Evolving Antibiotic Resistance in Bacteria by Controlling Mutation Rate

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1 Evolving Antibiotic Resistance in Bacteria by Controlling Mutation Rate Roman V. Belavkin 1 1 School of Science and Technology Middlesex University, London NW4 4BT, UK December 11, 2015, Tokyo University of Science Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

2 Introduction and motivation Introduction and motivation Mathematical models of DNA mutation Evolution as a search problem Optimal control of mutation Adaptive mutation in bacteria Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

Introduction and motivation Genetic Basis of Mutation Individual phenotypes are represented by their genotypes: ATCAGGACTCA Mutation of a genetic code means that

3 Introduction and motivation Genetic Basis of Mutation Individual phenotypes are represented by their genotypes: ATCAGGACTCA Mutation of a genetic code means that reproduction is not a perfect replication (i.e. changes of the phenotype): ATCAGG ATCGGG Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

4 Introduction and motivation Mutation and Natural Selection Definition (Fitness) A real-valued function f : Ω R such that f(a) f(b) means b has a better reproductive success than a. f(a) f(b) means b is better adapted to the environment than a. Fitness can be absolute or relative (e.g. may depend on other individuals or environment). Mutation of a into b can be: Deleterious if f(a) > f(b) Neutral if f(a) = f(b) Beneficial if f(a) < f(b) Remark Adaptation is an increase of fitness, which requires beneficial mutations. However, the majority of mutations in biology are deleterious or neutral. Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

5 Introduction and motivation Optimal Mutation Problem ATCAGG ATCGGG Question: 1 Can an organism increase the probability P + of beneficial mutations? 2 If yes, then do biological organisms optimize mutation? Related issues 1 Need for a mathematical model of mutation and exact formulation of optimization problem for mutation. 2 Need to develop experiments to test theoretical hypotheses. oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

Warwick : John Aston University of Keele : Alastair Channon & Elizabeth Aston University of

6 Introduction and motivation Evolution as an Information Dynamic System EPSRC Sandpit Math of Life (July, 2009): Three year project ( ) Middlesex University : Roman Belavkin University of Warwick : John Aston University of Keele : Alastair Channon & Elizabeth Aston University of Manchester : Chris Knight & Rok Krašovec oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

7 Mathematical models of DNA mutation Introduction and motivation Mathematical models of DNA mutation Evolution as a search problem Optimal control of mutation Adaptive mutation in bacteria Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

8 Mathematical models of DNA mutation Assumptions about Mutation Let P (b a) be transition probability for a b. Independence : mutations at different sites are independent: P (b a) = P (b 1,..., b l a 1,..., a l ) = Neutrality : selection does not act on sites: P (b i a i ) = P (b j a j ) l P (b i a i ) i=1 Models of DNA substitution Jukes and Cantor (1969) : one parameter µ = P (b i a i ) Kimura (1980) : two parameters µ 1 (transitions A G or C T ) and µ 2 (transversions) Tavaré (1986) : six parameters (plus four base frequencies). Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

9 Mathematical models of DNA mutation Random Mutation Point mutation Probability of r independent substitutions with equal probability µ in a sequence of length l: ( ) l P µ (r) = µ r (1 µ) l r r Mutation may involve: multiple parameters µ 1, µ 2. Random substitution, insertion, deletion, transposition Non-independent events (epistasis) ATAGGACTCA ATGGGGACTAC Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

10 Evolution as a search problem Introduction and motivation Mathematical models of DNA mutation Evolution as a search problem Optimal control of mutation Adaptive mutation in bacteria Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

11 Evolution as a search problem Optimization and Search Problems Ω a search space; f : Ω R objective function (fitness). Search problem: Find ω such that f(ω) = υ. Example: f(ω) = max f(ω). Or find: f(ω) = 0 and 2 f(ω) 0 Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

12 Evolution as a search problem Optimization and Search Algorithms Iterative Procedure ω t+1 = Λ(ω t ) Unless f(ω t ) = υ do ω t+1 = ω t + ω t, ω 0 = a The steps ω t can be random: ω t rand(ω) If f is twice differentiable (i.e. f, 2 f exist), then we can apply the gradient method: ω t = [ 2 f(ω t )] 1 f(ω }{{} t ) }{{} Step-size Direction Usually, f is not differentiable, and therefore we need gradient-free methods. Example: genetic algorithm (GA). Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

13 Evolution as a search problem Search in a Metric Space Random mutation: ω m t+1 = ω t+1 + ω t, ω t S(0, r) 3 ω n t+1 r is mutation radius (step size), controlled r via the mutation rate µ [0, 1] (probability ω t of mutation of one letter). The probability that r letters mutate is given by the binomial distribution: ( ) l P µ (r) = µ r (1 µ) l r r Question Does the probability of beneficial mutation depend on µ (or r), and if it does, which µ is optimal? oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

14 Evolution as a search problem Remark Fisher s work predates the discovery of DNA, and is based on Euclidean geometry. oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27 Fisher s Geometric Model of Adaptation Fisher (1930) considered Euclidean space m R l of l traits 3 b There exists an optimal individual R l n r Fitness = negative Euclidean distance to a Question How does P + := P (f(a) < f(b)) depend on mutation radius r? Main Results(Fisher, 1930): P + decreases exponentially with r for all n = d(, a) [0, ) Evolution is more likely to occur via small mutations

15 Evolution as a search problem Geometry of a Hamming space H l α The set H l α := {1,..., α} l is finite with the number of elements: α l This means that there always exist the top (best) element and the bottom (worst) element in {1,..., α} l. Hamming metric: d H (a, b) = l δ ai (b i ), δ ai (b i ) = i=1 { 1 if ai = b i 0 otherwise Hamming space H l α := {1,..., α} l has finite diameter l. Every point has a diametric opposite point. The complement of any ball B(a, r) := {b : d H (a, b) r} in {1, 2} l is also a ball. Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

16 Evolution as a search problem Probability of adaptation in H Probability of adaptation n = 90 n = n = 75 n = 50 n = Mutation radius r oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

17 Optimal control of mutation Introduction and motivation Mathematical models of DNA mutation Evolution as a search problem Optimal control of mutation Adaptive mutation in bacteria Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

18 Optimal control of mutation Optimal Control of Mutation Rate The fact that P + varies with mutation radius and distance to optimum means that we can find an optimal mutation rate control functions µ(n) (Belavkin, 2010, 2012): Maximum adaptation in no more than λ generations maximize µ(x) E{f s+t } subject to t λ Maximum adaptation in no more than λ bits between p s and p s+t : maximize µ(x) E{f s+t } subject to I(p s+t, p s ) λ Cumulative criterion: sup µ(x) λ E µ(x) {f s+t } t=0 λ t=0 sup µ(x) E µ(x) {f s+t } oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

19 Optimal control of mutation Optimal mutation rate control functions in H Step Linear n/l max µ P µ (m < n n) P 0 (m < n) Mutation rate µ oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

20 Optimal control of mutation Evolution of Fitness in Time Distance to optimum n = d(, a) Constant 1/l Step Linear n/l max µ P µ (m < n n) P 0 (m < n n) e + 06 Generation t oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

21 Adaptive mutation in bacteria Introduction and motivation Mathematical models of DNA mutation Evolution as a search problem Optimal control of mutation Adaptive mutation in bacteria Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

22 Adaptive mutation in bacteria Do Biological Organisms Control Mutation? DNA contains vast amount of information l base pairs. Damage to DNA is very high due to radiation and other mutagens. Mutation rates are extremely low µ 1/l Such a fair replication is due to the DNA repair mechanism (e.g. Hakem, 2008). Mutation rates are not minimized Mutation rates can be very different between closely related species (e.g. bacterium Deinococcus radiodurans Cox, Keck, & Battista, 2010) Mutation rate may change with environment for a single genotype (Bjedov et al., 2003) DNA repair mechanism is expressed through several genes, and their deactivation can reduce the repair by different values The existence of a control would require some form of feedback about fitness Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

of Rifamipicin Estimated mutation rates µ in E.

23 Adaptive mutation in bacteria Mutation Rate Control in E. coli Used strains of Escherichia coli K-12 MG1665 Fluctuation test using media 50µg/ml of Rifamipicin Estimated mutation rates µ in E.coli strains grown in Davis minimal medium with different amount of glucose. Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

24 Adaptive mutation in bacteria Results (Krašovec et al., 2014) Population density (10 8 c.f.u./ml) Strong relationship between µ and density of cells (p <.0001). No such relationship in the luxs quorum sensing mutant (p =.0234). Krašovec, R., Belavkin, R., Aston, J., Channon, A., Aston, E., Rash, B., Kadirvel, M., Forbes. S., Knight, C. G. (2014, April). Mutation-rate-plasticity in rifampicin resistance depends on Escherichia coli cell-cell interactions. Nature Communications, Vol. 5 (3742). Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27 Mutation rate (10 9 / gen)

25 Adaptive mutation in bacteria Summary We have formulated mathematical problem of mutation rate control and obtained its solutions in several important cases (Belavkin, 2011, 2012, 2013). The geometry of a discrete space of DNA sequences implies that minimal mutation rate is not always optimal, contrary to common perception based on Fisher s work. Mutation rate plasticity (MRP) has been observed in E.coli and shown to depend on quorum sensing (Krašovec et al., 2014). The observed MRP corresponds well to theoretical functions for adaptive optimal control of mutation rate. Hypothesis Bacteria use quorum sensing to measure population density and estimate their own fitness. This information can be helpful for various adaptive strategies. Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

26 Adaptive mutation in bacteria Introduction and motivation Mathematical models of DNA mutation Evolution as a search problem Optimal control of mutation Adaptive mutation in bacteria Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

27 References Introduction and motivation Mathematical models of DNA mutation Evolution as a search problem Optimal control of mutation Adaptive mutation in bacteria Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

28 References Belavkin, R. V. (2010). Information trajectory of optimal learning. In M. J. Hirsch, P. M. Pardalos, & R. Murphey (Eds.), Dynamics of information systems: Theory and applications (Vol. 40, pp ). Springer. Belavkin, R. V. (2011). Mutation and optimal search of sequences in nested Hamming spaces. In IEEE Information Theory Workshop. IEEE. Belavkin, R. V. (2012). On evolution of an information dynamic system and its generating operator. Optimization Letters, 6(5), Belavkin, R. V. (2013). Minimum of information distance criterion for optimal control of mutation rate in evolutionary systems. In L. Accardi, W. Freudenberg, & M. Ohya (Eds.), Quantum Bio-Informatics V (Vol. 30, pp ). World Scientific. Bjedov, I., Tenaillon, O., Gerard, B., Souza, V., Denamur, E., Radman, M., et al. (2003). Stress-induced mutagenesis in bacteria. Science, 300(5624), Cox, M. M., Keck, J. L., & Battista, J. R. (2010). Rising from the ashes: DNA repair in Deinococcus radiodurans. PLoS Genet, 6(1), Roman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

29 Adaptive mutation in bacteria e Fisher, R. A. (1930). The genetical theory of natural selection. Oxford: Oxford University Press. Hakem, R. (2008). DNA-damage repair; the good, the bad, and the ugly. Embo J, 27(4), Jukes, T. H., & Cantor, C. R. (1969). Evolution of protein molecules. New York: Academic Press. Kimura, M. (1980). A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences. Journal of Molecular Evolution, 16(2), Krašovec, R., Belavkin, R. V., Aston, J. A. D., Channon, A., Aston, E., Rash, B. M., et al. (2014, April). Mutation rate plasticity in rifampicin resistance depends on escherichia coli cell-cell interactions. Nature Communications, 5(3742). Tavaré, S. (1986). Some probabilistic and statistical problems in the analysis of DNA sequences. Lectures on Mathematics in the Life Sciences, 17, oman Belavkin (Middlesex University, London) Controlling Mutation Rate / 27

Mechanisms of evolution in experiment and theory. Christopher Knight

Mechanisms of evolution in experiment and theory Christopher Knight Evolution and molecular biology Nothing in Biology makes sense except in the light of evolution (Dobzhansy 1964 in Biology, molecular