Distinguishing Among Causal Explanations in Biology: Mechanical and Topological Explanations

1. Distinguishing Among Causal Explanations in Biology: Mechanical and Topological Explanations Philippe Huneman. IHPST (CNRS/ Université Paris I Sorbonne)

2. Models of explanation in philosophy of science • Causal explanations (Salmon, 1984) • Unifyingexplanations (Friedman 1963, Kitcher 1969) • The twomodels are rathertwopoles (combinations are possible (eg Strevens 2004))

3. Often causal explanationsmean : identifyprocesses or mechanisms) displaying the phenomena. • (Salmon 1984; Wimsatt 1982; Craver Machamer Darden 2000, Craver 2007 • In thisview the role of counterfactuals is downplayed • Veryaccurateconcerningmany aspects of neurosciences (Craver 2007), molecular biology (Darden 2003)

4. Here : in addition, define and consider « topologicalexplanations » • Present in ecology, evolutionary biology, social sciences, etc. • Do not pinpointmechanisms • Influentialwith the rise of network science, and the inflation of « neutral systems » (Hubbell, UNTB/ Stadler and Schuster, etc.) • Are a kind of causalexplanations in a counterfactualsense

5. Outline • Topologicalexplanations: nature and role • The nature of topologicalexplanations • The pervasivenessof topologicalexplanations in communityecology • Topological vs mechanisticexplanations • Mechanical and topological explanations reconsidered.- disjunct, combined • Disjunctexplanations : the case of robustness in living systems • Neutralspaces, neutral networks • Combiningmechanism and topologicalexplanations • A generalview of causal explanations in biology

6. Part I. Topologicalexplanations

7. I.A. The nature of topologicalexplanations

8. Example:the survival of the flattest(Wilke, Lenski et al. 2001) • Twostrains of bacteriawith distinct distributions of mutations in competition, varying the mutation rates m • two types of topological structure of populations in fitness space – sharpness, flatness

9. The probability that a mutation displays a large drop in fitness as compared to the current alleles is clearly a consequence of the shape of the curve, because it only depends on the fitness distribution of possible mutations. And this probability is what determines the evolutionary fate of each strain

10. What is a topological explanation • 1. Topological properties: System S, properties Ti, related to some space: how parts of the system are located regarding one another, etc. The “parts” can be parts of a more abstract space: • Phase space • Community ecology : parts are species/ cluster of species in a (foodweb) graph

11. S has elements, parts, features or capacities, moments of its regular behavior, or of the set of its possible behaviors, which are likely to be represented in a graph, a network or a variety S’ in a space E. S’ will have topological properties Ti, namely, properties which specify its invariance under some continuous transformations, and which will determine equivalence classes between all structures S* homotopic to S’. Or, if S is a graph, you can specify some properties of S’ (e.g. connexity; cyclicity, etc.) which will define an equivalence class, and distinguish S’ from other graphs S* not having those properties.

12. S has topological properties in virtue of its relation to S’ and its elements and relations • Eg. Lenski’s bacteria strains have the properties of “flatness” and “sharpness” in virtue of their essential relations to their mutations distribution.

13. Twoconsequences • S realizes the topological properties Ti, and many Sj can be said to be equivalent because they realize the same Ti (for example, belonging to the same equivalence class defined by homotopic paths in some space) • Ti have many consequences, especially, they may constrain the possible transformations of S’, for example because all continuous transformations should lead to some homotopic structure S’’.

14. 2. Topological explanation an explanation in which a feature, a trait, a property or an outcome X of a system S is explained by the fact that it possesses specific topological properties Ti.

15. Topology and processes • The topological explanatory relation implies that whatever possible process Bj occurs to S that involves some or all elements or parts of S, no Bj is sufficient to account for X, but the simple fact that S realizes Ti entails as a consequence the fact that S has X. • It may be that a given X causally results from some Bj, but that what explains X is not the Bj itself; eg. causal process Bk also produces X, but that the topological properties Ti which constrain in the same way all processes Bi, entail that whatever the process Bj, the outcome will be of type X -> the causal process Bj itself does not make any difference (conditionally on the fact of Ti) to the outcome and is therefore not explanatory.

16. --> Explanation of X goes like a relation of entailment between topological properties Ti and X or features of X, and not like the display of a mechanism from which X would be a temporal outcome. • Phrase « Topologicalexplanations »: frequent in maths; in quantum physics (Arcos & Pereira 2007) – but about alreadymathematicalentities (spin etc.) • Yet in biology, ecology etc. those are pervasive,

17. I.B. The pervasiveness of topologicalexplanations in communityecology

18. Ecology: the diversity-stabilityhypothesis « The more diverse a community is, the more stable it is in its composition » Classicaluntil the 70s May 1974: mathematicallyit’s not the case Thus: Need to definediversity (functional ? Number of species ? ) and stability (of composition ? Constancy of a biological property (eg biomass); persistence (no speciesgetextinct); resilience

19. -> Various modes of the hypothesis: Tilman 1996 (constancy of biomass); etc. Basically, May’s model assumedrandom connections betweenspecies; but this is not realistic. Hence: exploringvariousconsequences of specificconnectivityschemesuponkinds of stability

20. Networks of connections approach Variables: the number of species and their average number of links, the number of connections realized between species as compared to the number of possible connections (connectance), the distribution of the connections between species. Solé and Montoya 2001; Dunne et al. 2002; Mc Cann et al. 1998; Montoya and Solé 2006 Many ecological interactions (competition, predation, mutualism); some graphs are drawnwithoutconsideringeven the nature of the causal relation (Montoya and Solé 2006)

21. Network with 2 hubs and lots of poorlyconnectedspecies (464). Deleting one species: chances to deleteonly an isolatednode (hence not changing the pattern) are 232 times higherthan chances to alter the pattern S : the community S’: the network of interactions T: property of havingtwo hubs. Tentailsthat the probability of altering the pattern with one species extinction is verylow-> stability

22. Important networks Scale-free networks : distribution of connections varies according to a power law – hence, few big hubs, manyslightlyconnectednodes Production rule : preferentialatachment (richgetricher) (P(n) ~ n) Small worlds(twonodes are always close, high level of clustering); Production rule – addingrandom links

24. Scale free networks -deletingrandomly a species has many chances to affect poorlyconnectedspecies, hence the stability of the communitythrough conservation of quite all the links • Internet has the same property (Solé et al. 2001); alsosomefinancialsystems (Levin et al. 2008) Many ecological networks are often not exactlyrandom but almost (truncatedscale free networks) – hence the consequence of stability

25. Small world networks – disconnecttwospecies A & B (by deleting an intermediary one betweenthem ) thenthereexists a cluster where A is wherethere is a speciescloselyconnected to B. • S’ the interaction network • T: « Being a smallwork » Entailsstability (robustnessviz. Species extinction)

26. These are topologicalexplanations : theydeduce the stability property from the topological property of the network of connections, withoutconsidering the dynamics, the causal interactions betweenspecies, etc. • In both cases, if S1 has lots of preyingspecies and S2 lots of parasiticspecies but S’1=S’2, the same explanation for stabilityholds. • Stating the formal property – e.g., the relative number of small and large hubs – immediately yields the result, no matter what are all the temporal mechanisms which can happen in the community.

27. I.C Topological vs mechanisticexplanations

28. Topological and mechanisticexplanations • A mechanism is defined by entities and activities (MDC 2000); here no specificactivities and entities are identified. • For ex. in the « survival of the flattest » what’sexplanatory is the topology of the mutations, not the mechanisms (the same in both cases)

29. CommunitystabilityTopologicalepxlanations of stabilitycontrastwith the explanationswhichwouldidentify a restorationmechanism • A ->B->C->D • A ->B- / D C extinct, B decreases in frquency, D increases Vacant niche appears Otherspecies C’ replaces C A->B-> C’ -> D • The food web is a scale free network • Probabilitythat a randomspecies extinction alters the community structure is thereforelow Mechanistic explanation Topological explanation

30. Mechanisms • It i s in virtue of X’s y-ing, that Y and Z will for example be triggered to do J and K, and then make the system S likely to be f -ing as a result of combined J and K. • The explanandum of a mechanistic explanation is determined by stating the “set-up” and “termination” conditions. The chronology of sequences through which entities act is crucial to explain the production of this termination condition • Topological explanations There are no specific activities listed: no matter what the species do, whether they prey or not, whether they y or they f, and on whom etc. Stability rather occurs because of some network property of S’. • Another difference = the temporal sequence between events in topological explanations is irrelevant, since switching the order of events does not affect the explanandum

31. Suppose that two systems S1 and S2 have the same associated shape S’ in abstract space but that the relata of links in S1 are J and K, and in S2 are J2 and K2, with their distinct associated activities. Yet the topological properties of S’, realized identically by S1 and S2, will not be affected by this difference of activities and properties; therefore the identification of mechanisms in S1 and S2 is not relevant for explaining their outcomes

32. Part II. mechanical and topological explanations reconsidered.- disjunct, combined

33. 2.A. Disjunctexplanations : the case of robustness in living systems

34. Robustness A capacity to keep some parameters stable in the face of changing variables. • All modes of stability mentioned above in ecology satisfy this characterization. • Increasingresearches about robustness in biology (Kitano 2005, de Visser 2002, Wagner Robustness and evolvability in living systems) • Two more specific characterizations: a. “level” definition, in the sense that change in low level variables doesn’t involve change in high level variables – for instance changing the identity of several species does not affect some general properties like biomass or abundance pattern. b. “functional“ definition: robust are the systems able to maintain (some of) their functions in the face of perturbations.

35. Classicalkinds of robustness • Homeostasis (Cannon – Physiology) is an important kind of physiologicalrobustness • « Canalisation » (Waddington) meansrobustness of developmentagainstgenetic noise

36. Images of canalisation (waddington)

38. Robustness of genetic code • Redundancy : many codons yield the sameaminoacid -> often, changing A to C, G to T etc., does not change the functionalproduct • In general: Redundancy is an explanation of robustness (and also of evolvability) Otherkinds of robustness do not involveredundancy but « distributedrobustness » (most of them, Wagner 2005b) -> an example of distibutedrobustness (at the gene level): generegulation network of Endo16 (seaurchins)

39. 2 types of robustness issues • a. proximate : how is it ensured? (-> why does distributed robustness occurs?)? About a .Manykinds of explanations; they can be partitionedintomechanical vs. topologicalexplanations • b. evolutionary – b1), why has it evolved? b2) what role may it play in evolution About b1. Articulating genetic and environmentalrobustness.(What has been selected for ?)

41. DNA repair • The fact of diploidy and the mechanisms of recombination are such that if DNA is altered, another copy is intact which can be a template for DNA repair. In this sense robustness of DNA is produced by this ability to copy on the basis of a second haplotype as a template.

42. Modularity (Modularity is a highly clustered network) It reduces the effect of perturbations because those won’t break the whole systems, since they have chances to concentrate upon only one module, the others being likely to still function. An explanation for evolvability (Altenberg and Wagner 1996)

43. robustness of proteins in the face of recombination.

44. Why are topologicalexplanations of robustnesspervasiveat all levels ? • Manytopological properties : defined by invariance regarding to deformation • Stability : reactionviz. perturbation; in the S’ associated to system S , a « perturbation » is a « deformation » • Therefore, topologicalequivalence classes (ie shapes S* which are invariant troughcontinuousdeformation) determinekinds of stabilities • Hencetopological properties are oftenrelated to robustness properties

45. 2.B Neutralspaces, neutral networks

46. RNA sequence/structure (GP) maps

47. Many different sequences, due to the laws of chemistries and the requirements of minimizing free energy, fold into the same spatial structure (ie, with the same molecules at surface sites, hence functionally behaving similarly). • The space of structures is such that few structures are reached by several large sets of RNA sequences, whereas many structures are reached only by a few sequences.

48. « Neutral networks » • Defining a distance betweentwosequences G’G’ as the number of mutations needed to get from G to G’ • Defining the set G1 of sequencessuchthat f(H) = f (G) and d (G, H) = 1; sameoperationwitheach H in G1 etc. • Gives a neutral network – all Hs in it are accessible from G by a series of mutations withoutchanging the phenotype • Stadler and Stadler, 2001; Otalloni and Stadler, 2007; Van Nimwegen, Huynen and Crutchfield, 1996; Stadler, Fontana and Schuster 2001, Fontana et al., 1992 • Wagner 2005 • Gavrilets 2004 (neutral tunnels)

50. Explainsmutationalrobustness: if G is deep in a neutral network, mutations will not alter the phenotypes • Explainsalso evolution from low to high fitness with drift (through a neutral network) • Genetic robustness of some genotypes is the topological structure of the GP map, and the specific position of the genotype x in this structure. Once again, no mechanism of gene-protein interaction has to be hypothesized in order to understand such robustness. • And those neutral spaces allow biologists to formulate issues about the evolution of robustness (the more robust is a system, the larger are the neutral spaces in it) and the roles of robustness in evolution.