Master M2 Sciences de la Matiere – ENS de Lyon – 2016-2017

Phase Transitions and Critical Phenomena

Phase Transitions and Critical Phenomena

in Biophysics

Alexandre Torzynski

January 13, 2017

Abstract

Because many behaviors of living systems emerge from the interactions between numer-ous components, physicists have long hoped to describe such behaviors using statisticalphysics. A trending idea is the emergence of self-organized criticality in such systems,thought to improve the biological function. Through different scales, we review severalexamples of biological phenomena that exhibit a phase transition or a critical behavior,and illustrate the relevance of such methods and analogies.

A B

Figure 1: Eukaryotic cells. A: Sketch of an eukaryotic cell. Blue: nucleus. Gray: cytoplasm.Black: membrane. Pink: membrane-bound organelles. Orange: membrane-less organelles.B: Fluorescent microscopy image of eukaryotic cells cytoskeletons, from Wikipedia. Blue:nucleus. Green: microtubules. Red: actin filaments.

1 Introduction

Biophysics is an interdisciplinary science, that applies theories and methods of physicsto biological systems and questions. Its field of study ranges from molecule assembly toentire ecosystems. Whatever the scale, one key feature of the biological system is thatit is highly dynamic, involving numerous molecules, cells, animals or species. Statisticalphysics are therefore the tool of choice to study those systems. A major difference withphysical systems, however, is that biological systems are inherently far from equilibrium:from ATP hydrolysis to animal behavior, energy is constantly being injected in the system.

Consistent with the strong link between biological populations and statistical mechan-ics, some phenomena can be explained or modeled by phase transitions or separations.Examples include cell membrane fluidity, microtubule polymerization and organelles for-mation. In a less ‘thermodynamical’ vision, phase transitions theory, ideas and formalismcan be used to map biological data to physical models: for example, a neural network canbe described using an Ising model. This physical reasoning allows to realize the impor-tance of critical phenomena in several biological systems, such as flocks of birds, swarmsof midges or evolution itself. At each scale, self-organized criticality leads to improvedbiological functions.

2 Phase transitions in the cell

As explained by Brangwynne et al., “an essential aspect of biological function is the com-partmentalization of biomolecules” [6]. The cell is probably the most important exampleof compartment (fig. 1A): the intracellular medium, called cytoplasm, is separated fromthe extracellular environment by a membrane. But it is not the smallest one: in multi-cellular organisms, the cell’s genetic material is inside a nucleus, and therefore separatedfrom the cytoplasm. These cells are called eukaryotic, and will be our only concern inwhat follows. Other structures than the nucleus can be found, called organelles. Many ofthem, like the nucleus, are delimited by a membrane; but in the last decades membrane-less organelles were discovered, and are still an active field of research.

Like any good biological system, the cell is a highly dynamic medium: the cytoplasm isan aqueous solution of hundreds of different molecules and polymers, in constant physicaland chemical interaction with each other. The principle of biological metabolism is thatthe cell is out of equilibrium, and constantly exchanging molecules with the external en-vironment, while synthesizing or degrading polymers inside the cytoplasm. Putting asidethe complexity it brings, this dynamic aspect allows for many interesting phenomena,including phase transitions and phase separations. In the following subsections, we will

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Figure 2: Lipid molecules and bilayers, adapted from Wikipedia. A: Saturated lipid molecule.Red: hydrophilic head. Black: hydrophobic tails. B: Bilayer configuration. C: Unsaturatedlipid bilayer, less compact than the saturated one. D: Because unsaturated molecules occupymore space, the resulting bilayer is in a less crystalline state.

Tail lentgh (atoms) 12 14 16 18 20 22 24 18 18 18

Double bonds 0 0 0 0 0 0 0 1 2 3

Transition temperature (◦C) -1 23 41 55 66 75 80 1 -53 -60

Table 1: Transition temperature between gel and liquid phases for a membrane, for differenttail lengths and saturations (from [11], via Wikipedia).

describe more precisely some of the cell’s components, and see how these transitions andseparations play a strong role in the cell’s compartmentalization and organization.

2.1 Lipid membranes

The cell’s membrane consists of a lipid bilayer. Lipid molecules are composed of a hy-drophilic head and several hydrophobic tails (fig. 2A). In an aqueous medium, theytherefore adopt a bilayer configuration to avoid highly energetic interactions (fig. 2B). Inthis configuration, a lipid molecule can diffuse in the membrane plane, rotate or flip fromone layer to another. This fluidity of the membrane is an important feature for the cell’sfunctioning.

The lipid bilayer can undergo a phase transition between two states: a ‘liquid’ state,where a given molecule can diffuse laterally, and a ‘solid’ or ‘gel’ state, where it is trappedbetween its neighbors. Several techniques exist to determine the transition temperature;for example, carbon NMR gives a transition temperature of 18 ◦C for chloroplast – a typeof organelles – membranes [18]. Several criteria have an influence on the transition tem-perature; hydrophobic tails length and saturation are the most important (fig. 2C, 2Dand table 1).

In vivo membranes, however, are composed of several types of molecules, with differenttransition temperatures. The resulting membrane can exhibit an intermediate behavior,or undergo a process called phase separation and detailed below: different constituantswill spatially segregate, forming solid ‘islands’ inside an elsewhere liquid membrane.

2.2 Microtubules

In every cell is a complex network of filaments and tubes called the cytoskeleton (fig. 1B).This ‘skeleton’ gives the cell its shape and elastic properties, and is involved in many cel-lular processes. Microtubules are one of its components: they are long, hollow structures,formed by the polymerization of small dimers called tubulin (fig. 3A). One unique featureof these structures is called the dynamic instability: a microtubule randomly alternatesbetween assembling and disassembling (fig. 3B). This instability comes from a competition

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A B

Figure 3: Microtubules: structure and dynamic instability. A: Structure diagram of a micro-tubule, from Wikipedia. B: Length of a microtubule over time, from [8]. Transitions fromgrowth to shrinkage (resp. from shrinkage to growth, if any) are called catastrophes (resp.rescues).

between the addition of monomers at the tip of the microtubule, and a chemical reactionaltering recently bound monomers: if the reaction front catches up with polymerization,the microtubule starts to disassemble. This event is called a catastrophe, as opposed to arescue, which is when the microtubule polymerises again.

One can see the assembly or disassembly of a microtubule as a phase transition betweentwo states of the tubulin: a ‘gas’ state where monomers are solvated in the cytoplasm, anda ‘solid’ state where they are assembled in an almost crystalline arrangement [17]. Oneobvious parameter of this transition is the cytoplasmic concentration of tubulin monomers.If the concentration is greater than a critical value, polymerization goes on; if it becomeslesser, polymerization is not quick enough and the microtubule starts to shrink. Anotherparameter is the temperature, relevant at this Brownian scale. Using these two, is it pos-sible to draw a phase diagram for tubulin ?

To do this and complete the analogy with a phase change, one must first ask the ques-tion of nucleation. In vivo, preexisting structures help the nucleation and coordination ofmicrotubules inside the cell. Spontaneous polymerization, on the other side, would onlyoccur for sufficiently high tubulin concentration and low temperature.

Fygenson et al. managed to draw an equivalent phase diagram for microtubule as-sembly [8], distinguishing regions of spontaneous nucleation, growth from nucleation sitesand no growth at all (fig. 4A). Another transition was highlighted, between bounded andunbounded growth of the microtubule. This obsevation is consistent with measurementsof the microtuble characteristic length, which diverges following a power-law (fig. 4B) ata temperature fixed by the tubulin concentration.

2.3 Membrane-less organelles

Organelles are specialized subunits of a cell (fig. 1A), and posess specific functions, in thesame way that organs are present in our body. While many of them are delimited by amembrane, others are not (fig. 5): therefore, they must constantly be exchanging materialwith the surrounding cytoplasm. Most of them contain RNA and protein molecules, andare called granules. These membrane-less organelles were shown to exhibit a liquid-likebehavior [5] – spherical shape, liquid-like fusions, etc. Through several articles, the studyof these structures led Brangwynne and his collaborators into thinking membrane-lessorganelles were the fruits of a phase separation.

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A B

Figure 4: Microtubules: phase diagram and characteristic length. A: Phase diagram ofmicrotubule assembly, from [8]. I: region of no growth. II: region of growth from nucleationsites, divided between bounded and unbounded growth. III: region of spontaneous nucleation.B: Characteristic length l0 of microtubules vs temperature T , from [8]. Close to the criticaltemperature T0, this length shows a power-law divergence (l0 ∝ (T0 − T )−1), reflecting thetransition from bounded to unbounded growth. The value obtained here for T0 and theconcentration used in the experiment are consistent with the phase diagram.

Figure 5: Nucleoli – membrane-less organelles found in the nucleus – of an X. laevis oocyte,from [5] via [4].

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Figure 6: Phase diagram of a concentration-induced phase separation, from [4].

Phase separations are a type of phase transition in which the two ‘phases’ are in thesame physical state, here the liquid one. A solution of several components can either behomegeneous, or nucleate regions of high concentration for one component (fig. 6). Ananalogy for this phenomenon is the condensation, from a homogeneous vapor, of a dropletof water in which molecules are much more spatially concentrated. In the case of gran-ules, the phase separation of RNA and other proteins from the cytoplasm increases thelocal concentration of molecules of interest beyond a treshold value, which allows for theirinteraction in biochemical processes. This interpretation is consistent with the strong linkbetween granules and RNA metabolism [17].

Biological organelles and cytoplasm, however, contain numerous different molecules,and are active media. This gives the system complexity, but also richness: exchangeswith the extracellular medium or post-assembly modifications to the RNA add a dynamicaspect, allowing the cell to efficiently regulate itself and respond to changes in its envi-ronment.

These three examples illustrate how the auto-assembly of biological structures can beexplained by phase transitions or separations. These physical phenomena are subject toa wide range of biological parameters, thanks to the complexity of the cellular medium.This allows the cell to react to any event by adapting its metabolism. This adaptativebehavior, along with the link between physiological conditions and criticality, will also bediscussed in the next section.

One last feature that can be discussed here is the link between phase transitions andthe size scaling of several of the cells’s structures. The analogy suggested by Brangwynne[4] is the following: the size of a droplet of condensed water depends on the size of thevapour tank. In the same way, the size of phase separation-induced structures, suchas granules, depend on the size of the cell. This would give an explanation to severalsize-scaling observations, assuming the cell keeps the different molecular concentrationsconstant when its size varies.

3 Phase transitions and criticality at different scales

We saw on the previous examples that phase transitions, and more generally statisticalphysics, are the tools of choice to study biological systems at a microscopic scale. In the

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Figure 7: Sandpile model and self-organized criticality, from [9] via [12]. When adding grains,the slope increases until it reaches a critical value; adding other grains then triggers avalanchesand decreases the slope. At criticality, avalanche size and duration distributions follow powerlaw scalings.

last years, new experiments were designed for several systems and scales, and permittedthe obtention of large sets of data. This allowed for the construction of statistical physicsmodels directly from the data. Remarkably, the obtained models seemed to be poisednear or at a critical point in the parameter space.

These parellel observations revived and old hypothesis stating that biological systemswere poised at criticality. However, several questions remain. What sets the system tothis critical point ? To quote Mora and Bialek, “there is something special about thestates corresponding to functional, living systems, but at the same time it cannot be thatfunction depends on a fine tuning of parameters” [10]. And how does criticality benefitthe biological function ?

A trending answer to the first question is self-organized criticality. This concept isbest illustrated by the historical example of sandpiles (fig. 7): when adding grains, theslope of a pile of sand will increase or decrease to remain close to a critical slope. In thefollowing subsections, we will see biological evidence of criticality through different scales,and try to answer the questions ‘why’ and ‘how’.

3.1 Neural networks

In animals, the nervous system consists of a network of interlinked neurons. Neurons arecells that can communicate with each other through electrical pulses, called spikes. At agiven time, a neuron is either spiking or not. The activity of a network of N neurons cantherefore be described [10] by a list ~σ = {σi}i=1..N of the N neuron states, where σi = 1if the neuron i is spiking and −1 if it is not. Recordings of the optical neural activity insalamanders, macaque monkeys and cats give such states over time for networks of tensof neurons.

The construction of a model directly from data relies on the principle of maximumentropy from statistical physics. Given some constraints inferred from the sampled data,the simplest probability distribution that represents the phenomenon is the one with thelargest entropy. Following this principle, one chooses the least-structured model that fitsthe data, therefore making no assumptions. For instance, the maximum entropy modelconsistent with observed averages 〈σi〉 is a model of independently firing cells:

P1(σ) =∏i

pi(σi) =1

Zexp

[∑i

hiσi

]

This model, however, does not reproduce spikes obervations. A second model, improvedto take interactions into account, is asked to be consistent with observed averages 〈σi〉

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Figure 8: Heat capacity C(T,N) vs effective temperature T , from [15]. The different systemsizes N are obtained by considering subsets of the original network, or by extrapolation.

and pairwise correlations 〈σiσj〉:

P2(σ) =1

Ze−E(~σ) =

1

Zexp

∑i

hiσi +1

2

∑i 6=j

Jijσiσj

This model is mathematically equivalent to the well-known Ising model for magnetic spins,with local fields hi and couplings Jij . Here these parameters are directly determined fromthe experimental data.

Numerical study of this Ising model is henceforth possible. To explore the statisticalmechanics of the model, Tkacik et al. introduce an ‘effective temperature’ T by rescalingh and J [14]. T = 1 therefore corresponds to the actual network. T has nothing to dowith the actual temperature; it is only a parameter we vary to probe the parameter phasealong one direction. This allows for the computation of an equivalent ‘heat capacity’ fromnumerical simulations (fig. 8). The heat capacity shows a peak, that gets higher and closerto T = 1 as the size of the neural network N increases. Although the neural network is outof equilibrium, the striking resemblance with the heat capacity divergence in equilibriumthermodynamics is a step towards the criticality hypothesis: the neural network seems tobe poised at a critical state. Other observations, based on the link between equivalent‘energy’ and ‘entropy’ [13], also back this idea.

The biological interest of criticality is not clear for neural networks. A continuousmodel of the network dynamics [9] lead to the construction of a phase diagram of theneural activity (fig. 9). The brain is then thought to dwell at the critical point of atransition between ‘dead’ – any neural activity is doomed to disappear – and ‘epileptic’– neurons spike randomly and independently. With a brain poised at criticality, brainfunction would therefore be improved. Other reflections draw links between spiking prob-ability distributions and enhanced memory and reflex coding [13].

3.2 Flocks and swarms

Flocks of birds are a daily and striking example of cordination between animals: birds flytogether in the same direction, and take synchronized turns. However, order in a flockdoes not come from following a leader or an external stimulus. Instead, flocking behavioremerges from local interactions between neighboring birds. It is possible, as above, torecord flights of flocks and infer a maximum entropy model from the data [3]. Rather, we

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Figure 9: Number of activated neurons A in a neural network vs average number of connectionsper neuron z, from [9] via [12]. A phase transition is observed at z = z∗, between a subcriticalphase – any neural activity is doomed to disappear – and a supercritical phase – neurons spikerandomly and independently.

will focus here on correlation functions and size-scalings [7].

Let us consider a flock of N birds. For each bird i, we define the speed ~vi, and thefluctuation ~ui = ~vi − (1/N)

∑k ~vk around the mean velocity of the flock. Correlations in

these fluctuations are given by

C(r) ∝∑i,j ~ui · ~ujδ(r − rij)∑

i,j δ(r − rij)

where δ is a smoothed Dirac delta function an rij is the distance between birds i andj. Both fluctuations in orientation and magnitude of the speed vector were studied (fig.10A, 10B). The correlation function changes sign – which means fluctuations becomeanticorrelated – for a distance we note ξ. A striking result is that ξ scales linearly withthe size L of the flock (fig. 10C, 10D), which means no other scale than L is present inthe system. Using this property and approximating the correlation function by its leadingcontribution C(r) ∝ ξ−γf(r/ξ), we obtain the following form for the correlation function:

C(r, L) ∝ 1

rγf( rL

)which, in the limit of large flocks, gives C(r) ∝ r−γ . This power-law decay of the corre-lation function is characteristic of a critical point. In the present case, the smallness of γimplies unusually long-ranged interactions across the flock.

A similar study on midges [1] also reveals long-ranged interactions across a swarm(fig. 11A). Collective behavior therefore does exist in swarms, but is not associated withemergent order. The correlation function is defined in the same way as before, and onecan compute its volume integral Q(r) (fig. 11B). This function reaches a maximum atr = r0, corresponding to the sign change of C(r). An equivalent susceptibility is thendefined as χ = Q(r0). Numerical simulations show that the susceptibility is a peakedfunction of the density (fig. 11C). Again, the peak gets higher and closer to a criticaldensity as the number of midges N increases. This divergence is interpreted as the criticalpoint of a transition between an ordered phase (flock) and a disordered phase (swarm) [16].

Flocks and swarms poised near a critical point see their susceptibility maximized. Thisenhances the reactivity of the animals to external perturbations, for instance the attackof a predator. At the same time, interactions scale through the whole group, reinforcingthe collective behavior.

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Figure 10: Correlations between birds in a flock, from [7]. A, B: Correlation functions for theorientation and magnitude of speed fluctuations. C, D: Correlation lengths for the orientationand magitude of speed fluctuations. The correlation lengths scale linearly with the size L ofthe flock.

A B C

Figure 11: From correlations to susceptibility for a swarm of midges, from [1]. A: Correla-tion function for the speed fluctuations. B: Volume integral of the correlation function. C:Simulated susceptibilities vs nearest neighbor distance of midges, for different swarm sizes.

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A B

Figure 12: Evolution bursts and avalanches, from [2]. A: Evolution activity vs time (in unitsof number of mutations). B: Distribution of avalanche sizes in the critical state, that followsa power law (P (S) ∝ S−0.9).

3.3 Evolution

Evolution is the process through which species undergo changes and adapt to their envi-ronment, thanks to natural selection. Random modifications of the genetic information,called mutations, induce modifications of a living being’s features. If these modificationsare a benefit, the individual will be better adapted to its environment. It will therefore bemore prone to reproduce and pass on these modifications. On the contrary, modificationsthat are drawbacks will not be selected. Through long timescales, this process applies to awhole species, or gives birth to new ones. However, actual evolution is not a slow, steadyprocess; instead, intermittent bursts of activity separate long periods of stability.

To account for these observations, Bak and Sneppen propose a model of interactingspecies that self-organizes into a critical steady state [2]. Rather than modeling a geneticcode to mutate, it considers entire species, and their associate ‘fitness’ parameter: in thissense, the model deals with a ‘coarse-grained’ evolution. Different species are coupled inthe ecosystem, and the fitness of one species is sensitive to the fitness of coupled species.As a consequence, coupled species help each other mutate. This model reproduces burstsof activity (fig. 12A); while observed power laws point out to self-organized criticality.In the same way that the critical sandpiles can undergo avalanches of any scale, one candefine avalanches of mutations in this model, the size distribution of which also follows apower law (fig. 12B).

In these three examples, theoretical models or maximum entropy models based on largesets of data push the idea that corresponding biological systems are poised near a criticalpoint. Although still debated, these criticality hypotheses find more and more experimen-tal evidence, based on analogies with equilibrium statistical physics phase transitions. Ateach scale, criticality improves the system’s biological function.

A possible answer to how criticality is tuned in network interactions is evolution. Ifgetting closer to a critical point indeed improves biological function and fitting to theenvironment, natural selection will select individuals close to such a point. Everythinghappens as if evolution itself were an external operator, tuning systems to criticality [12].This naive reflection is of course not sufficient; for example, it does not explain how evolu-tion itself is poised at criticality. In every case, however, self-organized criticality emergesfrom the interactions between several constituents of the system.

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4 Conclusion

Whatever the scale, biological systems are very complicated, as they rely on the dynamicsof numerous constituents. The example of the cell illustrated this complexity, and how itturned out to be a richness for such a system. Indeed, cellular processes relying on phasetransitions and phase separations are sensitive to many parameter variations, which areas many stimuli for the cell to respond to by adapting its metabolism. In other words,the more dimensions there are in the complete phase diagram, the more parameters thecell can take into account. For this behavior to emerge, however, the cell’s compositionmust be poised near such phase transitions.

On other systems and through different scales, such proximity to a phase transition ora critical point in the parameter space was indeed suggested. The biological function wasimproved near critical points, highlighting the biological relevance of self-organized criti-cality, which arose from the interactions between constituants of the system. Even thoughthese ideas are still debated, more and more experimental evidence back such hypotheses,not to mention models inferred directly from raw data and with no assumption.

The examples presented here were unfortunately not well developed, but presented thefoundations of corresponding reasonings, more detailed in dedicated reviews or articles.There of course exist limitations to the hypotheses, results and interpretations discussed.We chose not to detail them in order to keep this report as concise as possible. On theother hand, several theoretical articles, much closer to statistical mechanics, phase tran-sitions and critical phenomena formalism, back such hypotheses by providing models andways to detect criticality. Again, developing such content here would have been too long,although it deserves attention.

Part of the work in biophysics is to adapt physical theories and methods to study bio-logical systems. Given the complexity of such dynamic and far-from-equilibrium systems,this is not an easy task. As often, what makes the richness of the biological system is alsowhat makes its study a challenge. However, new techniques and experiments designed inthe last years opened the door to new studies and reasonings, allowing for always betterinsights in such systems. We hope, if this work manages to raise curiosity, to see them inmore detailed and focused reports.

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