Saturday, February 18, 2006

Blind Watchmaker or Swiss Designer? (Part II)

Although it seems easy to identify robustness in biological systems, it is usually difficult to give a precise answer to the obvious question: robust compared to what? Ideally one would like to study many different, independently evolved creatures. However, these are rarely available. For example, the number of different, naturally occurring genetic codes is small, and they are not that different from the inaccurately named "universal" code. Instead, we usually have to make do with simulated creatures. This is the approach Andreas Wagner took to study the evolution of robustness in circadian oscillators.

He began by considering one of the simplest models for a circadian oscillator, known as a Goodwin oscillator. In a Goodwin oscillator a gene expresses a mRNA molecule R that is translated into protein P. This protein is then modified (e.g., dimerized or phosphorylated) to generate an inhibitor P' of its own expression.

This model was chosen because it can "exhibit stable circadian oscillations of an appropriate frequency", and because it can correctly predict "the response of circadian rhythms to stimuli such as temperature pulses, light pulses, and an inhibitor of protein synthesis".

Now consider the case of two coupled Goodwin oscillators forming the following two-gene six-product circuit:

Wagner points out that this circuit can take one of 378 different topologies where the six regulatory interactions represented by dashed lines can be activating, inhibitory or non-existent (the interactions represented by the solid arrows are always positive). He then asks two questions:
  1. Do different topologies result in different degrees of robustness?
  2. Can an evolutionary search in a space of possible circuit topologies reach highly robust topologies when starting from circuits with low robustness?
To address the first question, Wagner took each of the 378 circuit topologies and generated 5,000 random parameter combinations "from a parameter space within which circadian oscillations are known to occur for the basic Goodwin oscillator". He then used these parameter combinations to define a measure of robustness:
"For each of these parameter combinations, I examined whether the circuit adopts limit cycle oscillations with a period of ≈24 h [...] The fraction of randomly chosen parameters that yield circadian oscillations is an estimate of the fractional volume (P) of parameter space that admits such oscillations [...] P can serve as a proxy for a circuit's robustness to perturbations: changing parameters at random in a topology with high P is more likely to yield a parameter combination leading to circadian oscillations than in a topology with low P."
Of the 378 topologies, 176 show no random parameter combinations capable of generating circadian oscillations (P less than 1/5,000). Among the 201 topologies producing oscillations, robustness (P) varies by nearly 2 orders of magnitude among circuit topologies. For most of these topologies, only one or a few of the 5,000 randomly chosen parameter combinations produce circadian oscillations (low P). But for a small fraction of topologies, over 5% of parameter combinations yield such oscillations (high P). Here are nine of the topologies with the highest associated robustness:

These circuits show two important properties which give us some clues as to the mechanistic causes of robustness. First, most topologies contain a mixture of transcriptional (fast) and posttranscriptional (slow) regulation. Second, inhibitory interactions are more common than activating ones. "This is not surprising, if one considers that a closed positive regulatory feedback loop may cause the increase of a gene product without bounds, thus preventing stable oscillations".

The most robust circuits are ≈1 order of magnitude more robust (i.e., have a higher P) than a circuit composed of two decoupled Goodwin oscillators. This result suggests "that the increased complexity of interlocking oscillators involving two (or more) oscillating gene products" found in nature "may not be an accident of natural history: it may indeed provide greater robustness to mutation than single oscillators." However, when the first coupled oscillator evolved it was probably not very robust. So, to repeat the second question posed above, is it likely that a robust oscillator evolved by "numerous, successive, slight modifications" (Darwin 1859) to circuit topology? Wagner introduces an original way to attack this problem:
"This question is best posed by considering the following graph or network representation of oscillator topologies. Consider a graph where each node corresponds to an oscillator topology that is capable of displaying circadian oscillations. Connect two nodes (topologies) by an edge if the two topologies differ by only one regulatory interaction [see a, below]. Such neighboring topologies can arise from each other by genetic change that affects only one regulatory interaction. The question whether robust oscillator topologies can be found through a series of such changes, i.e., through gradual evolution, is a question about the structure of this graph. There is a spectrum of possibilities with two extremes. First, the nodes (oscillator topologies) of this graph may be disconnected. That is, a topology capable of circadian oscillations has no neighboring topologies also capable of producing such oscillations. This would mean that robust topologies cannot be reached from less robust topologies, because functional oscillators are isolated islands in this graph. At the opposite extreme, this graph might consist of one densely connected component, where any two topologies are connected by a path of edges. In this case, stepwise evolutionary alteration of a circuit topology could start from any one topology and reach any other topology via intermediate topologies that admit circadian oscillations.
Part b of the figure below shows this metagraph for the 201 topologies capable of producing circadian oscillations (P at least 1/5,000).

Two properties of this metagraph stand out: it consists of a single, highly connected component, and similar topologies also tend to have similar robustness (P). Thus:
"[G]radual evolutionary changes in circuit topology can generate any circuit topology from any other topology within such a component, without transitions through circuits that do not allow circadian oscillations."
The high evolutionary accessibility of robust topologies is not trivial: metagraphs composed of random samples of 201 topologies (i.e., picked independently of their robustness) tend to show, on average, over 20 disconnected components.

In conclusion, Wagner tested whether evolution could, in principle, discover a robust coupled oscillator circuit topologies and found that the answer is yes. This is an excellent example of why the standard position taken by intelligent design creationists when faced with the appearance of design is scientifically vacuous. No doubt Dembski and Behe will come up with some post hoc criticisms of Wagner's study. What I'd like to see are similarly rigorous attempts to test their claims, instead of the usual hand-waving about mousetraps and Mount Rushmore.

Can these results be generalized to other systems? It's impossible to tell at this stage. A recurring problem in the study of robustness is that it is difficult to judge how representative any biological system really is. For example, when Meir and colleagues (Current Biology 12: 778–786, 2002) discovered that the neurogenic network of the fruit fly Drosophila melanogaster is robust against large variations in the initial concentrations of gene products and the rate constants of molecular interactions they wondered:
"Perhaps this is a generic feature of genetic organization, but perhaps it reflects a coevolution between evolved networks, biologists and theorists: modular, robust networks are the easiest to get at experimentally. Thus, they are the best understood and are the best fodder for models."
There is only one way to find out... To keep working!