Biology & Philosophy

, Volume 26, Issue 3, pp 339–364

Historicity and experimental evolution

Authors

    • The Rotman Institute of PhilosophyThe University of Western Ontario
Article

DOI: 10.1007/s10539-011-9256-4

Cite this article as:
Desjardins, E. Biol Philos (2011) 26: 339. doi:10.1007/s10539-011-9256-4

Abstract

Biologists in the last 50 years have increasingly emphasized the role of historical contingency in explaining the distribution and dynamics of biological systems. However, recent work in philosophy of biology has shown that historical contingency carries various interpretations and that we are still lacking a general understanding of “historicity,” i.e., a framework from which to interpret why and to what extent history matters in biological processes. Building from examples and analyses of the long-term experimental evolution (LTEE) project, this paper argues that historicity possess three essential conditions: (1) multiple possible pasts, (2) multiple possible outcomes at a given instant, and (3) a relationship of causal dependence between these two sets. These criteria can be further specified in two general forms of historicity: dependence on initial conditions and path dependence. More attention is devoted to developing a rigorous account of the latter, which captures the type of historicity displayed by stochastic processes. This paper also highlights that it is often more productive to adopt an instant-relative approach and think in terms of degree of historicity instead of trying to maintain a rigid and absolute dichotomy between historical and ahistorical (completely convergent) processes.

Keywords

HistorictyPath dependenceContingencyUnpredictabilityExperimental evolutionLTEE

Introduction

Several prominent biologists during the last decades have emphasized “historicity,” i.e., the idea that history matters, in biological processes. 1 Take for instance Williams’ opening statement in his 1992 book, Natural Selection:

Successful biological research in this century has had three doctrinal bases: mechanism (as opposed to vitalism), natural selection (trial and error, as opposed to rational plan), and historicity. This last is the recognition of the role of historical contingency in determining properties of the Earth’s biota (Williams 1992).

The purpose of the present paper is to expand and clarify the meaning this “doctrine” of historicity. 2 I concur with Williams that “historicity” became increasingly influential in the field of biology during the twentieth century, but saying that there exist a doctrine of historicity, i.e., a set of established believes and principles taught by some expert authority, seems inaccurate. Biologists and philosophers have yet to develop a general and uncontroversial understanding of what “historicity” means (Beatty and Desjardins 2009).
Part of the difficulty with providing an account of historicity for biological sciences comes from the plurality of meanings associated with the notion of “historical contingency.” Beatty (2006) for example, shows that Gould (and many after him) used “contingency” in order to emphasize that evolution is both “unpredictable” and/or “causally dependent on the past.” As shown in the following passage from Losos et al. (1998), the two interpretations are often closely tied:

The theory of historical contingency proposes that unique past events have large influence on subsequent evolution. A corollary is that repeated occurrences of an evolutionary event would result in radically different outcomes (p. 2115).

The first sentence of this quote refers to what Beatty (2006) calls the “causal dependence” view of contingency. It means in the present context that current life forms are the result of a long series of causally related events and that changing their (evolutionary) past would (most likely) have resulted in different life forms. The last sentence on the other hand refers to the “unpredictability” version of contingency, which is entertained in a thought experiment titled “Replaying Life’s Tape” by Gould (1989), who imagines that if we could stop the course of life, press the rewind button and go back to any time in the evolutionary past,

any replay of the tape would lead evolution down a pathway radically different from the road actually taken … no finale can be specified at the start, and none would ever occur a second time in the same way, because any pathway proceeds through thousands of improbable stages (Gould 1989, p. 51).

In this passage, unpredictability follows from the vast number of paths evolution could have taken at one or another moment in the past. This becomes especially important during periods following episodes of massive extinction, during which periods a greater range of ecological opportunities are available for organisms to exploit. For example, according to Gould, the basic developmental patterns of bilaterians that arose quickly during the Cambrian explosion were mere possibilities (happenstances) among a “broad realm of alternatives (each “equally pleasing” to natural selection) … ” (Gould 2002, pp. 1159–1160) Had the diversity of these developmental patterns been constrained to a narrow range of workable states, evolution would have been much more predictable.

Note that there always remains some degree of unpredictability as long as the convergence is not perfect, i.e., as long as more than one possible outcome can occur from a given starting point. So the fact that there could have been thousands of improbable pathways only makes the process of evolution more unpredictable. I will argue however that the fact that evolution could have taken radically different pathways does not necessarily entail that history matters.

The importance of not confounding historicity and unpredictability can be seen from the dispute between Gould and Conway Morris on the likely outcome of Replaying Lifes Tape experiment. Conway Morris (2003), impressed by the morphological convergence displayed in fossil records and actual life forms, argues that evolution is much less contingent (i.e., unpredictable) than Gould seems to suggest. He even goes as far as claiming that human(ish) life was inevitable given the conditions of existence on this planet. 3 The disagreement here is not over the idea that history matters. In fact, Conway Morris’ argument relies one of the two general forms of historicity that I will present later in this paper, namely dependence on initial conditions. This is entailed by the claim that human-like beings are inevitable given the unique physical and biological conditions prevailing on earth. The fact that these conditions are unique explains why we are lonely in this universe, but this does not mean (for him) that we were not extremely likely to evolve on this planet. So we could say that both Gould and Conway Morris agree on the fact that history maters. Yet, their disagreement about the (un)predictability of evolution can have some implications on the way and the extent to which history matters. In order to see this, we will need a better understanding of the theoretical core and contour of “historicity,” which is the main objective of this paper.

By way of introduction, I will present some of the results obtained in the Long-Term Experimental Evolution project conducted by Lenski et al. (2010). Although rarely considered in the philosophical literature, this ongoing project gave rise to several interesting articles on the role of adaptation, chance and history in evolution. This will help set the bases for a more detailed account of historicity. The last section will develop a conceptual framework defining the conditions for “path dependence,” which is the form of historicity occurring in stochastic processes. 4 We will see there that having multiple possible outcomes is not a sufficient condition for path dependence, and that it is sometimes more useful to think in terms of degree of historicity.

Historical contingency in the long-term experimental evolution project

In 1988, a group of biologists at Michigan State University started off the long-term experimental evolution (LTEE) project with a single bacterium of E. coli that they cloned a few times in order to produce twelve genetically identical populations. These populations of bacteria were then introduced into a new environment containing a different kind of sugar for the bacteria to feed on. In a sense, each population can be conceived of as a “replay” from a given starting point, as if the group was replaying the evolutionary tape of one ancestral population 12 times. All populations were originally identical (or at least as identical as populations of clones can be) and they were facing the same selection regime: they were all going from environment type E1 to environment type E2. This shift in environmental conditions created a new ecological challenge (in this case alimentary) and thus promoted evolutionary changes.

The remainder of this section presents three contexts in which the notion of “historical contingency” has been used in the explanation of results gathered during the LTEE project. This analysis will set the stage for the next section, in which I provide a more complete and detailed theory of historicity for stochastic processes.

Chance and mutational history

Because they were not yet adapted to feeding on their new source of sugar, the bacteria used for the LTEE project did poorly at the very beginning of the experiment. But signs of adaptation appeared fairly early in the experiment. The fact that they were becoming increasingly adapted to their new environment was noticeable in their size and fitness. 5 Figure 1 shows that the fitness in all twelve populations has increased very early in the experiment. This result is not very surprising. An increase in average fitness was expected because the ancestral populations had evolved for many generations in a significantly different medium. So the increase in mean fitness was an expected reaction to the fact that the environment had changed significantly and that it took the bacteria some time to adjust to their new diet.
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Fig. 1

Trajectories for mean fitness relative to the ancestors in 12 replicate populations during 10,000 generations. From Lenski and Travisano (1994, p. 6811)

Although adaptation occurred in all populations, we can nevertheless observe some divergence in the mean fitness of these populations. This divergence is in itself a surprising result given the conditions of the experiment—identical genetic materials in identical controlled environments. In such situation, one would expected that all populations should follow the same evolutionary trajectory (as in Fig. 2a). But their results have proven otherwise.
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Fig. 2

Example of a parallel evolution, b transiently divergent evolution, and c sustained divergent evolution. The x-axis represents time and the y-axis represent mean fitness—although it could have been any mean trait value. These hypothetical scenarios can be used to evaluate the relative importance of adaptation, chance and history in evolution

Lenski et al. (1991) and Lenski and Travisano (1994) have considered different scenarios to explain the observed divergence in mean fitness of the replicate populations, and “historical contingency” is important in at least two of them (represented in 2b and c).

One of the scenarios suggests that these divergences are merely transient (see Fig. 2b). The same set of beneficial mutations would happen eventually but in a different order, such that at the end of the day, all populations would converge to the same genetic or morphological state and fitness level. The researchers could explain the observed divergence by the fact that the populations were not done adapting to their new environment, and that the divergences created by some chance variations would be temporary. Thus, this scenario envisages a global equilibrium (a unique adaptive peak in a fitness landscape, as in Fig. 3a) towards which all populations converge, but via alternative routes. But, Lenski and his team ruled out the transient divergence hypothesis as they realized that these divergences were maintained after 10,000 generations 6 (Lenski and Travisano 1994). They took this to confirm a different scenario (Fig. 2c), according to which different populations approach separate “adaptive peaks” of unequal fitness value on a stable adaptive surface (as in Fig. 3b). In effect, even if the populations all began their journey at the same point on the adaptive landscape, sustained divergence can occur if there exist multiple peaks to climb and if each population can integrate different mutations in different order.
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Fig. 3

Example of a adaptive landscape with a single peak and b adaptive landscape with multiple peaks of different fitness value. y-axis: fitness, x- and z-axes: values for different genetic factors. Landscape a is compatible with scenarios represented in Fig. 2a and b, whereas landscape b is compatible with scenario represented in 2c

At this point in the experiment, the researchers claimed that this maintained divergence in fitness after 10,000 generations “demonstrates the crucial role of chance events (historical accidents) in adaptive evolution” (Lenski and Travisano 1994, p. 6813). The effect of “chance” in this experiment is detected from the presence of a significant amount of variations between the populations, and is interpreted as being the result of random mutations, drift, or a combination of both. Chance most certainly plays a role, but as the exposition above suggests, history, or more precisely the history of mutations is a key explanatory element here.

We can see this more clearly in a paper from Johnson et al. (1995), who developed a model from which they were able to show that initially genetically identical populations evolving in identical environments can subsequently diverge when different beneficial alleles arise in a random order. 7 They divide the evolutionary dynamics for initially identical populations evolving in identical environments into two classes: coincident-event replacement and isolated-event replacement. Coincident-event replacement happens when all populations integrate the same mutations at the same time. This tend to produce parallel evolutionary dynamics (as in Fig. 2a). Alternatively, isolated-event replacement happens when different beneficial mutations are integrated in different populations at different instants. Populations governed by isolated-event replacement tend to present divergent evolutionary dynamics, which divergence may or may not be sustained (Fig. 2b, c). The divergence will be sustained if the adaptive landscape is rugged, i.e., if it admits of multiple isolated genotypes or phenotypes of high fitness value (peak). When this happens, a population might come to rest on one rather than then another peak for the simple reason that mutational changes in that direction happened to arise and were selected, whereas mutational changes in the direction of other peaks did not occur and could not be selectively accumulated. Thus, when there exist multiple stable equilibria, then history may never vanish.

It is also interesting to note that these mathematical models suggest that the type of dynamic a population will take is strongly influenced by the product of the size (N) of the population and the rate (μ) at which beneficial mutations occur. When Nμ is large, and selection is strong, then the entire genetic space is visited thoroughly and synchronically by all populations. Thus, at each generation, selection can favour the beneficial types coincidentally in all populations, thus resulting in coincident-event replacement and parallel evolution. This could characterize some but not all populations of small organisms such as viruses and some bacteria (Wahl and Krakauer 2000, p. 1447). On the other hand, when Nμ is small, the waiting time for a beneficial mutation to occur tends to be longer, which leaves room for alternative mutations to become fixed at different moments in different populations. Some beneficial mutations may arise, but they become lost by genetic drift before being selected. The latter situation would therefore result in isolated-event replacement and divergent evolution, either transient or sustained. The divergence will be transient if there exists but one genotype or phenotype with significantly higher fitness, but it will be sustained if the adaptive landscape is rugged, i.e., if it admits of multiple isolated genotypes or phenotypes of high fitness value (peak). When this happens, a population might come to rest on one rather than then another peak for the simple reason that mutational changes in that direction happened to arise and were selected, whereas mutational changes in the direction of other peaks did not occur and could not be selectively accumulated. As noted by Szathmáry (2006), species with larger individuals (like ours) tend to have smaller Nμ values, which also suggest a greater sensitivity to mutational history.

The mathematical analysis provided by Johnson et al. (1995) can help us see two things. First, the scenarios of sustained and transient divergence do imply that chance plays a role in the evolutionary dynamics, but more precisely, it is the random order of mutational histories that leads to this. Second, the relative importance of the Nμ factor in determining the type of dynamics a given population will present shows that the evolution of different types of organisms will be more or less dependent on the random order of their mutational history. In other words, this result suggests that we need an account of historicity that admits of degree.

Dependence on initial conditions

I will now consider a study by Travisano et al. (1995), also produced in the context of the LTEE project. This paper is especially interesting for the conceptual and operational distinctions it draws between the role of history versus chance in evolutionary processes. Unlike the situation entertained in the previous section, we do not have initially identical populations of bacteria evolving in identical environments. Instead, the researchers have decided to test for the role of history by comparing the evolutionary dynamics of initially different populations of bacteria evolving into identical environment. So, the form of historicity envisioned by the LTEE group in that paper corresponds to dependence on initial conditions.

The researchers achieved that by taking the twelve different genotypes derived after 2,000 generations and replicated each of them a certain number of times. 8 In summary, the whole initial set was formed of 12 different sub-groups. The members within each sub-group were identical (i.e., clones), but there were significant variations between the sub-groups due to previous divergent evolution. Then, they placed these populations into a new environment and let them evolve for another 1,000 generations. This allowed the researchers to assess the extent to which the final phenotypic states (relative fitness and cell size) were affected by the previous (unique) evolutionary history of these populations. 9

Different scenarios were hypothesized here again. If the initially different sub-populations were to converge towards the same fitness/size level, then they would conclude that history does not matter. This could happen if the situation was as in Fig. 3a. Initially different genotypes would eventually converge because all populations would climb the same fitness peak. Initial differences (and distinct mutational histories) would thus make no difference to the (long run) outcome. It also follows from this scenario that chance 10 would not be a relevant factor to include in the explanation of convergence towards a unique outcome. But the fact that convergent evolution has the same effect on “chance” and “history” does not mean that the latter two cannot be pulled apart from one another.

Chance typically produces a scattering effect on evolutionary trajectories. It plays a role when the variations between and among the subgroups tend to increase with time. A small effect will result in slight divergences between derived populations, whereas a large effect could result in populations becoming extremely scattered into the evolutionary space. How much divergence occur will also dependent on how many stable equilibria there are in the adaptive landscape, but the general effect of chance variations is assumed to be divergence, and to a certain extent unpredictability. A perfect convergence of evolutionary trajectories would mean that the outcome is rather robust, repeatable and thus (in principle) predictable. If chance variations lead to divergence, i.e., to a multiplicity of outcomes, then each outcome become merely possible and more or less difficult to predict from the start.

The impact of history, too, will result in some degree of divergence (or at least in the absence of complete convergence). But in order to perceive traces of history, the twelve sub-populations have to remain relatively distinct. According to the researchers’ conceptual and experimental framework, we can tell that history leaves a mark when the variations among the sub-populations remain significantly smaller than the variations between the sub-populations. In other words, we can say that history matters when type-1 populations evolve into type-1′ populations, type-2 populations evolve into type-2′ populations, type-3 populations evolve into type-3′ populations, etc. So, what matters is that the initial differences between the groups are maintained (or expanded), but that we can still “see” the presence of distinct subgroups.

This is not same as chance (or mere randomness). If all the populations became extremely scattered in the evolutionary space due to chance variations, then a new observer would be incapable of telling from which ancestor a given derived population comes from. Interestingly, this kind of situation would result in the same consequence for history as convergence. If only chance and not the historical (initial) state of the population were to affect the evolutionary dynamics, then the derived populations would cease to exist as relatively isolated clusters of organisms, i.e., there would possibly be as much variation between than among the groups, which would thus result in the impossibility of reconstructing the initial subgroups.

To summarize, both convergence and chance can erase history. When different populations adapt similarly to a given environment, history is erased because past differences in the value of a state variable ceases to exist. Chance on the other hand can make the derived populations more scattered and thus create a situation where it is impossible to see a relationship between changes in the initial states and the probability of different evolutionary outcomes. When this happens, having different evolutionary histories will not affect distinctively the probability of reaching one or another (set of) outcome(s).

As one might expect, Travisano et al. (1995) observed a mix between adaptation, chance and history. The relative fitness of all populations increased, suggesting that they adapted to their new environment. They also observed that history was important as several subpopulations remained relatively clustered. But historical differences were not perfectly maintained, as most derived populations displayed an increase in among- and betweengroup variation. This mixed result only proves that evolution is better conceived as a complicated process governed by several factors and that it is sometimes hard to tell them apart without a sophisticated set of hypotheses. The framework developed by Travisano et al. (1995) is very interesting in this respect. It allows for the delineation of conceptual and operational distinctions. The most interesting one for us right now is that history will matter as long as the derived populations diverge and that the initial similarities are sufficiently maintained. In other words, divergence in the derived populations is not sufficient to claim that history matters. The latter is true only if the state of a given derived population remains dependent on its ancestral state.

Before we move on to the next section, let us pause in order to abstract from this study the essential conditions required in order to obtain historicity. The analysis provided above suggests that in order for history to matter we need:
  1. 1.

    Multiple possible past states (in this case, different initial states)

     
  2. 2.

    Multiple possible outcomes

     
  3. 3.

    Causal dependence: the probability that a given outcome occurs must change as a function of the historical conditions realized at a given occasion.

     

These three conditions constitute the backbone of the framework I will develop in "Historicity as path dependence". The first condition (multiple possible pasts) is essential but rather trivial. 11 Most of the action happens in (2) and (3). Condition (2) entails that things could have been otherwise. This would fail for instance if there existed only one, robust stable evolutionary strategy for the kind of environmental problem faced by E. coli in the experiments designed by the LTEE group. The existence of multiple adaptive peaks and/or chance variations on the other hand would rather contribute to the realization of (2). Yet, this would be insufficient for history to matter. As suggested above, when evolution is extremely affected by chance variations, derived populations can become very heterogeneous and lose any evidence of clustering due to initial similarities. In this case, we are unable to claim that history matters, but only that things could have been otherwise. I believe that condition (3) captures what Travisano et al. (1995) where after when looking for maintained (or amplified) differences in derived populations. We are justified to claim that history matters if there exists some form of (probabilistic) causal dependence between past and present states, when past states constrain or enable certain futures states but not others. This is why condition (3) is also necessary for historicity.

An unexpected adaptation

We will see in this section that historicity also played a significant role in the explanation of what was qualified as one of the “most profound” adaptations observed during the LTEE project. I will first summarize what this adaptation is about and then suggest that the form of historicity entertained this time can be captured by the notion of path dependence instead of dependence on initial conditions. The next section will provide a detailed account of the former.

In Blount et al. (2008), the LTEE group reports the occurrence of a new and unexpected trait in one of the populations. As mentioned at the beginning of "Historical contingency in the long-term experimental evolution project", the bacteria were relying on sugar as a source of energy from the very first day of the experiment. This is in fact what E. coli typically does in aerobic conditions (i.e., in the presence of oxygen). But one of the populations has evolved a citrate-metabolizing phenotype (Cit+) after 31,500 generations. This new adaptation is rather significant, for the Cit+ phenotype achieved a several-fold increase in size and it possesses a much higher relative fitness than the rest of the populations. How come such a beneficial mutation took so long before manifesting itself? In effect, despite the potential of using citrate in anaerobic conditions and the ecological opportunity to evolve a citrate-metabolizing phenotype in aerobic conditions, none of the 12 populations has shown any sign of such a capacity before that, thus indicating that evolving a citrate-metabolizing phenotype is a hard thing to do. It was therefore a great surprise to discover that one of the twelve populations evolved that capacity.

Two hypotheses were formulated to explain the unique and late arrival of such a beneficial mutation:

The long-delayed and unique evolution of the Cit+ phenotype might indicate that (1) it required some usually rare mutation … that does not scale with the typical mutation rate. (2) Alternatively, the occurrence or phenotypic expression of the mutation that generated the Cit+ function might depend on one or more earlier mutation, such that its evolution was contingent on the particular history of that population. (Blount et al. 2008, p. 7900)

The first scenario is called the “rare mutation” hypothesis, whereas the second scenario is called the “historical contingency” hypothesis, and both are illustrated in Fig. 4. I see in these two hypotheses another way of distinguishing between two senses of contingency, i.e., mere chance and historicity. Some variations are rare but independent of the history of mutations that occurred before hand. These rare mutations are contingent (chancy) in the sense that they are not necessary but merely possible. We cannot predict wether (or when) they will occur in a given lineage. All we can say is that if we wait long enough, they should show up at some point (although we can obtain information about their general probability of occurrence by sampling a large number or populations). Other rare variations are more complex because their occurrence becomes more likely after a particular history of mutations has previously happened. These variations are contingent in two senses: (1) their probability of occurrence causally depends on the previous history of mutations, and (2) they remain to a certain extend unpredictabe because the order of mutations is random and possibly different for each lineage. This dual sense of contingency is well captured by the authors when they say:

[In biological evolution] random and deterministic processes become intertwined over time such that future alternatives may be contingent on the prior history of an evolving population. For example, multiple beneficial mutations will arise in some unpredictable order … thus constraining some evolutionary paths while potentiating other outcomes (Blount et al. 2008, p. 7899).

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Fig. 4

Alternative hypotheses for the origin of the citrate-metabolizing (Cit+) phenotype. The rare mutation hypothesis assumes a very low but constant probability of mutation from \(\hbox{Cit}-\,\rightarrow\) Cit+. The historical contingency hypothesis assumes a shift in the probability of mutation from \(\hbox{Cit}-\,\rightarrow\) Cit+ due to the acquisition of a certain set of mutations (Blount et al. 2008, p. 7901)

Note also that the language used by the authors, i.e. “constraining/potentiating mutations” suggests an analogy with the phenomenon of phylogenetic constraints in macroevolution. A phylogenetic constraint has two essential characteristics: (1) it is an evolutionarily maintained trait, i.e., that it persist in a lineage, and (2) it sets key aspects of the ecological interactions and selective regime for a taxon. For example it can be a developmental feature that limits (or enable) certain mode of alimentation, locomotion or reproduction of organisms of a same lineage. Unfortunately, pursuing further this analogy would take us beyond the scope of this paper, but it would be interesting to see if the historical contingency hypothesis promoted by Blount et al. (2008) could also explain the way in which macroevolution causally depends on such phylogenetic constraints.
How did the researchers managed to experimentally distinguish between “rare mutation” and “historical contingency?” Because they keep a frozen record of the different populations every 500 generations, the group was able to “replay” the evolution of this lineage before and after the arrival of the new Cit+ phenotype. They observed a higher proportion of citrate-metabolizing phenotype in the later generations, thus “support[ing] the hypothesis of historical contingency, in which a genetic background arose that had an increased potential to evolve the Cit+ phenotype” (Blount et al. 2008, p. 7903). So the numbers suggest that one lucky population went through just the right series of mutations, which made more likely the emergence of an ability to metabolize citrate in aerobic conditions. Their conclusion expresses this eloquently:

[O]ur study shows that historical contingency can have a profound and lasting impact under the simplest, and thus most stringent, conditions in which initially identical populations evolve in identical environments. Even from so simple a beginning, small happenstances of history may lead populations along different evolutionary paths. A potentiated cell took the one less traveled by, and that has made all the difference. (Blount et al. 2008, p. 7905)

This will be further supported if the investigations find the elements that arose in the mutational history of that lineage that actually favored that capacity. Yet, the historical contingency hypothesis offers an interesting point of view on the role of “small happenstances of history” that goes beyond mere chance. The point I wish to stress here is that the authors do not merely claim that that different lineages took divergent trajectories, that things could have been otherwise. Historicity emerges from their analysis because some past vagaries have shifted the probability of occurrence of a certain, more advantageous, evolutionary outcome.

It is also interesting to note that we can conclude from this analysis that “dependence on initial conditions” is not the only form of historicity taking place in evolutionary process. Unlike the experiment discussed in "Dependence on initial conditions", the populations compared in this section were all initially identical. According to the researcher’s report, the divergence observed in the evolutionary trajectories were not due to the fact that the populations were initially different, but because one of the populations went through a rather unique series of mutations. In other words, the difference maker in this case was not the initial conditions, but the variations that occurred along the way. I will argue that the notion of “path dependence,” is needed here. The next section will be devoted to explaining in detail what historicity amounts to in processes that diverge towards the future and yield different outcomes from the same starting point.

Historicity as path dependence

Definitions

It is useful to represent the situation created by the LTEE project with a branching tree.12 Let us represent such a “tree” as a partially ordered set of states si. What constitutes a “state” will often depend on the interests of the investigators. In the LTEE project, features like fitness, cell size, genetic and morphological constitution can be taken as “state variables.”

The tree is ordered by a “causal order,” <, which indicates the flow from past to future and can be understood as “earlier < later” or “backward < forward.” We also introduce the notion of “instant.” An instant is a set of states that result from a vertical cut in a tree. For example in Fig. 5, s1 and s2 are alternative states at instant i1, whereas s3s4s5 and s6 are alternative states at instant i2. This notion is relevant because, as we will see, I argue that inferences about path dependence consist of making same-instant comparisons: we consider what else could have happened at the instant at which a given state occurs. Note also that instants are temporally but not causally ordered. We signify the occurrence of state x at instant z as “sxiz.” This is also called an “outcome.”
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Fig. 5

Branching tree: partially ordered series of states si occurring at different instants i. p and q on branches represent probability of possible future states, where p + q = 1

An important concept here is comparability. We say that two states sx and sy are comparable if one of them lies in the past of the other, i.e., (sx < sy) or (sy < sx). Branching trees include incomparable pairs, and this is why we define them as partially ordered set of states. In Fig. 5 for instance, s1 and s2 form an incomparable pair, whereas s1 and s3 are comparable.

We define a history h as a complete course of events (or a process), i.e., as a maximal, totally ordered set of states in a tree. That is:

Totally ordered: any two states s1,s2 in h are comparable.

Maximal:If s is any state not in h, then h ∪ {s} is not totally ordered.

Figure 5 encompasses four histories: h1 = [s0s1s3]; h2 = [s0s1s4]; h3 = [s0s2s5]; and h4 = [s0s2s6]. To be more precise, the graphs used in this paper encompass processes that are portions of history for which we specify a particular initial moment along with the conditions characterizing it. It is often convenient in science to make up certain conditions as being “initial” for different reasons (e.g., because some entities are put in different controlled conditions). So, even if entities often preexist an experiment or an observation, we decide to take some states as “initial” because they correspond to the conditions of existence at the beginning of an investigation. We then treat what follows as different (possible) processes, constituted of maximal and totally ordered sets of states, given some specified initial conditions. In this sense, a process can coincide with a particular history if their respective initial moments also coincide, but it can also be a portion of a full history if the process’ “initial” state happens later in that particular history.

We say that all states in a tree have a historical connection, i.e., a point in their past up to which their histories intersect. Formally, we say that for any two states si and sj, there exists a state sk, such that sk < si and sk < sj, and there is no later state sn > sk such that sn < si and sn < sj. In Fig. 5, the state s1 is the historical connection between s3 and s4, but s0 is not. On the other hand, s0 is the historical connection between s3 and s6, and between s4 and s5.

This condition allows us to introduce another fundamental element of branching trees: the branch point, i.e., a node where histories diverge. We say that si is a branch point if and only if it belongs to two histories, hx and hy, as long as there is no sj such that sj > si and sj belongs to hx and hy. In contrast to a branch point, a convergence point is a node where histories fuse into one another and become indistinguishable.

We can now define a path p as a complete, totally ordered set of branch points in a tree. A path p is complete if, for any branch points sxsy in p, if sx < s < sy, and s is a branch point, then s is also in p. Completeness makes sure that there are no in-between branch points missing, but it does not necessarily include the first branch point (if there is one). This opens the possibility for considering only a (complete) subset of a history, without having to take into consideration every event that came before. This definition of path also leaves out the last nodes on the tree (which are not branch points). Figure 4 encompasses two paths: p1 = [s0, s1] and p2 = [s0, s2]. This definition of path focuses our attention on the moments where something non-trivial (branching) occurs.

With these basic elements of ontology in place, we can define path dependence and explain what it means to say that history matters in stochastic processes. A process (or a history) that leads to state sx at instant iz is path dependent if and only if:
  1. 1.

    There is a path p which has sx at iz as possible outcome, and some state other than sx as another possible outcome at iz;

     
  2. 2.
    There exists at least one alternative path p′ that intersects p such that either:
    1. a.

      sx does not occur as a possible outcome of p′ at iz; or

       
    2. b.

      sx does occur as a possible outcome of p′ at iz, but \(\hbox{Pr}(s_x i_z \mid p) \neq \hbox{Pr}(s_x i_z \mid p^\prime).\)

       
     

We can immediately recognize the conditions elaborated in the general definition of historicity mentioned above. More informally, the fundamental conditions stated in this definition are that alternative outcomes must exist at a given instant (things could have been otherwise) and the probability of a particular outcome should change when we switch from one path to another (probabilistic causal dependence). But unlike the definition provided in "Dependence on initial conditions", the multiple possible pasts are not alternative initial conditions but paths.13

Following our definition, one can say that processes in Fig. 6 are path dependent because the probability of a given outcome varies as a function of the path taken:
$$ Pr(s_3 i_2 \mid p) > Pr(s_3 i_2 \mid p^\prime); Pr(s_4 i_2 \mid p) < Pr(s_4i_2 \mid p^\prime), $$
(1)
where p = [s0, s1] and p′ = [s0, s2].
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Fig. 6

Path dependence

To the contrary, processes in Fig. 7 are path independent despite the fact that alternative outcomes are possible at the latest instant. Processes in Fig. 7 are path independent because the probability of reaching a given outcome, let’s say s3, remains the same for any admitted path:
$$ Pr(s_3 i_2 \mid p) = Pr(s_3 i_2 \mid p^\prime); Pr(s_4 i_2 \mid p) = Pr(s_4i_2 \mid p^\prime), $$
(2)
where p = [s0, s1] and p′ = [s0, s2].
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Fig. 7

Path independence

This clearly shows why multiple possible outcomes without causal dependence does not entail historicity, and at the same occasion reinforces the point that contingency as unpredictability may be compatible with historicity but does not capture its theoretical core. Even if the tree in Fig. 7 had included many possible outcomes at i2, any process in such tree would have been path independent if the probability of each of these outcomes remained identical given one or another path.

This account of path dependence also admits of different senses of degree. The first one has to do with the extent to which different paths yield divergent vs. convergent outcomes. Consider Fig. 5 again. This case corresponds to the most extreme degree of path dependence, where every path diverges to a different outcome. More generally, if an outcome sx occurs only once at a given instant, it means that the probability of reaching state sx at that instant via any alternative path is null. It is certain that the process will end up in a different outcome each time it changes path. A lesser degree of path dependence will admit multiple occurrences of a state at a given instant—provided that condition (2b) is met, i.e., there exist at least one path for which the probability of a given outcome differs from the other paths (as in Fig. 6). Pushing this phenomenon of recurrence of states at a given instant to its extreme would result in complete converge and path independence (as in Fig. 8). In effect, in the case of complete convergence, the same state occurs in every admitted history at a given instant. So, an interesting feature of this definition is that we have a continuum between historical and ahistorical processes. It also follows from this sense of degree that the zone between full path dependence and path independence includes a wide range of cases of partial convergence, where more repetition of states at a given instant means a lesser degree of path dependence.
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Fig. 8

Complete convergence corresponding to path independence

One could add an extra dimension to this by also considering the degree of similarity of these outcomes. The degree of path dependence could be greater if the alternative outcomes are very different, whereas a path leading to alternative but more similar outcomes would present a lesser degree of path dependence. Take Fig. 1 for instance. A large difference in fitness between the groups would suggest that alternative evolutionary paths made a big difference, whereas closer fitness values between groups would mean that history matters to a lesser extent.

There is yet another sense of degree implicit in my account that is not typically discussed by theorists of path dependence (Arthur 1994; Bassanini and Dosi 1999; Castaldi and Dosi 2006; David 2001; Pierson 2004; Szathmáry 2006).14 It relates to just how small or large the difference is between conditional probabilities in the second condition. For example, the processes illustrated in Figs. 9 and 10 have the same structure: both trees are partially convergent because the two final states appear twice in the final instant. However, the processes in Fig. 9 are weakly path dependent, whereas the ones in Fig. 10 are strongly path dependent. More precisely, the processes in 9 show a low degree of path dependence because the probability of a given outcome at the latest instant show very little changes when we switch from one path to another. Conversely, in Fig. 10, the difference in probabilities of outcome is much greater. Note also that we can push this variation to its path-independent extreme by making invariant the probability distribution of outcomes. This would be the case of Fig. 7, where the weights on the branches all equal to p = 0.5. Reaching alternative outcomes in this situation would be a mere result of chance, not of history.
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Fig. 9

Weak path dependence

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Fig. 10

Strong path dependence

This account makes path dependence an instant-relative property. Theoreticians of path dependence do not typically integrate this type of relativity. They treat a process as either path dependent or independent, but it cannot be path dependent at some instant and path independent at another instant. This can be problematic if one wants to count transiently divergent dynamics (see Fig. 2b) as examples where history matters, as did Lewontin in some early papers on “historicity” (Lewontin 1966; Lewontin 1967; see also Beatty and Desjardins (2009) for an analysis of this example). Using a simple mathematical model, Lewontin has shown that two initially identical populations evolving in the same average environments, but in which environmental conditions occur in different order, can have very different evolutionary pathways.15 But these populations take significantly different paths to very similar genetic end-states. In fact, it can be shown that a greater stability in environmental conditions can result in perfect convergence at the end of the day (Desjardins 2009). Thus, these populations show only transient divergences. So, Lewontin’s results would hardly count as path dependent if all we cared about were the final (long-run) outcomes.

Yet, history matters for these transiently divergent dynamics in a way that does not apply to parallel dynamics (compare Fig. 2a, b), namely that history matters along the way. This is exactly what Lewontin understood as historicity in the stochastic setting he implemented. But transiently divergent dynamics are in opposition to path dependence only if we can “see” from the beginning that the fate of populations is to converge in the long run. In an instant-relative framework, which compares outcomes at a given instant, history can matter along the way, and yet cease to have an influence once all admitted processes (or pathways) have actually reached a global equilibrium.

Furthermore, I believe that adopting an instant-relative approach reflects the epistemological status we are in when observing and comparing real processes. What we call “outcomes” in real life observations are always relative to an instant. Only mathematical models or simulations can give a global (almost omniscient) view. These projection tools make possible inferences based on what comes at the “end.” The problem is that we seldom reach this end point in real systems, where observations and data are always instant relative.

The distinction made between “dependence on initial conditions” and “path dependence” may seem unnecessary to some. Ben-Menahem (1997) for example suggests that one important notion of “contingency” found in the literature is captured by the phenomenon of “sensitivity to initial conditions.” This sensitivity can be extreme (as in deterministic chaos) or more moderate, but the core of her notion is that of “history as a difference maker.” And she contrasts contingency with “necessity,” as applied to situations where “the same type of final outcome results from a variety of different causal chains.” (Ben-Menahem 1997, p. 100) I agree with many points made by Ben-Menahem (e.g., we both suggests that historical contingency does not opposes, but embraces causal dependence, and that historians and historical scientists need an account of historical contingency that admits of degree). However, I make an explicit distinction between dependence on initial conditions and path dependence, whereas her analysis focuses essentially on dependence on initial conditions.

What counts as a “initial” condition can sometimes be vague (if not arbitrary). Ben-Menahem for example mentions that “ if the initial conditions themselves are necessary effects of earlier circumstance, the situation becomes more complex” (p. 102). This kind of language reflects a rather loose usage of “initial”, according to which any past conditions can count as initial. This may have been appropriate for her analysis of contingency, which focused on a certain interpretation of contingency as opposed to necessary. This does not mean that we do not need a further distinction between situations where the difference maker is not the initial conditions but what happens along the way. I believe that the latter can find several applications, in particular the situations created in the LTEE project. The bacteria were not spontaneously generated at the beginning of the experiment, and the elements entering into the petri dishes existed before (although perhaps under different forms). In that sense, the beginning of the experiment is not absolutely initial. However, I believe that it is not arbitrary to conceive as “initial” the state(s) of the cloned bacteria when they are introduced in their new environment, and consider the subsequent changes in state as part of the path taken by the populations.

Note also that dependence on initial conditions can be represented in one temporal step, whereas path dependence requires at least two temporal steps. In deterministic processes for example, the outcome is perfectly determined once the initial conditions are set. The only form of historicity occurring in such processes is dependence on initial conditions (if we admit that the latter can change). All we need to represent this form of historicity is alternative initial conditions mapping into different outcome (e.g. \(s_0 \rightarrow s_f\) and \(s_0^\prime \rightarrow s_f^\prime\)).16 But historicity in stochastic processes with a single initial state requires at least two temporal steps. If we had a single branch (e.g., \(s_f \leftarrow s_0 \rightarrow s_f^\prime\)), then we could only say that there are multiple possible outcomes from a given starting point, but we could not say in a non trivial way that one outcome is more likely to occur given the history of the process. The inference of path dependence makes sense in this case only if another branching point is added prior to s0, and if there was a difference in the probability of sf or \(s_{f}^{\prime}\) given the occurrence of s0, instead of something else (as represented in several graphs in this section). In other words we need minimally two subsequent branching points.

One can even see a similar kind of extended temporal dependence in Ben-Menahem’s (1997) explanation of degree of necessity. She says: “But it is perfectly acceptable to speak of degrees of necessity […], as long as the final outcome is relatively insensitive both to initial conditions and to potentially disruptive intervening events” (p. 100). This clearly suggests that the difference in the initial states is not always the sole difference maker. I believe that path dependence is what we need in order to capture the form of historicity that occurs when there are no difference in initial conditions, but differences in the path taken.

Application

My account of historicity as path dependence applies well to the experimental evolutionary processes considered in "Chance and mutational history" and "An unexpected adaptation. Recall first how Johnson et al. (1995) infer that the populations in the LTEE project have reached different fitness levels because they evolve in a landscape with multiple peaks and because they have “isolated-event” replacement (mutations arising in different order), as opposed to “coincident-event” replacement (same mutations in the same order). As they say:

when collective replacement is isolated-event, the random origin of mutations, either along or in concert with genetic drift, can theoretically lead to sustained divergence of formerly identical populations in identical environments, even for selectively important traits. In contrast, if collective replacement is coincident-event, then the most fit allele will usually appear and win in every population, so that there is no opportunity of sustained divergence among the populations of the metapopulation. (Johnson et al. 1995, pp. 128–129)

So coincident-event replacement results in parallel evolutionary dynamics (as in Fig. 2a), whereas isolated-event replacement produces divergent evolutionary dynamics (as in Fig. 2b, c). In order to see how path dependence applies to the latter and path independence to the former, we need to go slightly deeper into their analysis.

Their mathematical model assumes haploid organisms (i.e., organisms with one set of chromosomes) and three possible alleles or genotypes (i.e., three possible states at a given locus), that we will represent by: s0, s1 and s2.17 All populations are initially identical and begin their journey in state s0. They also establish that each one of these genotypes has a different fitness value, w(sx), ordered thus: w(s0) < w(s1) < w(s2). In order to create two stable equilibria, i.e., two disconnected adaptive peaks, the authors further assume that the mutation rate from si to \(s_j (i, j \in \{0,1,2\})\) is μ01 = μ02 = μ > 0, and all other μij = 0.

Figure 11 represents what happens when we take such model and implement isolated-event replacement. Clearly, path dependence applies here: there exist alternative outcomes at the latest instant and the probability of reaching s1 or s2 changes as we go from one path to the other.
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Fig. 11

Dynamic governed by isolated-event collective replacement

Conversely, coincident-event collective replacement, produced by high mutation rate and large population size, would rapidly and synchronically converges towards the same globally stable evolutionary outcome, i.e., all populations will proceed thus: \(s_0 \rightarrow s_2\). In this case, the evolutionary dynamic is not path dependent but linear.

Although the divergence of initially identical populations evolving in identical environments has been observed in experiments with living organisms (e.g., fruit flies and bacteria), the model used by Johnson et al. (1995) makes several strong and unrealistic assumptions. These include for example the restriction on the mutation rates and the fact that populations are perfectly isolated from one another. The authors acknowledge that in nature there will be migration between populations and that some suboptimal local adaptations could be wiped out by the immigration of organisms with more advantageous traits. This only means, however, that the phenomenon of sustained divergence is going to be less frequent and perhaps more difficult to detect in nature. So perhaps we should expect to find weaker forms of path dependence in nature, but it does not invalidate the claim that path dependence can account for the situation where populations of the same species evolve independently different solutions to the same adaptive challenge.

The evolution of the Cit+ phenotype in one of the lineages can also be interpreted in terms of path dependence. I will not represent the situation graphically, but the reader can imagine a tree with twelve branches leaving from a unique starting point, and after 31,500 generations, one of these processes results in the occurrence of a phenotype capable of using citrate as a source of energy. Clearly, this process is path dependent. There were other alternative outcomes at the same instant and only one corresponded to Cit+. Moreover, recall that several mutations have preceded this phenotype, and it was suggested by the authors that this series of changes has actually potentiated the evolution of Cit+. In other words, this particular outcome actually needed a specific mutational history in order to arise. Had the lineage followed a different mutational history, the probability of Cit+ would have been much lower (as suggested in Fig. 4).

Concluding remarks

Inspired by the work and analyses provided by the researchers involved in the LTEE project, I have identified and defined two general forms of historicity: dependence on initial conditions and path dependence. Although different in the details, both share the following three essential components: (1) multiple possible pasts, (2) multiple possible outcomes at a given instant, and (3) a relationship of causal dependence between these two sets, i.e., the probability of a given outcomes changes as a function of the past. The fundamental difference between the two forms of historicity discussed here resides in what is included into the expression “past”. The latter can either refer to a set of initial conditions or to the entire series of stochastic events (branch points) separating a given starting point and an outcome.

I have also argued that it is more productive to think in terms of degree of path dependence, and that claims to the extent that history matters should be context and instant relative. Unspecified claims about the idea that history matters (or not) are to a large extent vacuous and are perhaps another reason behind the ongoing debate about the (ir)relevance of history in evolution. The type of structure one decides to follow will also affect the degree of path dependence s/he will discover. Handford (1999) for example suggests that morphological evolution is much less likely to be divergent and retain information from the past than genetic evolution. So, we need to specify what system and what properties are relevant, along with what (initial) conditions and what time scale apply.

My definition of historicity remains compatible with the view that the outcome of a particular evolutionary process is (to a certain extent) unpredictable. This follows from the fact that historicity applies when there exist multiple possible outcomes, when things could have been otherwise. Yet, we have seen that the incapacity to predict what outcome will obtain from a given starting point does not necessarily mean that history matters. For example, the situation where every possible outcome is equally likely would also result in the impossibility to conclude that historical variations have any impact on the future (because the probability of these outcomes would remain independent of past variations). Furthermore, one can embrace historicity as dependence on initial conditions, but still believe that evolution will be mostly predictable and virtually path independent. This would characterize Conway Morris’ position for example, according to which many of the life forms existing on earth (including us) are inevitable given the conditions of existence that prevailed since the early stages of chemical and biological evolution on this planet. This is equivalent as saying that the number of viable evolutionary strategies were very few and that many pathways have (and will) eventually converge to the same outcome. And as we saw, more convergence also means less path dependence. Conway Morris’ position may not go as far as saying that things could not have been otherwise at all, but it surely entails a very low degree of path dependence. Gould in comparison would stand on the other end of the continuum. Note however that these conclusions are only highlighting trends in these authors writings and that a fuller analysis of their respective position would require that we attend the specific lineages, structures and time length covered by their researches.

Finally, I would like to conclude with a parenthesis about a possible advantage of using experimental evolutionary setting in investigating about the role of history in evolutionary processes. One of the inherent difficulties arising with this kind of project comes from the need to follow and compare variables in multiple populations of organisms living in identical conditions. There exist at least three ways in which we can go about this. One consists of using a comparative method in natural systems, i.e., looking in nature for conditions of the same type and see if species (or some of their features) tend to evolve in the same way or diverge. However, these conditions may be extremely rare, depending on how much similarity in the initial conditions and environment one is looking for. Another difficulty arising with the comparative method, if the purpose is to understand the relationship between changes in the historical series and outcomes on the long run, is that we cannot manipulate the variables we are interested in. Nor can we easily follow these variables for a long enough time. Using fossil records surely can help to solve the latter issue, but reconstructing the past comes with another set of difficulties and limits that I will not address here. 18

A second alternative is to use computer simulations and models. It is easy to set values of variables and parameters in mathematical models or to run a simulation many times in order to produce a large set of time series and thus see if the system describes parallel/convergent or divergent trajectories. However, mathematical models never achieve the complexity of living systems, and this lack of realism is a critique that any simulation must be ready to accept. Nevertheless, we have seen that mathematical models can be very useful in generating insights about what could be happening in living populations.

The third avenue is to conduct lab experiments with organisms having short generation time (as in the LTEE project). The experimental approach, I think, offers a nice compromise between realism and the need to manipulate some parameters. Moreover, it can at least partially solve the issue of reconstructing the past by creating replicated (cloned) populations that are subsequently and simultaneously propagated in identical environments. These lab experiments also offer the possibility for a better control on the conditions in which the populations evolve as well as on various parameters judged important in the evolutionary dynamics. One may object that observing path dependent evolutionary dynamics in microbial organisms in a laboratory does not necessarily mean that the same will occur in all kinds of natural populations. Yet, the LTEE project has shown that even in controlled, identical environments, initially identical populations can take divergent trajectories, and more importantly for the present paper, it provided a mean to gain a clearer understanding of the phenomenon of historicity.

Footnotes
1

Just to cite a few ones: Brown (1995), Strong (1984), Gould and Lewontin (1979), Gould (1970, 1980, 1989, 1991), Lenski et al. (1991), Lenski and Travisano (1994), Lewontin (1966, 1967), Pickett et al. (1994), Ricklefs and Schluter (1993), Szathmáry (2006), Travisano et al. (1995), Williams (1992) and Wilson (1992).

 
2

For reason of space, I will not be able to include all the relevant discussions of historicity in biology in this essay. I will focus on experimental evolution and thus leave aside many relevant examples from evolution and ecology (e.g., studies reporting that evolutionary and ecological processes are sometimes affected phylogenetic constraints (Price 2003); see also Sterelny and Griffiths (1999) for a more philosophical discussion on the role of history in ecology). I will also not discuss the position defended by Wimsatt (2001), according to which that history matters in (macro)evolution essentially because of the phenomenon of generative entrenchment. These topics are addressed more in detail in another, more extended manuscript (Desjardins 2009).

 
3

See also de Duve (1995) for a similar view, or Handford (1999) for another, milder, critique of Gould’s contingency thesis.

 
4

Note that there is a growing literature on “path dependence” in the social sciences (Bassanini and Dosi 1999; Castaldi and Dosi 2006; David 2001; Hodgson 1993; Mahoney 2006; Page 2006; Pierson 2004; Mahoney 2000), and that it has been recently applied in biology (Szathmáry 2006). Some differences between my and some of these accounts will be highlighted along the way.

 
5

The fitness of a derived (evolved) population is obtained by allowing the population to compete against the ancestral type. The relative fitness is then obtained by calculating the ratio of the competitors’ realized rates of increase.

 
6

And note that the populations have still not converged after 50,000 generations.

 
7

A very similar analysis was performed by Wahl and Krakauer (2000).

 
8

Note however that the variations in genotypes were not directly measured at this point. What they proved is that the fitness of all twelve populations were significantly different when introduced in a maltose limited environment. Although different genotypes could very well reach the same level of fitness, the reverse would be very surprising (Travisano et al. 1995, p. 88).

 
9

More precisely, two experiments were discussed in this paper, one with 24 (12 groups of 2) populations put in lower-temperature environment, the other with 36 (12 groups of 3) populations put in environment with different nutrient contents. But the idea was the same for both experiments: create different initial states (genotypes) and see whether these historical differences affect the evolutionary dynamics.

 
10

Recall that “chance” in these studies is usually interpreted as random mutations, drift or a combination of both.

 
11

This does not mean as we will see below that different initial states is necessary.

 
12

I base the following analysis of “tree” on Belnap et al. (2001), although the nodes are not “moments” in my account, but “states.”

 
13

Note also that, unlike many accounts of path dependence, this definition does not assume that outcomes are equilibrium states or stable attractors.

 
14

But see Ben-Menahem (1997) for a similar notion.

 
15

In this case, the order of environments was simply reversed for the second run of the model.

 
16

Dependence on initial conditions can also occur in stochastic processes. I chose the case of deterministic processes because they are simpler to represent.

 
17

Note that identifying alleles and genotypes is possible only for haploid organisms.

 
18

For interesting discussions about this issue, the reader can look at the following: Handford (1999), Sober (1988) and Tucker (2004).

 

Acknowledgments

I wish to thank John Beatty, Paul Bartha, Christopher Stephens, Robert Batterman, Gillian Barker and Christopher Smeenk, and the reviewers of Biology and Philosophy for their comments on earlier drafts of this paper.

Copyright information

© Springer Science+Business Media B.V. 2011