Local peaks and clinical resistance at negative cost

Last week, I expanded on Rob Noble’s warning about the different meanings of de novo resistance with a general discussion on the meaning of resistance in a biological vs clinical setting. In that post, I suggested that clinicians are much more comfortable than biologists with resistance without cost, or more radically: with negative cost. But I made no argument — especially no reductive argument that could potentially sway a biologist — about why we should entertain the clinician’s perspective. I want to provide a sketch for such an argument in this post.

In particular, I want to present a theoretical and extremely simple fitness landscape on which a hypothetical tumour might be evolving. The key feature of this landscape is a low local peak blocking the path to a higher local peak — a (partial) ultimate constraint on evolution. I will then consider two imaginary treatments on this landscape, one that I find to be more similar to a global chemotherapy and one that is meant to capture the essence of a targetted therapy. In the process, I will get to introduce the idea of therapy transformations to a landscape — something to address the tendency of people treating treatment fitness landscapes as completely unrelated to untreated fitness landscapes.

Of course, these hypothetical landscapes are chosen as toy models where we can have resistance emerge with a ‘negative’ cost. It is an empirical question to determine if any of this heuristic capture some important feature of real cancer landscapes.

But we won’t know until we start looking.

Therapy as a transformation of simple fitness landscapes

Usually, when I prove results about fitness landscapes, I like to focus on the unintuitive subtlety and complexity that hides in high-dimensional spaces. You can catch me in many a Slack channel berating people for drawing simple 2D or 3D representations of fitness landscapes as mountain ranges (instead of high-dimensional mazes). So, of course, that is exactly the piece of hypocrisy I’ll engage in this post: drawing a 2D fitness landscape. It is simply the best tool for the job in this illustration. However, if we must interpret it within a framework of high-dimensional spaces, we can see it as a 1D cut through that space, or as the average over paths of various Hamming distance from a given point. The details will not be essential.

Let’s imagine a genotype (or phenotype) space with 9 types: T, U, V, W, A, B, C, D, E and a notion of locality (i.e. mutational nearness) such that adjacent letters in the previous list are adjacent in the mutation network. Associate each of the 9 types with a fitness value (given by the solid line in the figure at left). Finally, suppose that evolution prior to treatment has taken our cells to the cancerous state W (for wild-type) which has fitness 6. I have chosen the landscape such that W is a local maximum that is separated by a valley from a higher local maximum at D. This will be our DMSO fitness landscape.

Now, if we wanted to look at treatment, the default seems to be to generate another fresh fitness landscape for the drug on the same space of types. This seems a bit unreasonable to me, and not a workable approach in the long-term for predicting drug’s fitness landscapes. In particular, if we fully embrace this approach then our only option will be to measure the landscape for each new drug.

An alternative is to try to build a simple theory of how drug transforms a DMSO fitness landscape. After all, the fitness of a cell in drug is not completely unrelated to its fitness outside of drug, since the drug affects only certain pathways of the many that determine fitness. This is in keeping with theories that view compact representations of fitness landscapes as descriptions of genotype-phenotype maps. In this way, we can see a drug as affecting some but not all of the interactions in the phenotype that lead to the cell’s fitness.

Now, in practice, this transformation of a fitness landscape (or its compact representation) due to drug could be incredibly difficult to measure. And, again in practice, we would only learn what these transformations are like by first measuring many DMSO and in-drug fitness landscapes. But the hope is that a pattern would fall out. An even more ambitious hope, maybe shared by some in developmental biology, is that we could learn the way compact representations of fitness landscapes are encoded and decoded by nature and then use biochemical analysis of the drugs to see how they’d effect the fitness.

Until then, as is common in the theory-heavy and experiment-poor fitness landscape literature, I think it is reasonable to make some educated guesses at how our toy landscapes will transform. Note that we already do this when making theoretical claims about how different treatments transform games (instead of, but closely related to, fitness landscapes).

Standard chemo and the cost of resistance

If we suppose (unreasonably) that the fitness gains in the DMSO landscape are due primarily to increases in reproductive rate then we would expect chemotherapy to hit the more fit genotypes harder than the less fit ones. This is an argument I’ve made before in the context of chemo for bone-metastatic prostate cancer; comparing it to a kind of time-reversal:

As a simple caricature, let f be the strength of our chemo: when a cell tries to undergo mitosis, it will be killed with probability f. This means that at f = 0.5, there will be no growth in the affected population, since half the time that a cell tries to divide, it succeeds and becomes two cells and half the time it dies and becomes 0 cells. For 0 < f < 0.5, the growth rate is slowed; and for 0.5 < f < 1 the growth rate is reversed.

In terms of a fitness landscape, a strong (f > 0.5) chemotherapy of the above kind would look like a mirror of the landscape through some constant fitness line. So in the case of our toy-model, we can consider something like the dashed line in the left-hand part of the below figure. Notice that it turns hills into valleys and valleys into hills. In particular, it destabilizes the fitness peak of W making it into the bottom of a valley.

Now, imagine that chemo is applied and it transforms the fitness landscape from solid to dashed. The ideal situation is that the resulting drop in fitness (from 6 to -1 in our example) produces such a rapid decrease in the tumour’s size that the cancer cells are all killed before a resistant mutant can emerge. But suppose this ideal situation doesn’t happen and cells arise that mutated to type A and eventually even B. For an example dynamic, see the right-hand figure above: in red we see the evolution of the resistant cell line, with its (hypothetical) DMSO fitness in solid and drug-fitness in dashed lines. Note that the later genotypes have higher fitness than W in drug and maybe high enough to no longer be controlled by therapy. At this point, both the clinician and biologist would consider the type B cells as resistant.

In particular, due to the definition of chemo, the local fitness peak of the drug landscape correspond to local fitness valleys of the DMSO landscape. Thus, in DMSO type B has a lower fitness than type W: 4 versus 6. And as such, we can associate the benefit of type B in drug with a cost of 2 in DMSO. This is what we would expect from a standard cost of resistance model.

In terms of reading the figure at right, we can see the amount of benefit that resistance provides by looking at how much higher the dashed red line is above the dashed black line (wild type fitness in drug). The cost can be read as the difference between the solid black line (wild type fitness in DMSO) and the solid red line.

No surprises so far.

Targetted therapy and a negative cost of resistance

But how should our models of targetted therapy differ from chemo?

I think that the key term here is ‘targetted’. Whereas chemo was a mirror image of all fitness points, I think that targetted therapy should be seen as a depression of particular genotypes (the target) and those nearby. I show this in the dashed line of the figure below left. Targetted therapy greatly decreases the fitness of W, from 6 all the way to -3 (compared to just – 1 for chemo) and also nearby mutants types like V and A, although it affects them less than W (taking both down to just 1, a decrease of 4 compared to W’s decrease of 9). Most importantly, it leaves genotypes further out unaffected. This seems to capture the goal of targetted therapy: hit only the bad genotype associated with a particular cancer.

Notice that this has transformed the local peak around W into a really deep fitness valley. And unlike the chemo landscape, the top of this valley leads us to our old DMSO landscape. Thus, after climbing to B, the tumour can continue climbing to the higher fitness peak at D. Cost of resistance becomes a question of timing.

A potential evolution of the tumour is summarized, as before, on the right hand side of the above figure in red. Note that the dashed and solid red lines merge together after time = 2.

If the patient relapses while the majority of the tumour is still at type B (time = 2), then comparing the resistant tumour versus wildtype will produce a cost of resistance in DMSO of the same type as with my chemo example. However, if the patient relapses when the tumour has evolved to type C (time = 3) or D (time >= 4), then we will see either no cost of resistance (type C) or a negative cost of resistance (type D: fitness 8 in DMSO versus 6 for wildtype W).

In other words, by transforming a local hill into a valley, the targetted therapy inadvertently eliminated the fitess valley that allowed evolution to keep W from advacing to D. With the therapy eliminating that evolutionary barier, the tumour can evolve a form of clinical resistance that is a more aggressive tumour all around.

Of course, this is all based on theoretical caricatures of fitness landscapes that were designed to illustrate the point. If real world landscapes do not resemble these caricatures at all then this is not a danger that will become actualized. But until we are sure of this, it is a nice mechanism to keep in mind for understanding differences in resistance between chemotherapy and tageted therapy.

For any of my clinical readers, I am especially curious on your experience if post-relapse tumours are all around more aggressive and quicker growing that pre-treatment tumours. A naive reading of cost-of-resistant would place them as less aggressive, but my impression was that clinically the opposite is seen: the post-relapse tumour is all around more aggressive than the original (even outside drug). If that is the case, how do clinicians currently make sense of this?

Those are the empirical questions, but there is also plenty of fun theory that could be done.

In future posts, I’ll consider a fitness landscape subject to a full ultimate constraint and thus not at a local peak prior to treatment; and also consider treatment strategies that include drug holidays or adaptive therapy. I hope all of this interests you, dear reader. And if you’ve read something like this already then please point me to the source.