Q-bio 2017 CSU FC

Marek Kimmel [email protected] Follow-up to Natalia Komarova lectures (with a twist)

• Two general talks – Stem cells – Crossing valley • Two hours of class – Very brief intro of cancer as an evolutionary process, including , selection, multistage carcinogenesis, and periods if growth and plateau. – Stochastic modeling of drug resistance in cancer . Brief intro to the of CML . Continuous time birth death process, derived the Kolmogorov forward equation . PDE for generating functions by the method of characteristics . Applications, such as combination therapies, the number of drugs required, and modeling cross resistance. Outline

Purpose: Understanding of cancer mutations based on data and theory of genetics and

BS-1 Monday Basic stochastic models of molecular evolution and

BS-2 Wednesday Applications to modeling and analysis of leukemic and lung cancer data

General Talk Thursday Modified Griffiths-Tavaré coalescent applied to The Cancer Genome Atlas Data (Estimation of growth and characteristics?) Vignette: Non-Darwinian evolution in tumors

What should we know about molecular evolution and population genetics?

• models of molecular evolution (infinite population) • Moran model (finite but constant population)

Later on (BS-2 and General Lecture) • Branching processes (finite growing or decaying population) • Wright-Fisher model and coalescent (later on) Elementary Introduction to Markov Chains Stochastic (random) processes

X (t,) :T   R or Z in discrete time

Sequence of random variables

X 0 , X1,, X n ,

P[X n  A| X 0 , X1,X n1]  P[X n  A| X n1]

If the process is nonnegative integer-valued, then

P[X n  j | X 0  i0 , X1  i1, X n1  i]  P[X n  j | X n1  i]

 Pij where i, j S the state space of the process (chain). Transition probabilities Transition probability matrix (TPM) Attention, state space S finite or denumerable

P11 P12  P1 j  P P  P   21 22 2 j  P  [Pij ]i, jS          Pi1 Pi2  Pij      

TPM is a

i, j  S : Pij  0

i  S :  Pij 1 jS Marginal probabilities Marginal probability of the process at time n follows from the Bayes formula for total probability

P[X n  j]   P[X n  j | X n1  i]P[X n1  i] iS or, in abbreviated notation

p j (n)   pi (n 1)Pij , j  S, n 1, 2,  iS where p j (n)  P[X n  j]. In matrix - vector notation

p(n)  p(n 1)P, where p(n)  [ p1(n), p2 (n),,] Example 1: Irreversible mutations in discrete generations

Let us assume that state 1 can mutate into state 2 but not conversely 1   P     0 1 p(n)  p(0)Pn

n n n (1 ) 1 (1 )  P     0 1  After a long time, nothing is left in state 1 (state 2 is absorbing) Example 2: Reversible mutations in discrete time

Let us assume that state 1 mutate into state 2 and conversely. 1    P      1  We expect existence of a stationary distribution, which is invariant wrt multiplication by P (and Pn) 1    p  pP  ( p1, p2 )  ( p1, p2 )    1  This 2-equation system is under-determined (since TPM is stochastic), so a probability norming condition is needed

p1  p1(1 )  p2 , p1  p2 1   p  , p  1   2   Markov property in continuous time

Assume the following sequence of time points

0  t0  t1  t2    tn2  tn1( s)  tn ( t) The Markov property now reads

P[X (t)  j | X (t0 )  i0 ,X (t1)  i1,, X (s)  i]  P[X (t)  j | X (s)  i]

 Pij (s,t) Pij (t  s)

Where the last equality follows from time-homogeneity (time-shift invariance) Transition probabilities in continuous time As in the time-discrete case, marginal distributions evolve by multiplication by TPM p(t)  p(s)P(t  s) A more general (in fact, fundamental) property is the Chapman-Kolmogorov equation (aka the semigroup property) P(t)  P(s)P(t  s) alternatively

Pij (t)   Pik (s)Pkj (t  s), i, j  S kS Transition intensities

How to build a time-continuous Markov chain? Let us specify infinitesimal transition probabilities (1 jump at most per t) Q t  o(t) j  i  ij Pij (t)  P[X (t  t)  j | X (t)  i]   1  Qijt  o(t) j  i  jS , ji o(t) lim  0 o(t) is called a “small” of t (generic Landau’s symbol) t0 t

Qij are transition intensities. Define the diagonal element

Qii    Qij  Qij  0 jS , ji jS We now have

Pij (t)  ij  Qijt  o(t), i,j  S Or in condensed notation P (t)  I  Qt  o(t) Matrix of transition intensities

Q11 Q12  Q1 j  Q Q  Q   21 22 2 j  Q  [Qij ]i, jS          Qi1 Qi2  Qij      

i, j  S, i  j :Qij  0

i  S :Qij  0

i  S : Qij  0 jS Differential equations for TPM Infinitesimal transition equation P(t)  I  Qt  o(t) Can be multiplied from the left by the TPM P(t)P(t)  P(t)  P(t)Qt  o(t) P(t)(...)

Applying the Chapman-Kolmogorov P(t  t)  P(t)  P(t)Qt  o(t) P(t  t)  P(t) o(t)  P(t)Q  t  0 t t dP(t) / dt  P(t)Q, P(0)  I This is called the forward equation (why ?)

Similarly, multiplying from the right by the TPM (...) P(t)  dP(t) / dt  QP(t), P(0)  I we obtain the backward equation Differential equations for TPM

Both forward equations dP(t) / dt  P(t)Q, P(0)  I and backward equations (remember these are matrix equations!) dP(t) / dt  QP(t), P(0)  I can be sometimes solved. If this is the case then the solution has the form P(t)  P(0)eQt  eQt  exp(Qt) which is formally the same as in scalar case, except the matrix exponent  exp(Qt)  I  Qt  (Qt)2 / 2! (Qt)3 / 3!  (Qt)i / i! i0 is a square matrix itself. Another useful expression is the following  lim t0  P(0)  Q Stationarity in time-continuous processes

Conclusion from Chapman-Kolmogorov: p(t  s)  p(s)P(t) For the stationary distribution p p(s)  p(s  t)  p  p  pP(t); all t

Since we have Q2t 2 Q3t 3 exp(Qt)  I  Qt    2! 3! an equivalent condition is pQ  0 Example 3: Mutations in continuous time

Now, μ and ν are intensities not probabilities

    Q       

P(t)  exp(Qt) 1    exp[(  )t]    exp[(  )t]        exp[(  )t]   exp[(  )t] In the limit, the transition probability matrix has rows being stationary distributions ()           p P()            p      Models for DNA substitution

Nothing in Biology Makes Sense Except in the Light of Evolution

Theodosius Dobzhansky (1900-1975) Substitutions

Purine Purine Pyrimidine Pyrimidine Transitions AG, G A, C T, T C (more likely)

Purine Pyrimidine Pyrimidine Purine Transversions AT, T A, A C, C A (less likely) GT, T G, G C, C G Hypotheses

Substitution of nucleotides in the evolution of DNA sequences can be modeled by a Markov chain – time-discrete, or – time continuous Usually – stationary, and – reversible (why?) Transition matrix

a g c t

a paa pag pac pat

g p p p p P = ga gg gc gt c pca pcg pcc pct

t pta ptg ptc ptt Jukes – Cantor model (~1960) Neutral evolution theory: most mutations have no selective value

All substitutions are equally probable

1 3       1 3    P       1 3         1 3  Stationary distribution

  P 

a g c t  0.25 0.25 0.25 0.25 Spectral decomposition of Pn

0.25 0.25 0.25 0.25  0.75  0.25  0.25  0.25 0.25 0.25 0.25 0.25  0.25 0.75  0.25  0.25 Pn     (1 4)n   0.25 0.25 0.25 0.25  0.25  0.25 0.75  0.25     0.25 0.25 0.25 0.25  0.25  0.25  0.25 0.75  Parameter estimation Jukes – Cantor model

The following 3 graphs are equivalent due to reversibility:

Ancestor A t t D1 D2 D2 D1 A A D1 D2 2t 2t Descendants Probability that the nucleotides are different in two descendants

p  p(t)  0.75(1 exp(8t)) Estimating α

We have two DNA sequences of length N

D1: ACAATACAGGGCAGATAGATACAGATAGACACAGACAGAGCAGAGACAG D2: ACAATACAGGACAGTTAGATACAGATAGACACAGACAGAGCAGAGACAG Number of differences p = N

1 4 t   log(1 pˆ) 8 3

Estimated product of mutation rate and time only ! But how does this work in finite populations? • Is there really a problem? Moran Process with discrete time and directional selection Assumption: Mutants are already there!

t t + 1

One dies (randomly chosen) Another reproduces Transition probabilities if i is the number r 1 s of Orange Mutants i N  i N  i ri p  p  p 1 p  p i,i1 N ri  N  i i,i1 N ri  N  i i i,i1 i,i1

Final effect: Extinction or fixation of an evolutionarily favorable mutant Expressions for probability of fixation and expected time to fixation 1 (1 s)i P[T  T | Z(0)  i]  N 0 1 (1 s)N

2ln N  lni E[T |T  T ;Z(0)  i]  N N 0 s (this latter approximate: large N, small s) Anatomy of a Moran Process conditional on mutant fixation (Durrett-style)

N N - Subcritical bp N - N/lnN

Deterministic

   N/lnN 1 2 3 i Supercritical bp 0 TN