Saturday, April 26, 2014

Criticality and phase transitions in biology

My piece just published in New Scientist on phase transitions in biology has had one of the most difficult gestations I’ve ever encountered. No one’s fault, it is just that it’s a very tough job finding the right way to tell a story like this. For one thing, what I came to realise during the editorial process is that, if you talk about criticality, people outside condensed-matter physics are likely to imagine you’re talking about self-organized criticality, and that they don’t generally know that critical points have a long, long history going way back beyond this. Neither, it seems, is the connection between the notion of criticality, with its scale-free phenomena, and phase transitions well known. The real point of the ideas I discuss in this piece is not that there’s something wonderful about being poised at a critical point, on the edge of order and chaos etc., but that it can be useful for a biological system to situate itself near to some phase transition and to draw on the fluctuations and sensitivity to external conditions that this engenders. It doesn’t have to be a critical transition – I am coupling together here the current discussions of near-critical biology with work on first-order phase transitions in protein hydration, where again the value seems to be that one can draw on large fluctuations to attain a big response to a small stimulus. This latter material didn’t make the final cut in New Scientist, and I can see that it made for an even more complicated story. But I do think that there is important common ground between the two ideas. What’s more, no one previously has made the link to Eigen’s ideas about natural selection coming from a phase transition – a notion that he has set out in full in his immense, dense but fascinating recent book, which I reviewed here.

Anyway, this version is based on my original draft, but with some of the later material mixed in. Hopefully it will give some indication of the bigger picture.


It’s not the midges that were the problem, says Andrea Cavagna, but the kids. You’d think his efforts to record the movements of midge swarms in the public parks of Rome near sunset would be fraught with risks of being eaten alive by the little beasts – but these were a non-biting variety. Keeping away the children who gathered to watch what these folks were up to with their video cameras, generators and thickets of cabling was another matter. That, and the problem of finding a parking space in central Rome.

It’s not easy, he realised, for a physicist to turn field biologist.

The reason why Cavagna, based at Sapienza University in Rome, and his colleagues went midge-hunting sounds strange, perhaps even bizarre. The researchers wanted to know if midges behave like magnets. More specifically, if they act like magnets close to the point where heat flips them between a magnetic and non-magnetic state: a so-called critical phase transition.

Cavagna is one of a small and diverse group of scientists who have begun to suspect that critical phase transitions play vital roles in a wide variety of biological systems. Not only might they underpin the swarming of midges and the flocking of birds, but they might enable neurons in the brain to encode a picture of our environment, some protein molecules to fold up and bind their target molecules, and cell membranes to attract molecules that trigger cell-to-cell messaging. They might even explain how evolution itself works.

It’s good to be critical

Staying alive might seem to be a question of keeping calm and carrying on in the face of whatever comes along. But it’s often important to be able to respond and adapt to challenges rather than stoically riding them out: if you’re a small creature about to be eaten by a big one, you’d better get out of there. The trick is to keep your options open, maintaining easy access to a wide range of actions. It’s a delicate balance: you need stability, but also responsiveness.

In 2010 two physicists suggested how this might be possible. Thierry Mora (now at the École Normale Superieure in Paris) and Bill Bialek at Princeton University argued that many biological systems, from flocking birds to neural networks, might be “poised close to a critical point”. The idea drew on a well-established notion from statistical physics: the critical phase transition, where a system of many interacting components switches suddenly from one global state of organization to another, typically from an orderly to a disorderly state. The magnetic transition of iron, where it changes from having the magnetic orientation of its atoms random and disorderly to all lined up as the material is cooled, is the classic example. The switch happens abruptly at the so-called critical point – for iron, at a temperature of 1,043 Kelvin.

What have magnets got to do with biology? The point is that the critical transition of a magnet isn’t anything to do with magnetism per se. It is an outcome of the fact that each atom is interacting (here via magnetic forces) with its neighbours, and that they all have to come to some ‘collective decision’ about how to organize themselves. Because of that collective aspect, phase transitions happen all at once when a threshold value of some control parameter such as temperature is surpassed. They occur in all manner of physical systems, from superconductors and the Big Bang to polymer mixtures. So why not in biology?

The proposal of Mora and Bialek didn’t spring from nowhere. It echoes suggestions made in the 1990s that many natural systems, including some in biology (such as mass extinctions), display ‘self-organized criticality' (SOC), meaning that they undergo disruptions and fluctuations at all possible scales of size. The archetypal example of SOC was a pile of sand, which can have avalanches of all sizes as new grains are added at the top of the slope. This wide range of fluctuations scales is just what is found at an ordinary critical point – for example, a magnet at its critical point is a patchwork of domains of all different sizes with different magnetic orientations.

But, says Cavagna, it was never really clear that SOC had a deep connection to the older notion of critical phase transitions. The key feature of SOC is that it is indeed ‘self-organized’, which means that it will return to the critical state after a disturbance like an avalanche. So there’s no actual phase transition at all. “It’s just a point of great instability”, Cavagna says – and not one that is reached, like a true phase transition, by tuning a parameter like temperature. What’s more, says Bialek, “there was a huge amount of ideology about why criticality was a good thing in biology” – but no good argument for why. These two researchers and others are now trying to clarify what advantages criticality might convey on a wide range of biological systems, regardless of whether it is achieved by self-organization, natural selection or something else.

A critical magnet is poised on a knife-edge, where the smallest nudge can tip it into becoming wholly magnetic or non-magnetic. This knife-edge character of traditional (rather than self-organized) critical points means that it is all but impossible for a system to stay there. But the proposal of Mora and Bialek is that biological systems might benefit from operating close to critical points. This could provide access to a wide range of fluctuations involving different configurations of its components. The striking thing about near-criticality is that the rarity of specific, seemingly unlikely configurations is exactly compensated for by the fact that there are many more variants of such states than there are of common ones. “There’s a small number of very common configurations, a large number very rare configurations, and everything in between”, says Mora. “Being close to a critical point means that you're as likely to find yourself in any of these configurations.” As a result, he says, “being critical may confer the necessary flexibility to deal with complex and unpredictable environments.”

Another key feature of a critical system is that it is extremely responsive to disturbances in the environment, which can send rippling effects throughout the whole system. “At the critical point, everything is about to go crazy”, says physicist Jim Sethna of Cornell University. “So you get massively more sensitive behaviour.” That, Sethna says, can help a biological system to adapt very rapidly to change. The sensitivity stems from the long-ranged correlations in the behaviour of the system’s components that develop near criticality: a tweak here has an influence right over there, so that each component can ‘feel’ what all the others are doing.

Crucially, this flexibility and adaptiveness is achieved not by some incredibly complex and fragile set of interactions between the components, but taking advantage of the universal and robust characteristics of all systems made up of many interacting components. If a system evolves to be close to critical, says Sethna, it then has something like a set of general-purpose knobs that can allow it to adapt to environmental changes without having to reconfigure genomes.

Recent work by physicists Amos Maritan and Jayanth Banavar and their coworkers gives a clearer picture of why criticality in particular is useful. They have calculated how a system of agents that can gather information about their environment, and whose fitness depends on their ability to locate the source of environmental stimuli, evolves over time. They found that such a collection of evolving cognitive agents settles naturally into a critical state. “Being poised at criticality provides the system with optimal flexibility and evolutionary advantage to cope with and adapt to a highly variable and complex environment”, says Maritan.

In effect, this critical state allows the system to ‘sense’ what is going on around it: to encode a kind of ‘internal map’ of its environment and circumstances, rather like a river network encoding a map of the surrounding topography. “A key ingredient to the success of a living system is its ability to capture relevant information from the richly varying external world, synthesizing its most prominent features into manageable maps”, says Maritan. If this is indeed a feature of a near-critical state, the activity of neurons would be expected to operate in such a state just as Mora and Bialek proposed, because what our brains ‘show us’ will then be a good approximation to what is really ‘out there’.

There’s now mounting evidence that brains really are organized this way. One signature of criticality would be long-ranged correlations between the ‘spiking’ activity of neurons – something that Bialek and his coworkers have found in their models of neural networks. These correlations mean that the state of each neuron is to some degree encoded in the state of the rest of the network, providing a mechanism for error correction and recovery of lost information.

And it’s not just all theory. Dante Chialvo of the National Council for Scientific and Technological Studies in Buenos Aires, Argentina, and colleagues have shown that dynamics characteristic of a critical state in the activity of the human brain can account for some of the key features seen in MRI brain imaging, such as the coherent operation of many neurons clustered together in space. And Nir Friedman of the University of Illinois at Urbana-Champaign and his coworkers have found that avalanches in the firing of neurons show the same kind of size–probability relationship as those in self-organized critical sand-piles. It’s not hard to imagine that this apparently general operating principle of neural networks might bring some structure to the mass of data soon to emerge from the large-scale projects recently launched in the US and Europe to map out the connectivity of the human brain.

Superfluid starlings

Responsiveness has an obvious utility to a herd or flock of animals looking out for predators: if a few individuals spot one, the rest of them can gain that information almost at once. And they do – just think of schools of fish darting around in unison to avoid sharks.

It was this sort of flocking behaviour that partly stimulated Mora and Bialek’s proposal in the first place. Theoretical modelling of flocking over the past decade or so has shown that coordinated motion requires each animal simply to respond to its nearest neighbours’ movements by trying to align itself. This is similar to the way magnetic atoms get aligned, and in fact some flocking models are directly analogous to models of magnetism. Mora, Bialek, Cavagna and their collaborators have recently shown that the graceful, orderly motion of flocks, familiar from watching starlings at twilight, is most easily maintained if the flock is close to a critical point. Further from this point, a flock might stay coordinated but loses the ability to respond quickly and coherently to outside disturbances such as predators.

In other words, says Cavagna, flocking isn’t just about orderly motion. Too much of it and you end up regimented like a crystal, slow to respond to anything. The responsiveness comes instead from the correlations between individuals – how one affects another.

Fine in theory. But do real flocks work this way? In a happy confluence of ideas and observations, Cavagna and his coworkers in Rome began their studies of flocking in 2010 just as Mora and Bialek were presenting their ideas on biological criticality. The Italian team found that flocks of starlings have scale-free correlations in the velocity fluctuations of individual birds. In other words, if one bird in the flock changes course, others will tend to do so too almost instantaneously, no matter how far apart they are.

Cavagna and colleagues placed video cameras on top of the National Museum of Rome in the city centre, which overlooks a major roosting site for starlings in winter. They filmed the birds during their flocking displays at dusk, and then used computer-vision methods to turn the footage into records of the three-dimensional movements of individual birds in the flock, which typically contained between a hundred and several thousand birds. They analysed this data to figure out how each bird deviated from the average velocity of the entire flock, and to measure the correlations: how closely these deviations for any pair of birds shadow each other as the distance between the pair increases.

“We found that correlation was very strong”, Cavagna says. In other words, the birds seem to be tuned into one another’s movements even over scales beyond which they can see each other. The influence of one bird is transmitted to others far away through neighbour-to-neighbour interactions, in just the same way as the magnetic poles of atoms of iron in a magnet can ‘speak’ indirectly over long distances close to the critical point.

What’s more, these observations showed that realignment of the birds’ orientation as the flock changes direction spreads much faster than the standard theories of collective movement permit. This behaviour can be explained by adding an extra ingredient to the theory: a ‘symmetry rule’ which reflects the fact that all directions of flight are equivalent. With this included, it turns out that the movement of the flock becomes mathematically equivalent to that of a superfluid such as liquid helium, which can flow essentially without losing any energy through viscous drag. In other words, a flock of birds can be considered a kind of living superfluid.

Midges don’t exhibit the orderly swarming motions of birds and fish. Might they, nevertheless, display the long-ranged correlations expected on the disordered side of a critical phase transition? “Some biologists insisted there is no collective behaviour in midges”, Cavagna says, and he expected his observations to confirm that view. But after painstakingly filming the midges swarming around park landmarks, reflected in the setting sun, he and his coworkers couldn’t avoid the conclusion that there were very strong correlations here too.

“It’s physically exhausting work”, Cavagna says: lugging all the equipment into a park, filming for several hours, then immediately going back to the lab well after dusk to download the data. “Still, at least it was summer, and the Roman parks are lovely.” Filming birds is harder, he says, since they only flock in the cold winter.

But why would evolution tune midges to behave that way, given that predation isn’t an issue for them? Cavagna thinks that this might be looking at the question the wrong way. Perhaps they can’t help being near-critical. The researchers found that the reach of the correlations was always about the same size as the swarm: the bigger the swarm, the longer the correlations. So maybe the swarm size isn't an adaptation, but is a side-effect of some other factor that determines how the midges interact. This factor - the range of neighbouring midge interactions, say - sets the correlation distance for midge motions, so that if the swarm gets bigger than that size, it will automatically shed midges.

Quick drying

The idea that biology makes use of phase transitions and their associated correlations and fluctuations could go far deeper than these large-scale networks and communities, and might be applied even at the level of individual cells and molecules. Protein molecules, for example, often carry out their functions as enzymes by switching from one shape to another. That needs to happen easily when the right signal is given, for example when another molecule binds to the protein to activate it. These conformational changes are, like phase transitions, cooperative, meaning that they involve interactions between all the component parts. Tweak this bit of a protein, and the whole thing tips into a new shape.

Cooperative transitions have also long been thought to govern the way protein chains fold up into their functional shapes in the first place. But recently David Chandler at the University of Berkeley at California and his coworkers have argued that both this process and the way several protein molecules stick together into many-component assemblies could be controlled by a transition that occurs not in the protein itself but in the water that surrounds it. They believe there may be an abrupt ‘drying transition’ in which all the water suddenly exits from the space between two water-repelling parts of proteins. Chandler argues that these drying transitions, which have been seen in computer simulations of some proteins, draw on the strong fluctuations that exist in the water, whereby the water molecules organize themselves into ever-changing regions of high or low density – not unlike a midge swarm, in fact. These fluctuations make it easier for the gap between the protein segments to tip over from a ‘wet’ to a ‘dry’ state, just as they make it easier for a critical magnet to tip over into a magnetic or non-magnetic state. Not all, or even most, proteins seem to fold or aggregate via these drying transitions. But Chandler and colleagues argue that most of them may be fine-tuned by evolution to be close to such a transition, some lying on one side of that boundary and some on the other.

Drying transitions have also been found in computer simulations of the docking of small molecules into the ‘binding cavities’ of the enzymes they activate. Some proteins in thermophilic organisms, which thrive in hot environments, have cavities lined with water-repelling chemical groups that seem poised right on the brink of expelling the water and becoming dry at the organism’s normal working temperature. The docking of the ‘plug’ into its ‘socket’ would be made easier by this ease of emptying. Meanwhile, some protein channels that sit in cell walls and regulate the flow of other molecules or ions in and out are also poised to undergo drying transitions within their conduit pores, so that they can be easily switched from an ‘open’ state (where the water-filled pore lets dissolved substances pass) to a ‘closed’ state (where the pore is dry and denies passage).

Another benefit of being close to a phase transition has been suggested by Sethna and his colleagues. Some biological membranes are patchworks in which different types of lipid molecule are segregated into liquid-like ‘rafts’, phase-separated like immiscible droplets of oil and water. Because these patches have a wide range of fluctuating sizes, rather like the domains of a near-critical magnet, Sethna’s team argued that they are close to a critical phase transition at which the molecules become fully miscible.

They say that the value here is not in the phase transition itself, but in the domain size fluctuations that accompany it. Such fluctuations in immiscible fluids were shown in the 1980s to give rise to a force analogous to the so-called Casimir force that pulls together two closely spaced metal plates in a vacuum. The normal Casimir force is caused by electromagnetic fluctuations in the vacuum, themselves a consequence of quantum physics: because the size of these fluctuations is restricted between the plates, this produces a pressure that draws them together. Likewise, constraints on the ‘near-critical’ fluctuations of lipid patches between protein molecules embedded in the membrane give rise to a ‘critical Casimir’ attraction that might help molecules to bind together and trigger chemical reactions involved in cell signalling. In effect, says Sethna, it means that proteins at the membrane surface can talk to each other via the lipid rafts. “Here again criticality allows the system to access structures over a wide range of scales”, says Mora.

The physics of evolution

Phase transitions and criticality might turn out to be important in the operation of gene networks, which currently seem absurdly baroque and yet somehow generate stable and robust organisms. Bialek and coworkers recently reported an indication of criticality in the gene regulatory network that determines the spatial patterning of the fruit fly embryo – the so-called gap gene network. They found long-ranged correlations in the fluctuations of gene expression levels at well-separated parts of the embryo. It’s possible that these critical-like fluctuations might help to improve the signal-to-noise ratio of the information transmission in the regulatory network.

Mora and Bialek have suggested that phase transitions in the ‘information space’ that relates a protein’s structure to its shape and function through the collective interactions of its chemical building blocks might account for the appearance of distinct ‘families’ of protein structures. This would imply that the evolution of protein sequences (and hence gene sequences) is significantly constrained by the limited number of ‘stable states’ in sequence space – in other words, that nature’s profusion is regulated by an order even deeper than natural selection.

In fact, not only does evolution seem likely to make use of phase transitions – it might actually be one. Chemist Manfred Eigen, who won the 1967 Nobel Prize for his work on fast chemical reactions, has argued that natural selection appears in a system of self-replicating, information-bearing entities as an abrupt phase transition at certain threshold values of the rates of replication and mutation. In other words, it is not just ‘something that happens’ in reproducing systems, but is a physical law that arises from the way information itself is organized. In Eigen’s theory, neutral selection – in which mutations get fixed in a population even though they have no adaptive benefit – injects fluctuations analogous to those at a critical point. These are essential to prevent natural selection from getting ‘stuck’ in minor valleys of the evolutionary landscape – or as a physicist might say, to prevent the system settling into a metastable phase, which is provisionally stable but not the optimal arrangement of the components. That would fit with the recent suggestion of evolutionary biologist John Tyler Bonner at Princeton University that the random fluctuations of neutral evolution could account for the immense variety of forms found in organisms such as diatoms.

Criticality and the critics

“I knew from the beginning that I wanted to do something in between physics and biology”, says Bialek. The question is, he says, “can you talk about these things that biologists usually study in the way that physicists do?” He suspected “that there’s some collection of phenomenon that people didn’t realise were related to each other, or some part of the biological world that nobody has looked at from a physicists’ point of view” – in other words, the big question was “whether aspects of particular [biological] models can be derived from some more general principle.” If Bialek and Mora are right, criticality could emerge as one such general principle.

But these ideas have yet to be embraced by most biologists, whose agenda is often now dominated by fine details rather than a search for over-arching principles. Getting these ideas a hearing in biology is likely to be a struggle. “There’s a big difference in culture”, says Sethna. “Biologists tend to be skeptical of anything that involves a lot of math.” In an effort to bridge this ‘two cultures’ divide, in 2010 Bialek spearheaded an interdisciplinary centre called the Initiative for the Theoretical Sciences at the City University of New York, where he is now director. Here physicists can discuss these ideas with neuroscientists, ecologists and other biologists – Cavagna was recruited as a visiting professor last year, and has been collaborating with Bialek and Mora to refine the understanding of critical flocking. But it will take time and patience, both to figure out how widely phase transitions and criticality really are used in biology, and to persuade life scientists that, as Sethna puts it, cells, and perhaps proteins, animals and entire ecosystems, “do a lot of interesting physics.”


JimmyGiro said...

When introduced to liquid crystals, it was pointed out that most developing cells, and even DNA itself, showed liquid crystalline behaviour. And as you say, life has to be both stable, to live, and yet unstable, to evolve. Thus the ambivalence of 'life' requires an ambivalent phase, such as liquid crystals.

And as for starlings, why do you suppose they swarmed as a consequence of predation? By late autumn, most of the hawks and falcons have gone down to Africa. My belief is that after the flush of summer, the overpopulation of local starlings are apt to weed out the weaklings by performing the ultimate river-dance knock-out competition, whereby each starling pair-bonds with other good performers. Indeed, if you hang around long enough, you may witness some of the birds drop dead with exhaustion.

But then socialist eugenics is strictly dancing with Stalins.

Burt Voorhees said...

I've published several papers on the concept of "virtual stability" which relates to this topic. Easy reference is Voorhees, B. (2009) Virtual Stability: Constructing a Simulation Model. Complexity 15(2) 31 – 44.