SFI Bulletin Articles

Many Are Claimed, Few Are Validated:

Report on the Spring 1998 Science Symposium ``Scientific Models: Claiming and Validating''

by Cosma Rohilla Shalizi

Differs slightly from the version published as ``Scientific Models: Claiming and Validating,'' Santa Fe Institute Bulletin, vol. 13, no. 2 (Summer 1998), pp. 8--12
It is easier to make a theory of everything than it is to make a theory of something.
---Aharon Katchalsky, as quoted by George Oster
This spring's science board symposium consisted of three talks and general discussions, and while they weren't chosen to exemplify Katchalsky's aphorism, they might as well have been. I'll describe each in turn, and close with a few thoughts on their relation to the ostensible theme of the symposium, the claiming and validation of scientific models.

I. George Oster, ``The World's Smallest Rotary Motor''

The general topic which Oster's work is meant to address is how cells turn chemical energy into mechanical force, and in particular how ATP synthase pulls off the trick. To begin with, what does ATP synthase do?

Principally, of course, it synthesizes ATP, adenosine triphosphate, from ADP, adenosine diphosphate, and phosphate; and ATP is then used as the energy source for almost all cellular reactions. (This is why ATP synthase is sometimes billed as ``the most important protein in the universe,'' a description Oster modestly admits ``may not be entirely hype.'') The energy for putting ATP together from its components is, in the case of ATPase, provided by proton gradients --- as the protons flow down the gradient, shedding their potential energy, they pass through the synthase, which, in a manner not totally transparent, uses the potential energy to build up ATP. In bacteria (but not eucaryotes), ATP synthase also gets run backwards, fed with ATP to produce proton gradients; a slightly modified version, vesicular ATP synthase, is used to regulate the pH of cellular compartments by pumping protons in or out. All in all, it's a tolerably important and very ancient protein, so ``if any molecule has been optimized by evolution, this is it.''

The ATPase has two components, rejoicing the evocative names of F0 and F1. F0 deals with the proton gradients, F1 with the synthesis or hydrolysis of ATP; each can act as a rotary motor, the pair of them being connected by a shaft, giving the whole affair the look of a lollipop on an ornamental stand.

Physicists like Oster (and myself) are conditioned to respond to ``engine'' with ``heat'', since we're drilled from the start to tackle motors as problems in thermodynamics. When one measures the efficiency of the synthase, however --- which we can do, thanks to a recent series of remarkable experiments in Japan, where the head (F1) part of the protein was detached, fed with ATP, connected to a (comparatively) large and heavy fluorescent filament, and actually filmed under a microscope in action, and the work done calculated from the known properties of the filament --- it turns out to be over ninety percent, with no temperature gradients worth mentioning. ATP synthase can't be a heat engine, and Carnot is of no use in understanding what it does.

What is of very great use is the way the structural biologists have gone over ATP synthase atom by atom in its different conformations. Very detailed simulations of molecular dynamics (using ``about a mole of Crays'') show that, while there's a lot of jitter, the important conformational changes of proteins can be decomposed into shearing motions and hinged bending motions. Using this, and the known detailed structure of ATPase, Oster was able to write down a very simple model --- a ``tinker-toy model'' --- for the mechanical and elastic properties of the synthase, incorporating only those hinges and shears. The tinker-toy is only for illustration; the real model is a set of mechanical equations, fairly straight-forwardly deduced from Newton's laws, in which viscous drag balances a stochastic driving force --- for F1, which, recall, is the end of the molecule which deals with ATP, the driving force comes out of chemical kinetics; for F0, from the Brownian motion of the protons. The arrangement of two key charged amino-acid groups in F0 turns out to rectify that Brownian motion, effectively only allowing them to flow one way, and in the process imparting a torque to the protein. In F1, the binding of a sub-unit to the ATP molecule creates an elastic strain, which is released when ATP dissociates, and it is the release of this strain which provides the torque.

The solutions of both sets of equations are in extremely good agreement with experiment, qualitatively (e.g., correctly showing that F1 makes a revolution in three steps, consuming one molecule of ATP at each step) and quantitatively (e.g., matching the load-velocity curves). Moreover, when run in reverse --- when provided with outside torque, and used to either synthesize ATP or build up a proton gradient --- the equations again are in quantitative agreement with experiment. (In the normal use of the protein, to synthesize ATP, F0 runs as a motor, and supplies torque to F1.) This is, as Bob May said in the discussion afterwords, ``mathematical biology at its best.''

Two related evolutionary issues occupied that discussion. The first was the optimality of ATP synthase. There is extraordinarily little variance in the protein across species --- you can take parts from E. coli, and parts from cows, and put them together to get a working protein, and even those organisms which use sodium ion gradients instead of proton gradients don't change the protein much. This, together with its very high mechanical effiency, suggests that ATP synthase is about as good as proteins working on its general lines can get; but we can't rule out something differing, not by one or three mutations, but fifty or a hundred, working as well. The second issue was whether those general lines are the only way of doing ATPase's job, whether the little grey men from Zeta Reticuli also have it. Partly this depends on how many different ways there are to do exactly the same job as ATP synthase (nobody knows; but Oster said there are known to be workable alternatives to the designs of some other molecular motors, like the one which drives bacterial flagella), and how many ways there are to change that job by tweaking other aspects of biochemistry, e.g. by using something other than ATP (nobody knows; and, even more remarkably, nobody was willing to speculate).

II. Per Bak: ``Punctuated Equilibria in Physics, Biology and Economics''

``Punctuated equilibrium'' is, as nowadays every school-child must know, a phrase coined by the paleontologists Steven Jay Gould and Niles Eldredge to describe a pattern they saw in the fossil record, of species remaining pretty much unchanged for long periods of time, and then going through a (geologically) brief spurt of change. Bak believes he can see this pattern in many places other than the fossil record, and even give it a mathematical characterization. The signatures of punctuated equilibrium, he says, are, first, a power-law distribution of event sizes (suitably defined), where there is no characteristic size for events, but the number of events over a certain size is inversely proportional to some power of that size; and second, ``1/f noise,'' where, again, events are distributed over all time-scales, but the power or size of events is inversely proportional to some power of their frequency; really big changes are rare, but not exponentially so. Finally, he claimed that punctuated equilibrium involves the assertion that ``catastrophic'' events are internally generated, as opposed to being the effects of exogenous causes, like large rocks falling from the sky.

Power laws are well-established for some phenomena, like the strengths o f earthquakes (the Gutenberg-Richter law and the Richter scale); somewhat more shakier for others, like the extinction rates and lifespans of fossil genera (or perhaps families or species; Bak wasn't sure what the paleonotologists had measured); and even a bit controversial in cases like the prices of stocks and other securities. (If it is true that changes in securities prices follow a power law, and the power law shows that the changes are internally generated, this undercuts one of the important traditional justifications for securities markets, namely that they effectively pool and evaluate all the available external information about the true worth of commodities and companies...). Acting on the assumption that punctuated equilibria are common, it is natural --- at least for a physicist --- to ask about a ``general mechanism'' which will produce them, one for which ``the details don't matter.''

Bak suggests that this general mechanism is to be found in statistical physics's notion of the critical point, at the boundary between two radically different states (for instance, between liquid water and steam, where the critical point is 100 degrees centigrade at sea level, and noticeably lower in Santa Fe). At critical points, the response to perturbations can (Bak says) be extremely non-linear, and the power-law distributions and 1/f noise depend on this; in fact, power-laws are generically found in physical systems close to the critical point. What Bak has proposed are models which are not just critical, but ``self-organized critical'' --- instead of needing to have parameters (like temperature and pressure) tuned to get them to the critical point, their own dynamics will take them there, keeping them in (as the TV news says) stable but critical condition.

``You cannot make a realistic model of California to explain earthquakes '' --- so simple models will have to do. One of the very simplest models which exhibits self-organized criticality is the so-called ``sandpile'' model, where we imagine ourself to be dribbling sand from above onto a small pile. Friction between the grains can balance gravity only up to a certain slope, so we imagine that, when enough sand has built up at one location to exceed that slope, it shifts to some down-hill locations. If these are already near the critical slope, sand will flow from them as well, and so on, generating an avalanche; all of this can be formalized with two or three rules and just integer values for the height. This displays all the power laws (avalanche size, time between avalanches at a site, etc.) one could want, the 1/f power spectrum, and so forth.

Now, what makes us think that this toy model can teach us anything about the real-world systems with similar statistical features? Bak answers: ``Our intuition from physics.'' There, powerful mathematical methods have established that many properties (including the power-laws) are the same for all critical systems in the same ``universality class,'' regardless of their detailed physical construction. This being so, we use the simplest model in the right universality class, since trickier, more realistic models give no more ``insight into the type of beast we are dealing with.''

So: what does Bak say falls within the universality class of this ``theorist's sand''? Well, actual granular materials (at least sometimes --- depending on physical details, like the ratio of the length of the grains to their width); earthquakes, as described by a simple mechanical model with ``stick-slip'' motion; evolution, where avalanches get mapped onto extinctions, and the different sites of the sandpile model to different species; and the actions of the many separate traders on the stock-market.

The discussion following Bak's presentation was wide-ranging and, to be diplomatic, heated. The two most important issues were those of exogeneity and model construction. There is a strong consensus among biologists that at least the five great mass extinctions were caused by exogenous events --- most notoriously, that at the end of the Cretaceous by a meteorite impact. Bak expressed a certain skepticism about this consensus and its representatives: ``How does he know? He was not around!'' A second point which was urged was that there are really very many ways, many statistical mechanisms, which generate power-law distributions, which would undermine the arguments about universality classes and the like (the fact that the chest-widths of nineteenth century Scottish soldiers and the values of g my undergraduate students used to come up are both Gaussian doesn't mean the same mechanism was at work in both cases).

Neither issue came to anything like resolution.

III. John Padgett, ``Banking Markets in Renaissance Florence''

That is the ``Florentine history'' title; the ``Santa Fe Institute'' title would be ``The Co-evolution of States and Markets''; and his heart is with the Florentine title. That is, he doesn't look at Florence as a particular instance or test case of a general model of how states and markets co-evolve; rather he wants to understand that co-evolution ``primarily to understand Florence.'' The research began with the goal of understanding political parties in Florence, and he was ``driven by the state itself'' to look at the influence of markets on that politics, and of changes in the state on markets, especially in banking.

Now, there is a long and very well-established body of work on the relation between markets and state structures, namely political economy, which Padgett accepts. However, being largely about the effects of state policies on markets, it's inapplicable to the present case. Not only was the Florentine state made up of exactly the same people as its business firms, serving their few months in the government of the republic, meaning that there was no distance, really, between those who made policy and those affected by it, but political economy is silent on the ``constitution of social agents,'' the way agents (like firms or political divisions) get built up, which is precisely what Padgett wants to know about.

When one examines the data from the records of the Florentine guilds, principally that of the bankers, says Padgett, one can see four distinct kinds of firms --- all of them partnerships --- succeeding each other in turn, and rather abruptly at that. At the beginning of the fourteenth century partnerships were made within large, patrilineal families, grew to great sizes and competed with each other viciously (e.g., conspiring on the city council to have business rivals exiled and their property expropriated). These were followed by much smaller partnerships arranged on guild and neighborhood lines, between masters and apprentices, pooling together resources for large projects and all in all more ``solidaristic''; then ``multi-divisional firms,'' where one businessman would join one partnership in (say) wool, another partnership in silk, a third in banking proper, and so on; and, finally, rentier firms, with a separation between owners and managers. In each case the different organizational form was ``socially embedded,'' each ``mobilized'' a different sort of social raw material --- first families, then guilds and neighborhoods, then social classes and marriage alliances within them, finally patron-client networks and political factions.

What Padgett set out to explain is why, since Florence at all these time s had families, guilds, social classes and factions, now one and then another of these provided the raw material for building economic agents. The answer, ``driven very much by what I see in my data,'' is state formation. The lines along which firms were formed were also the lines of division of Florentine politics. The advantage of forming partnerships with political allies are that, first, you can have more trust in your partner, and spend more time thinking about how to make money, rather than about whether he's going to have you exiled or poisoned, and, second, there are institutionalized recruitment channels --- you can form a partnership with your nephew or the person on the next street over or your brother-in-law, with little effort at searching out a suitable candidate.

Is this so? That is, do the changes in organizational form track the changes in politics? It seems that they do; at the beginning of the period, Florentine politics was dominated by the rivalry of the great families, and the shift from family to guild firms took place at the same time as these elites seized control of the guilds. The shift to ``multi-divisional firms'' coincided, not just with Florence becoming the mediator for the whole European economy, but with ``classic Marxian class warfare'' as well. The elites were exiled and expropriated, a worker's republic was set up, and when the elites returned there were massacres in the streets, and, unsurprisingly, the inter-class partnerships of the guild period were gone. So it looks rather like changes in market structure grew out of changes in organizational form, which grew from changes in politics, and especially in changes in how the state was constituted.

Now, these changes in the state were, some of them, triggered by external crises, but in what Padgett says is a ``predictable'' way. When the crises arrive, and previously existing divisions (say, between families) are exacerbated, then political actors mobilize some other network as an instrument in their struggle, as an extra source of power. In time, these other networks move from being tools to being identities, the things on which political divisions are based; and the wheel is ready to turn again.

IV. Reflections

If the goal of the symposium was to show the diversity of the things which get lumped together as ``models,'' it succeeded, almost to excess. Both Oster and Bak are, for instance, physical scientists trying to colonize biology, but in completely different ways. Bak is a study in the conquistador spirit of really hard-core mathematical physics --- ``details don't matter,'' just a single, unifying general mechanism; resistance by biologists is futile, and they will be assimilated to the Journal of Statistical Physics in due time. Oster, on the other hand, goes native, immersing himself in biological detail --- not just the structural studies which say where every single atom is, but knowledge of the function of a single enzyme, and exceedingly detailed genetic work which enabled him to identify key amino acids in a particular sub-unit. Padgett's work is something else again, even more detail-dependent but without the formal, mathematical structure of the physicists.

Some of the speakers professed to be mere slaves of their data, others confessed to prolonged trial and error; nobody had anything original to say about how models get built, or how they should be. Probably it's unreasonable to expect otherwise --- good musicians are rarely good musicologists --- but one was left with the feeling that the ``claiming'' side was rather neglected.

Validation for models like Oster's is, in principle, straightforward: if it fits the known structure and kinetics of the enzyme, and agrees quantitatively with experiment, then the model does all it can be asked to do. Things are trickier for Padgett, since most of the data which the model is supposed to fit were used in putting it together, making agreement between the two less than compelling evidence in the model's favor. But this is a common problem for historical hypotheses, and the solution is new data, e.g., for guilds other than banking, or even cities other than Florence; within this comparatively thick description of one state and one set of markets there is a thin theory of states and markets in general hinting politely to be let out. (The model would probably need to have some formal backbone put into it first, though, rather than leaving it in its present spineless verbal form.)

It is not at all clear what validation would even begin to look like for as all-encompassing and ambitious a notion as Bak's self-organized criticality. If it was the only way to get certain statistical properties, and earthquakes, evolution, etc., showed those properties, then that would be strong evidence that an SOC-like mechanism was at work; but it seems that there are other ways of generating such distributions, which involve neither self-organized criticality, nor, as Bak put it, ``conspiracies'' (for instance, the Newman-Sneppen model of extinction, where all extinctions are caused by exogenous stresses). The only hope for figuring out which mechanism is active in particular cases would seem to lie in studying the details of that particular system, squashing the hope of finding general, details-independent mechanisms. As a theoretical physicist with the regulation degree of hubris, I think that's a crying shame but what can you do?