Here follow some notes, observations and references that were discussed in or are relevant to the 596b Discussion Group that meets at 12:00 in Room 302 before the Noon Seminar. The discussion group is open to all students (undergraduate and graduate) attending the seminar, but not to faculty, post-docs, etc.

09/20/05

We discussed complexity theory, its meaning and implications. Conversation also turned to design vs. selection and the notion of irreducible complexity.

• I only have a few books on complexity theory. One of them is Solé, R. and Goodwin, B. 2000. Signs of Life. Basic Books, N.Y. The biology is naive in places, but you'll find the basic elements of the theory:

  1. Nonlinearity;
  2. Bifurcations;
  3. Phase transitions.

  If these terms are unfamiliar, here's a primer:

  1. Figure 1. The time evolution of a dynamical system is determined by internal feedback and the environment.

     Complexity theory is the study of dynamical systems applied to real world systems.

  2. Dynamical systems are mathematical objects. As such, they can be studied without reference to nature. Still, interest in them has been stimulated enormously by their applicability to real world phenomena.

  3. When we study dynamical systems, we study the "evolution" (mathematicians use the word in a non-Darwinian sense) of quantities called state variables in time and space. To do this, we formulate governing equations (also called equations of motion or evolution equations). An example is (dx/dt) = f(x,p), where x is a quantity in which we have an interest and p is a parameter (see below)

  4. In the "well-stirred" case (all particles have an equal probability of interacting), we use ordinary differential equations to model the dynamics. In the "spatially extensive" case (particles interact principally with their neighbors), we use partial differential equations.

  5. In either case, rates of change depend on the values of quantities called state variables - the things in which we are interested - and on other quantities, called parameters, reflective of the state of the environment. So, for example, if you're studying temperature regulation of a rat in a box, the state variable is the animal's body temperature, and the parameter (there may be others), ambient temperature.

  6. Nonlinearity enters, because if the governing equations are nonlinear, the response to continuous changes in parameter values can entail abrupt changes in behavior.

  7. In the well-stirred case, these changes are called bifurcations, and they are marked by qualitative changes in temporal dynamics. For example, equilibrium behavior (the system sits still) can give way to a cycle. Or cycles of a particular period can give way to cycles of another period. And of course, there can be transitions to chaos, in which, case the fluctuations are aperiodic.

  8. The spatially extensive analogs of bifurcations are called phase transitions, because they were first studied in physics and chemistry - the most familiar example being different states matter: solid, liquid, gas. In this case, varying a parameter - temperature, pressure, volume - produces qualitative changes in spatial structure.

  9. Figure 2. Human femur in section.

     Applied to biology, the idea is that continuously changing a quantity on a can lead to qualitative changes in system behavior, often at a higher level of organization. An example is Kaufman's idea of a self-catalyzing network of biochemical reactions. As one increases the number of enzymes, the system becomes "self-catalyzing," at which point, every protein is a product of a reaction catalyzed by some other protein.

• For an introductory discussion of adaptive vs. developmental explanations of biological structure, go here.

• For further discussion with regard to the structure of cancellous bone, see If bone is the answer, then what is the question? (R. Huiskes. 2000. J. Anatomy. 197: 145-156).

09/27/05

We discussed the fixation of advantageous mutants, omitting reference to stochastic effects, and focusing on the implicit assumption that mutant and wild type are ecologically equivalent. The consequences of relaxing this assumption were considered - the bottom line being that if mutant and wild type diverge sufficiently, the requirement for mutant persistance is not that the new variety work better, but simply that it work. Reference to the work of Margulis on eukaryotic origins was made as an example of an evolutionary transition in which the new variety was ecologically distinct from its antecedents.

• Optimization Theory: Optimization theory presumes the existence of an objective function, J(u), the value of which depends on one or more parameters, u = [u₁, ... , u_n], called controls. The goal is to identify values, u^* = [u₁^*, ... , u_n^*], of the controls that maximize J(u). In biology, the controls are often allocation schedules - root, shoot, stores, for example - which, given limits to what can be allocated - time, energy, etc. - leads to notions of trade-offs and strategies in a natural way. As to the objective function, J(u) is often fitness, or some surrogate thereto - rates of caloric intake, numbers of offspring fledged, etc. Depending on the application, calculating u^* can be more or less difficult. A good introduction to the various methods can be found in [i71]. As the title suggests, this book was written for economics students, with the consequence that the mathematics is reasonably readable.

• The Best of All Possible Worlds: The use of mathematics to compute optimal evolutionary strategies traces to the work of G. P. Bidder [b27, b37], who studied oscular diameters in stillwater sponges. Bidder's criterion of optimality, i.e., his objective function, was maximization of the distance travelled by ejected water before it re-enters the sponge through the pores, his control, the diameter of the osculum and his conclusion, that there is a diameter (not too large, not too small) that maximizes said distance. Within experimental error, he found that oscular diameters are optimal. More familiar examples of the the application of optimality theory to evolutionary biology include MacArthur and Pianka [mp66] and Charnov [c76] and Gadgil and Bossert [gb70]. For a recent review [s04] of the theory applied to life history evolution, go here. For a look see at Gould and Lewontin's famous / infamous critique of this approach, go here.

• Figure 3. Eukaryotic origins according to Margulis.

  Eukaryotic Origins: Margulis imagines that eukaryotes arose by symbiogenesis whereby a sulfur-reducing Thermoplasma-like archaebacteria and a sulfide-oxidizing eubacterium (spirochete) entered into a symbiotic relationship ("Thiodendron" stage) that eventually became obligate. In Margulis' view, the karyomastigont, a structure connecting the nucleus and flagellum in amitochondriate eukaryotes and other protists, is "the morphological manifestation of the chimera genetic system that evolved from a Thiodendron-type consortium." Here are links to a recent paper by Margulis and another by Gupta.

• References:

  [b27] Bidder, G. P. 1927. The relation of the form of a sponge to its currents. Q. J. Microscop. Soc. 67: 293-325.

  [b37] Bidder, G. P. 1937. The perfection of sponges. Proc. Linn. Soc. 149: 119-146.

  [c76] Charnov, E. L. 1976. Optimal foraging, the marginal value theorem. Theoret. Pop. Biol. 9: 129-136.

  [gb70] Gadgil, M. D. and W. Bossert. 1970. Life historical consequences of natural selection. Amer. Natur. 104: 1-24.

  [i71] Intrilgator, M. D. 1971. Mathematical Optimization and Economic Theory. Prentice-Hall. N. J.

  [mp66] MacArthur, R. H. and E. R. Pianka. 1966. On optimal use of a patchy environment. Amer. Natur. 100: 603-609.

  [s04] Schaffer, W. M. 2004. Life histories, evolution and salmonids. Pp. 20-51. In, Stearns, S. and A. Hendry (eds.) Evolution Illuminated: Salmon and their Relatives. Oxford University Press.

10/04/05

Three topics were discussed: 1. Relevance of large scale patterns to local interactions; 2. Methods of malaria control – not sure how we got off on that one; 3. Definitions of fitness and, more generally, can fitness be defined independent of evolutionary outcomes.

• Large Scale Patterns. With regard to macroevolutionary patterns, your faithful correspondent is something of a sceptic. So much is swept under the rug in a log-log plot. Moreover the hypotheses profered to account for the "patterns" (such as they are) often range from the weak to the rediculous. Good examples of the abuses that ensue can be found in (Brown J. H. 1995. Macroecology. Univ. Chicago Press).

• Malaria. For discussion of malaria in the context of diploid genetics (this is an Ecol/MCB 182 supplement), go here (just scroll down). For discussion of malaria models and their implications for control strategies, go here. Nakul Chitnis, a graduate students in Applied Math, will be giving a talk on modelling malaria this coming Friday, 10/7 at Noon in Math 402.

• Fitness. Conventional population genetics equates fitness with rates of exponential increase of individual genes growing according to N_i(t+1) = w_i N_i(t). Note that in the diploid case,

  w_i = ∑w_ijp_ij(t)

  (the summation ism over j) and is therefore generally not constant over time.

  An alternative to the (often misplaced) emphasis on rates of exponential growth was provided by MacArthur (R. H. 1962. Some generalized theorems of natural selection. Proc. Nat. Acad. Sci. USA. 48: 1893-97.) who reformulated some of the basic results in population genetics, e.g., Fisher's theorem, in terms of carrying capacities. Although rarely commented upon – ecologists instead got hung up on "r-" vs. "K-" selection – the real import of MacArthur's calculations is that they make explicit contact with Darwinian evolution, which theory is quintessentially one of competition ("nature red in tooth and claw"). Both approaches assume all individuals of a given genotype to be the identical, which is manifestly not the case. Genetically identical individuals differ according to age, size, developmental history, ecology and where they happen to occur in time and space. Incorporating all such considerations (or even a representative few) into a workable definition of fitness is daunting. This is especially the case when one realizes that populations in nature are subject to an enormous number of spatial and temporal forcings, hence the reference to "time and space." It follows that defining fitness independent of evolutionary outcomes is difficult – possibly even impossible. The best one can do is to make simplifying assumptions, derive a formula and deduce predictions therefrom. Trouble is, when the predictions fail – and they always do if one looks closely enough – one doesn't know what to blame: the simplifying assumptions or the hypothesis in question that is lacking. The issue was famously (infamously?) explored by Peters (R. H. 1976. Tautology in evolution and ecology. Amer. Natur. 110: 1-12) back in the mid-seventies when optimality theory was much in vogue.

  PROGRAM MakeData       DIMENSION x(n)     DO i = 1, n, 2       x(i) = 1       x(i+1) = 0   ENDDO     END

  More generally, the problem can be viewed from the perspective of algorithmic complexity (Jackson, E. A. 1985. Perspectives of Nonlinear Dynamics. Cambridge Univ. Press. Vol II., pp. 514 ff; Li, M. and P. Vitanyi. 1997. An Introduction to Kolmogorov Complexity and Its Applications. Springer Verlag. Berlin), which, crops up in the theory of data mining. Basically the idea is this: given a data set, e.g., a time series, DNA base sequence, etc., how long a computer program is required to generate the data? Obviously, if the sequence is an alternating set of 1's and 0's, the program is quite simlple: In FORTRAN, one might write as shown in the accompanying text box.

  Here n is the length of the data set to be generated. More complex data sets, i.e., ones containing multiple patterns, require more complex programs. In the extreme case, the smallest program is the data itself, and one says that the data are maximally complex. One imagines that evolutionary paths manifest just this property – in which case, the articulation of predictive theory becomes quite impossible.

• Algorithmic Complxity (AC). What sorts of data manifest AC? One class is the output of chaotic dynamical systems. To see why, we recall that chaotic sets are densely populated by periodic oribits and that motion on such a set can be viewed as an elaborate choreography whereby evolving trajectories visit the neighborhoods of many such orbits (all of them, if you let time go to infinity), shadowing each for a while and then moving on to the next - sort of like fads in education and the social sciences (real science, too)! Obviously, the more orbits, the more patterns in the data and the greater its AC. In fact, the periodic orbits on a chaotic set are countably infinite – hence the fact that chaotic motion is maximally complex.

It's important to bear in mind that algorithmic complexity is mechanism-free: It makes no reference to the possibility that a simple rule might be generating the data. Take for example, the logistic map (difference equation) that can be written as

X_i+1 = rX_i(1-X_i).

With r = 4.0, there is a sense that the data so generated are indistinguishable for a Bernouli process. So if the program, in question, is concerned only with pattern, the data have maximum AC. If, on the other hand, the programmer knows à priori that he is looking at the output of this equation, only a few lines of code are required.

What this has to do with evolution is as follows: If we could write down the exact governing equations, we could correctly predict the history of life. But we can't – even to first order &ndash from which it follows from sensitivity to initial conditions, that our predictions will be wrong, very wrong, indeed. A good example of how you need to know everything is the work on Darwin's finches by the Grants (Grant, P. 1986. Ecology and Evolution of Darwin's Finches. Princeton University Press. Princeton, N. J.) who concluded, among other things, that Geospizid evolution cannot be understood absent knowledge of the geological history of the Galoapgos.

10/11/05

We continued last week's discussion as to whether or not fitness can be defined independent of evolutionary outcomes. It was suggested that consideration of "components of fitness" was a useful approach, and we agreed to think about that during the coming week.

• Spandrels. I believe we made mention some weeks back of the attack by Gould and Lewontin on what they characterize as the Panglossian excess of "pan-adaptaionism." Recently, I came upon an interesting review of that paper and some of the controversy that ensued, which clearly bears on the question of optimality vs. contraints in evolution.

  Actually, the discussion goes back to Darwin and his predecessors - see Chapter 5 of the Origin, in particular, the section on Compensation and Economy of Growth, in which he references work by Goethe and Geoffrey de Saint-Hilaire. The quote from the former is notable for the way it gets to the crux of the matter with a minimum of verbiage: "In order to spend on one side, nature is forced to economise on the other side."

• The Tangled Bank. The final, and arguably the most famous, paragraph of The Origin is best known for its concluding sentence:

  "There is grandeur in this view of life, with its several powers, having been originally breathed by the Creator into a few forms or into one; and that, whilst this planet has gone circling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved."

  Here, we have the principle of Descent with Modification (DWM), independently proposed by Wallace in 1855, and which both critics and defenders of Darwinism often equate with the "theory of evolution."

  Equally important are the sentences that precede it. In them, Darwin enumerates the mechanisms that lead to DWM. Surprisingly – to the reader couched only in the dogma of contemporary evolutionary theory – he lists three: natural selection plus two others. Darwin's eclecticism reminds us that a central problem in contemporary evolution is determining the extent to which selection, which is the sole creative force in the synthetic theory, can account for biological diversity and complexity.

10/18/05

Citing the arguments of Jenkin and Lord Kelvin, we discussed the proposition that as a logical and self-consistent theory, Darwinism was dead within a decade of the Origin's publication. The reaction of geologists and evolutionists, especially to Kelvin's seemingly unassailable criticism was considered.

• Jenkin pointed out that the laws of inhritance, as they were then understood, necessitated a 50% reduction in variation every generation; Kelvin, that the laws of physics, as they were then understood, placed upper limits on the age of the habitable earth and the sun that were far less than the hundreds of millions of years estimated by Darwin as being necessary for earthly life to evolve. Of course, both criticisms were incorrect: particulate inheritance preserves variation; radioctive decay heats the earth; nuclear fusion drives the sun. Still Darwin's response to the criticisms is instructive. While retainging his opinion that natural selection was evolution's prime mover, he cast about for devices that would speed the process, especially with regard to the creation of new variation. To this end, he invented the theory of pangenesis, whereby environmental influences perceived by the body could be transmitted to the germ cells where by some means or other they induced adaptive changes that were heritable. He also mused about saltation and the inheritance of acquired characters.

• Fast forward to the present and what is widely referred to as "evo-devo" Rather than a collection of traits, each coded for by its own genes, the developmental view of evolution is one of tool kits and switches: vastly different organisms use the same structural genes to make legs or eyes or whatever, the differences being in the way structural genes are turned on and off. One of the consequences is that evolution is made substantially easier: structures don't have to be conjured up from nothingness; new elaborations are more likely to "work," because they are modified versions of older structures that did, saltatory variations are less likely to have negative consequences to reproduction and survivial, etc. Neo-Darwinians might claim that there is nothing conceptually new in this - just another version of the "getting the monkey to type Shakespeare problem." But when the same genes crop up in taxa as distant as man and Drosophila, one is forced to conclude that there really is something new here.

• Back to making evolution easier. Let's suppose the evo-devo folks are substantially correct – but see Coyne for a more measured assessment. It follows that the view of evolution that preceded was wrong in the sense that omitting the role of regulatory genes makes it impossible to account for the principle features of evolution. But, of course, no one imagined that to be the case. The Modern Synthesis, we all believed, accounted quite nicely for what Hutchinson called the "evolutionary play." This leads us to enquire as to why we could have been so very wrong without realizing it. The answer, I submit, goes back to Peters. Lacking the ability to predict with anything remotely resembling precision, the "theory" of evolution was (and remains) a chronology conjoined with an enumeration: the history of life and a list of mechanisms that can influence its course. That being the case, we never know when we are wrong – until the next round of empirical discovery triggers the next round of re-assessment.