Novel physics arising from phase transitions in biology
Abstract: Phase transitions, such as the freezing of water and the magnetisation of a ferromagnet upon lowering the ambient temperature, are familiar physical phenomena. Interestingly, such a collective change of behaviour at a phase transition is also of importance to living systems. From cytoplasmic organisation inside a cell to the collective migration of cell tissue during organismal development and wound healing, phase transitions have emerged as key mechanisms underlying many crucial biological processes. However, a living system is fundamentally different from a thermal system, with driven chemical reactions (e.g., metabolism) and motility being two hallmarks of its nonequilibrium nature. In this review, we will discuss how driven chemical reactions can arrest universal coarsening kinetics expected from thermal phase separation, and how motility leads to the emergence of a novel universality class when the rotational symmetry is spontaneously broken in an incompressible fluid.
Summary (4 min read)
- Collective phenomena are intimately linked to the phenomenon of phase transitions in physics.
- By a universal behaviour, the authors mean certain properties of the system that are highly independent of the system’s microscopic details.
- In the salad dressing example, such property can be the power law exponent that governs how the average size of oil drops changes with time; in the example of magnetisation, it can be the power law exponent that governs how the correlation function of two atomic spins decays with respect to their distance.
- Recently, phase transitions in living systems have also been under intense attention.
- Finally, the authors will end with Conclusion & Outlook.
2.1. Membrane-less organelles
- Biological cells organise their contents in distinct compartments called organelles, typically enclosed by a lipid membrane that forms a physical barrier and controls molecular exchanges with the surrounding cytosol.
- Membrane-less organelles have attracted an intense interest from the biology community as they are present in many organisms from yeast to mammal cells, and are critical for multiple biological functions.
- And stress granules assemble during environmental stress and protect cytoplasmic RNA from degradation  (Fig. 1 a)).
- In particular, the authors show that the threshold for macroscopic phase separation is altered by the elasticity of the polymer network, and they highlight the role of correlations between nuclei positions in determining the drop size and polydispersity.
- The authors will then review the latest progress on phase separation driven out of equilibrium by energy-driven chemical reactions in Sec. 2.3.
2.2. Equilibrium phase separation
- Interactions between molecules can cause a homogeneous system to undergo a phase separation, i.e. the spontaneous partitioning of a system into multiple phases of distinct properties such as concentration .
- Inside the phase boundary (“♦” symbol) the system phase separates into two phases (“in” and “out”) of distinct concentrations (P̂in,out, Ŝin,out), given by the intersections between the tielines (straight lines) and the phase boundary.
- The smaller the drop, the larger the concentration outside which is a consequence of the Laplace pressure .
- There exists a steady state radius R∗ (Jout→in = 0, purple disk) that is unstable, called nucleus radius.
- Therefore Ostwald ripening occurs until, in a finite system, a single drop survive (Fig. 3 c)).
2.3. Phase separation in presence of non-equilibrium chemical reactions
- The presence of non-equilibrium chemical reactions have been proposed recently to explain multi-drop stability in the cytoplasm, as well as being a mechanism to control the formation, dissolution and size of membrane-less organelles [22, 23, 29, 30].
- Molecules P are then transported by diffusive fluxes toward drops (red arrows).
- Since drops are small compared to the gradient length scale ξ (Eq. (11)) the diffusion coefficient D is large enough so that the excess of S is quickly evacuated outside drops by diffusion, leaving the drop concentrations unperturbed.
- The chemical reaction-induced term ξ/R therefore tends to stabilise a multi-drop system against Ostwald ripening.
2.4. Spatial organisation
- Another interesting phenomenon resulting from this type of non-equilibrium phase separation is the potential spontaneous spatial organisation of drops on a lattice, as observed in Monte Carlo simulations shown in Fig. 11a .
- In this section the authors provide a simple intuitive argument that accounts for the observed lattice organisation.
- Let us consider a drop approaching another one.
- On the side where inter-drop distance is reduced, concentration gradients become shallower leading to weaker solute influx into the drop (small red arrow).
- Therefore chemical reactions in their multi-drop system tend to distribute drops on a lattice structure.
3. Active matter: motile organisms in the incompressible limit
- Active matter refers to physical systems in which some or all constituents of the system can exert forces continuously on their surrounding environment .
- In the case of a bird flock, the birds fly by flapping their wings to move the air around them; in the case of a cell tissue on a substrate, the cells move via coordinated and ATP-driven remodelling of biopolymers beneath their cell membranes .
- Active matter constitutes a non-equilibrium system and the energy is provided either through a continuous supply of fuel or by energy already stored in the system.
- Here, the authors will focus exclusively on active matter in the condensed state, to the extent that the system can be viewed as incompressible.
- Such an EOM can generically be written down based on symmetry consideration alone and the associated universal behaviour of the system can then be analysed using analyical methods such as dynamical renormalisation group (DRG) methods [40, 41], or numerically.
3.2. Incompressible active fluids
- The authors will focus exclusively on the so-called “dry” active matter [38, 33], in the sense that there exists a fixed background in the system for the active constituents to exert forces on.
- Experimentally, the active constituents can be motile cells and the fixed background can be a gel substrate that the cells crawl on.
- In contrast, wet active matter describes motile organisms in a fluid medium in which organisms move by exchanging momentum with the surrounding fluid, and the resulting fluid flow can in turn affect the motion of the organisms [51, 52].
- Ignoring the blue terms in Eq. (23) for the time being (whose omissions will be justified later), and focusing on spatially homogeneous states (so all terms involving ∇ become zero), the simplified EOM can be written as ∂tv = − δH δv (25) where H(v) = −av2/2 + bv4/4.
- The transition between these two phases is continuous and thus constitutes a critical transition.
3.3. Universal behaviour at the critical point
- To understand the emergence of scale-invariant structures at the critical point when the system transitions from the disordered phase to the ordered phase, the authors will first analyse the EOM at the linear level and then incorporate the nonlinear effects using DRG methods.
- What the authors have seen is that in the linear theory, by suitably re-scaling the field variable and time, the coefficients in the EOM will remain invariant under spatial rescaling, which leads to a power-law behaviour of the correlation function.
- To proceed, the authors will first employ the scaling exponents from their linear theory to gauge the importance of the additional terms in their full EOM.
- Under a DRG transform, fluctuations associated with the short distance behaviour of the system are averaged over and the effects of the averaging are then incorporated back into the EOM.
- Since these two dimensionless quantities themselves vary with `, the authors can study their own flow equations.
3.4. Ordered phase in two dimensions
- The authors have seen that at the critical transition, the scaling behaviour of a generic incompressible active fluid constitutes a novel universality class in non-equilibrium physics.
- Here, the authors will describe how in two dimensions, the ordered phase in incompressible active fluids also exhibits universal behaviour, albeit with scaling behaviour that belongs to a well known universality class: the Kardar-Parisi-Zhang (KPZ) universality class that originated from modelling surface growth in the nonequilibrium regime . (48).
- In a smectic liquid crystal, the liquid crystals (depicted as red ellipsoids in Fig. 14) formed a layered structure in which the layers are parallel to x-axis on average and h(x, y) describes the height deviation of the layers from the expected location.
4. Conclusion & Outlook
- Motivated by recent studies focused on phase transitions in biological systems, the authors have discussed how novel physics can arise from the generic non-equilibrium nature of living matter, be it driven chemical reactions or self-generated mechanical forces.
- A recent discovery found that a biologically relevant active polymer network under fragmentation can self-organise itself to exhibit a scale-invariant signature of a critical system [74, 75].
- (ii) In Sec. 2.4 the authors have provided intuitive arguments to explain the appearance of a lattice structure of phase-separated drops in their Monte Carlo simulations.
- (iii) In Sect. 3, the authors have studied the simplest kind of symmetry: the rotational symmetry and the associated universal behaviour when the symmetry breaks spontaneously in an active system.
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"Novel physics arising from phase tr..." refers background in this paper
...ripening that causes large drops to grow and small drops to evaporate [26, 27], and coalescence caused by the fusion of drops upon contact ....
...The diffusive profile leads to a flux Jout→in = D∇Pout|r=R of molecules P entering the drop at the interface :...
...The second ingredient driving Ostwald ripening is the existence of a diffusive concentration profile between drops, which can be approximated by an ideal gas diffusion profile in the case of small concentration outside drops :...
"Novel physics arising from phase tr..." refers background in this paper
...At thermal equilibrium the chemical conversions are non-driven and the energy required to overcome the barrier ∆U is provided by thermal fluctuations alone, and therefore the reaction rate constant k decreases exponentially with ∆U ....
"Novel physics arising from phase tr..." refers background or methods in this paper
...The roughness and dynamic exponents of the KPZ model are known exactly: zKPZ = 3/2 and χKPZ = 1/2 ....
...Here, we will describe how in two dimensions, the ordered phase in incompressible active fluids also exhibits universal behaviour, albeit with scaling behaviour that belongs to a well known universality class: the Kardar-Parisi-Zhang (KPZ) universality class that originated from modelling surface growth in the nonequilibrium regime ....
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Frequently Asked Questions (2)
Q1. What contributions have the authors mentioned in the paper "Novel physics arising from phase transitions in biology" ?
In this review, the authors will discuss how driven chemical reactions can arrest universal coarsening kinetics expected from thermal phase separation, and how motility leads to the emergence of a novel universality class when the rotational symmetry is spontaneously broken in an incompressible fluid.
Q2. What are the future works mentioned in the paper "Novel physics arising from phase transitions in biology" ?
In terms of outlook, the authors believe the following future directions will expand the horizon of both biology and physics. ( i ) In Sec. 2 the authors have studied how driven chemical reactions can stabilise a multidrop, ternary system. As the cell cytoplasm is a complex mixture of thousands of different molecules [ 82, 83 ] it will be interesting to see how these results may be modified in a many-component mixtures. Such a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Novel physics arising from phase transitions in biology 32 structure naturally suggests a kind of repulsive interactions between drops, which may serve to stabilise a multi-drop system against coarsening via coalescence due to drop diffusion.