by taking random x(0) andα, drop histories in which the AAE were extinguished,calculate y1(T ) by numerically solving the ODEs and retainhistories in which y1(T ) lies in the experimental interval.


Probability density distribution for α

0

2

4

6

8

10

12

14

0 0.1 0.2 0.3 0.4 0.5

cond

ition

al p

roba

bilit

y de

nsity

interbreeding rate

αmax ≈ 0.013, αmean ≈ 0.083, 106 events with y1(T ) in theexperimental interval.


Correlations between x(0) and extinction time andbetween x(0) and α.

Sample of 790 histories satisfying the experimental constraints.The mean extinction time in the sample is 0.58.

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0.2 0.3 0.4 0.5 0.6 0.7 0.80

1

2

3

4

initial fraction of Africans

extin

ctio

ntim

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