Evolutionary non-commutativity suggests novel treatment strategies
February 14, 2015 10 Comments
In the Autumn of 2011 I received an email from Jacob Scott, now a good friend and better mentor, who was looking for an undergraduate to code an evolutionary simulation. Jake had just arrived in Oxford to start his DPhil in applied mathematics and by chance had dined at St Anne’s College with Peter Jeavons, then a tutor of mine, the evening before. Jake had outlined his ideas, Peter had supplied a number of email addresses, Jake sent an email and I uncharacteristically replied saying I’d give it a shot. These unlikely events would led me to where I am today — a DPhil candidate in the Oxford University Department of Computer Science. My project with Jake was a success and I was invited to speak at the 2012 meeting of the Society of Mathematical Biology in Knoxville, TN. Here I met one of Jake’s supervisors, Alexander Anderson, who invited me to visit the Department of Integrated Mathematical Oncology at the Moffitt Cancer Center and Research Institute for a workshop in December of that year. Here Dr. Anderson and I discussed one of the key issues with the work I will present in this post, issues that now form the basis of my DPhil with Dr. Anderson as one of two supervisors. Fittingly, the other is Peter Jeavons.
Jake was considering the problem of treating and avoiding drug resistance and in his short email provided his hypothesis as a single question: “Can we administer a sequence of drugs to steer the evolution of a disease population to a configuration from which resistance cannot emerge?”
In Nichol et al. (2015), we provide evidence for an affirmative answer to this question. I would like to use this post to introduce you to our result, and discuss some of the criticisms.
Jake’s hypothesis was formed from the results of two papers. In the first, Weinreich et al. (2005) showed that if the genome of a pathogen exhibits sign epistasis then there can exist inaccessible mutational trajectories. These inaccessible trajectories form the latter half Jake’s hypothesis, the inaccessible trajectory to high resistance. The second paper was by Tan et. al. (2011) which showed that adaptive mutations gained under the selective pressure of one drug are often irreversible when a second is applied. Forcing these irreversible mutations to occur was Jake’s concept of steering.
We began with the model used by Weinreich et al. and Tan et al. to derive the results which served as our motivation — the fitness landscape. Fitness landscapes have featured extensively on TheEGG and in fact long enough ago that I used some of the posts here to solidify my own understanding. For simplicity we assume that an individual has a bialleic genome in loci, given by , and that mutation occurs as point mutations which flip a single bit of . Thus the genotype space can be represented by an N–dimensional (hyper-)cube with genotypes as vertices and edges representing mutations. A fitness landscape is then a function which assigns to each genotype a single real, positive valued fitness:
Gillespie (1983) showed that if the population size and mutation rate of an evolving population satisfy , and if we assume that each mutation is either beneficial or deleterious, then evolution proceeds according to Strong Selection Weak Mutation (SSWM) dynamics. Under SSWM the population can be modelled as monomorphic and inhabiting a single vertex in the genotype space. This population undergoes a stochastic up–hill walk moving at each step adjacent vertex which corresponds to a genotype of higher fitness. When no more moves can be made the population is at a local optimum of fitness.
In our work we model this stochastic walk on the hypercube graph as a Markov chain, , defined by:
This is based on the selection rule Artem considered at the end of his discussion of semi-smooth fitness landscapes; the parameter determines the extent to which the fitness increase of a mutation affects its likelihood of fixing within the population. If each fitter neighbouring genotype is equally likely, if then only the fittest mutant can fix and if then we have Gillespie’s (1983) selection rule with a fitter neighbouring genotype has probability of fixing with probability proportional to the fitness increase it creates. You can see an example Markov Chain produces this way (with r = 0 in the figure below.
Using the Markov Chain encoding we can explore the possible evolutionary trajectories of a population under a given fitness landscape . We define a collection of population row vectors for each , where has length and component which gives the probability that the population has the genotype at time (the genotypes are ordered numerically according to their binary value). The distribution of a population at time is related to its initial distribution, , by . Since the Markov Chain is absorbing we know that there exists some such that Consequently, we know that the matrix
exists. In particular, we can compute the probability that the population evolves to each of the peaks of by a simple matrix multiplication .
This Markov Chain encoding associates global properties of the fitness landscape with the algebraic properties of the transition matrix. As a simple example, suppose there two drugs and with fitness landscapes and and associated transition matrices and respectively. Then our model predicts that the order in which drugs are applied will make no difference to the final population probability distribution if, and only
If the initial genotype distribution is unknown then we would require that this condition holds for all and hence that . As we know, matrix multiplication is not commutative and hence different orderings of drugs are likely to produce different results. Specifically, different orderings of the same set of drugs can have different likelihoods of resistance emerging. This result, whist mathematically simple, has profound clinical implications. Prescriptions of sequences of drugs occur frequently in the clinic, and often without any guidelines as to which orderings are preferable. As such, drug orderings are often decided arbitrarily, or from an individual clinician’s historical experience.
We sought to test what effects arbitrary drug orderings have on clinical outcomes. Using the fitness landscapes for 15 commonly used –lactam antibiotics (derived by Mira et. al. 2014) we performed for each of the drugs an exhaustive in silico search of all single, double and triple combinations of steering drugs (allowing the drug itself to appear as a steerer) applied to an unknown starting population, , followed by a final application of that drug. We found that in 57.3% of cases applying a single steering drug before a final drug increased the likelihood of resistance when compared to applying that final drug alone. Further, we found that 64.1% of steering pairs and 65.6% of steering triples increased the likelihood of resistance emerging when compared to application of the final drug alone. In fact, for 12/15 of the antibiotics in our study a random steering combination of one, two or three drugs was more likely to increase the probability of resistance emerging than decrease it. These results serve as a cautionary warning for clinicians prescribing sequential multidrug treatments — that we may be inadvertently selecting for highly resistant disease populations through irresponsible drug ordering in the same way that highly resistant disease can emerge through irresponsible drug dosing.
In our paper we go to test our steering hypothesis by performing for each of the 15 empirically determined drug landscapes a test of steering using combinations of one, two and three drugs to prime the population. We found that for 3 of the 15 drugs there exists another which when applied first steers an initial population to a configuration from which the trajectory to the global fitness optimum is inaccessible. This number rose to 6 when pairs of drugs applied sequentially are used to steer the population and to 7 when triples applied in sequence were considered. These results answer Jake’s original question in the affirmative.
Criticism of the work presented here has come from two angles and primarily from two people. The first criticism, primarily raised by Artem, is a result previously outlined on this blog. Using the limit matrices to simulate evolution assumes that a drug is applied for sufficiently long that the population converges to a local fitness optima. As shown by Kaznatcheev (2013), there can exist evoutionary trajectories through genotype space which have length exponential in the number of loci. As such, this assumption of convergence is a strong one. We offer the retort that the number of loci associated with antibiotic resistance is often low and further that as we get to choose the steering drugs we may be able to avoid long convergence times. Of course there is a simple way to find out — to test the predictions of our model through in vitro experiments.
The second criticism is one which is applied to many results derived from the fitness landscape model, that we are ignoring too much of the complexity at work in evolution. In particular, in fixing a fitness landscape we entirely ignore the phenotype scale and assume a static disease environment. These issues were raised by Dr. Anderson at rooftop bar in the centre of Knoxville a few days before my talk. He argued that, whilst our results are interesting, our assumptions were too strong for our results to translate to biological reality. Sandy argued that our hypothesis could be correct and that our model provided some evidence in favour of it, but that we would be unlikely to find effective steering combinations without considering more of the complexity inherent to the system. Sandy’s criticism has now become my research. In my DPhil we are aiming to build models of the genotype–phenotype map and to examine them through the algorithmic lens. I hope some of my results will find themselves onto this blog in future.
Over the course of this project I’ve learned many things. From our model I’ve learned that conventional wisdom about how to treat disease might not always be correct. This is a theme that continues in my research today where we aim to build evolutionary models and determine better treatment schedules. On a personal level I’ve learned much more. Most importantly to not be afraid of being wrong or engaging with your critics. The two biggest critics of our work so far have been Artem and Sandy. Through Artem I’m able to appear on this blog and to potentially reach a larger audience with our work. Through my conversation with Sandy in Knoxville I earned an invitation to an IMO workshop and eventually a supervisor for a DPhil in which we’re trying to build models which are a little less wrong.
Weinreich, D. M., Delaney, N. F., DePristo, M. A., & Hartl, D. L. (2006). Darwinian evolution can follow only very few mutational paths to fitter proteins. Science, 312(5770): 111-114.
Tan, L., Serene, S., Chao, H.X., & Gore, J. (2011). Hidden randomness between fitness landscapes limits reverse evolution. Physical Review Letters, 106 (19) PMID: 21668204
Gillespie, J. H. (1983). A simple stochastic gene substitution model. Theoretical population biology, 23(2), 202-215.
Mira, P. M., Crona, K., Greene, D., Meza, J. C., Sturmfels, B., & Barlow, M. (2014). Rational Design of Antibiotic Treatment Plans. arXiv preprint arXiv:1406.1564.
Nichol, D., Jeavons, P., Fletcher, A.G., Bonomo, R.A., Maini, P.K., Paul, J.L., Gatenby, R.A., Anderson, A.R.A., & Scott, J.G. (2015). Exploiting evolutionary non-commutativity to prevent the emergence of bacterial antibiotic resistance. biorXic preprint: 007542.
Kaznatcheev, A. (2013). Complexity of evolutionary equilibria in static fitness landscapes. arXiv preprint: 1308.5094.