Novel physics arising from phase transitions in biology

TL;DR: In this paper, the authors 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.

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

1. Introduction

• 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 [10] (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 [17].
• 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 [17].
• 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 [29].
• 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 [33].
• 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 [34].
• 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.

3.1. Hydrodynamic theory of incompressible passive fluids – Navier-Stokes equation

• For an equilibrium system, symmetry constrains the allowable form of the Hamiltonian of the system [42].
• In an incompressible fluid, the obvious field variable is the velocity field v(r, t), whose dynamics can be written as: ∂tv = F ρ , (22) where ρ is the density field and F corresponds to the local force density.
• This symmetry means that experimental results do not depend on which direction the experimental apparatus are positioned towards.
• The EOM is invariant under spatial inversion, hence forbidding terms like ∇ × v, also known as (iv) Parity invariance.
• The existence of long-time tail of various correlation functions of thermal fluids, first discovered via simulations [47, 48], have been confirmed using diverse analytical methods such as kinetic theory [49, 50] and DRG analysis [41].

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 [61]. (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.

Figures

Figure 7. Non-equilibrium concentration profiles in a multi-drop system at steady state. The profiles of P (red curve) and S (blue curve) (Eq. (10)) are shown along an axis connecting the centres of two drops of radius R. The interfaces are at local thermal equilibrium so the coexistence solute concentrations (Pin,out(R)) are given by the Gibbs-Thomson relations (Eqs (1) and (2)). The concentration S is continuous at the interface (Sin(R) = Sout(R)). The non-equilibrium chemical reactions coupled with diffusion and phase separation create a circulating flux of molecules P and S between drops and cytoplasm. Drops are rich in P so the chemical conversion P →k S dominates, leading to accumulation of an S molecules inside drops (downward red/blue arrows). The excess of S is then transported by diffusion toward the cytoplasm (blue arrows). In the cytoplasm the reverse reaction dominates, leading to creation and accumulation of P molecules between drops (upward blue/red arrow), which diffuse toward the drops (red arrows).

Figure 10. Stability diagram of a multi-drop system for varying forward rate constant k and drop radius R. A steady state exist within the continuous curve. Outside this region all drops dissolve (downward arrows). Outside the grey region, drops are unstable against Ostwald ripening and coarsen (upward arrows). In the grey region drops are stable against Ostwald ripening. The stability-instability boundary is shown by a dashed curve. The analytical results for the critical radii in the small drop regime are showed by the upper dotted line (Ru, Eq. (14)) and lower dotted line (Rl). The analytical results for the critical rates kc, kl and ku (Eqs (15), (20) and (21)) are showed by arrows. Large drops dissolve beyond ku, all drops dissolve beyond kc, and large drops are stable against Ostwald ripening beyond kl. The large drop regime is coloured in blue and the boundary between the large and the small drop regime is R = ξ (Eq. (11)). Parameters: h = 10−2s−1, lc = 10−2µm, D = 1µm2s−1, P̂in = 10 −1ν−1, P̂out = 10−4ν−1, φ = 5 × 10−4ν−1, where ν is the molecular volume of P and S and can be chosen arbitrarily. Reprinted figure with permission from [Wurtz J.D. and Lee C.F., Physical Review Letters, vol. 120, 078102, 2018]. Copyright (2018) by the American Physical Society.

Figure 15. Our discussion in Sect. 3.4 demonstrates that in two spatial dimensions, the large distance properties of an incompressible active fluid in the ordered phase, e.g., incompressible flock, has the same scale-invariant properties as a two-dimensional smectic liquid crystals and the KPZ surface growth model. The findings thus suggest that instead of the apparent equivalence between an incompressible flock and a two-dimensional lattice depicted by the landscape in Escher’s Day and Nights [top figure (awaiting approval of use from the copyright holder)], the flock should instead be mapped onto a smectics-like layered structure such as sedimentary rocks [66] [Top half of bottom figure: photograph by Tommy Hansen. Public domain. Bottom half of bottom figure: photograph by Ryan McGrady, distributed under a CC-BY 4.0 license.].

Figure 3. Ostwald ripening in equilibrium systems. a) The influx Jout→in of molecules P at a drop interface for varying drop radius R and constant supersaturation ∆ (Eq. (5)). The steady state R∗ (J = 0, purple disk) is unstable: drops larger than R∗ grow while smaller ones shrink. Insert: as ∆ decreases, the critical radius R∗ increases. b) Schematic of P concentration profiles along an axis connecting the centres of two drops of different radii R1 > R2. The GibbsThomson relation (Eq. (2)) dictates that the solute concentration is lower close to the large drop (Pout(R1)) than close to the small drop (Pout(R2)). This causes a diffusive flux from small drops to large drops (red arrows, Eq. (5)). c) A multi-drop system is therefore unstable against Ostwald ripening: small drops evaporate and large drops grow. As fewer drops survive the supersaturation ∆ decreases causing more drops to dissolve (see insert in a)). Eventually a unique drop remains in a finite system.

Figure 6. Monte Carlo simulation of phase separation in a ternary mixtures with non-equilibrium chemical reactions [29]. Molecules P (red dots) form energetic bounds with neighbouring P , while molecules S (blue dots) and C (not shown) do not form bounds. Chemical reactions transform P into S and vice versa with fixed reaction rate constants (Eq. (6)). In the initial state molecules are randomly distributed (leftmost snapshot). As time progresses drops form and undergo Ostwald ripening leading to increase of the average drops radius (middle snapshot). Eventually a multi-drop steady state is reached (rightmost snapshot). At the steady state drops have roughly the same radius and are evenly distributed. Simulations details can be found in [29]. Reprinted figure with permission from [Wurtz J.D. and Lee C.F., Physical Review Letters, vol. 120, 078102, 2018], Copyright (2018) by the American Physical Society

Figure 9. Concentration profiles in the large drop regime. The profiles of P (red curve) and S (blue curve) are flat and therefore at chemical equilibrium far from the interface (Pin(0), Pout(∞)). Concentration gradients of length scale ξ (Eq. (11)) are localised at the interfacial region only. Hence the reaction-diffusiondriven circulation of molecules P and S between drop and cytoplasm (blue↔red circular arrow) is also localised at the interfacial region only. See Fig. 7 for details about this circulation.

Figure 1. Cytoplasmic phase separation and tissue dynamics as active matter. a) In many distinct types of cells, certain proteins can phase separate from the cytosol to assemble membrane-free organelles, such as the stress granules (yellow drops) shown here in human epithelial cells (HeLa) [6], akin to oil drop formation in an oil-water mixture (b). c) In a monolayer of Madin-Darby Canine Kidney (MDCK) cells, the cells in the tissue can undergo dynamical rearrangement as shown by the snapshot of the velocity field shown in (d) [7]. Figure a) is adapted from [J.R. Wheeler et al., eLife vol. 5, pp. e18413, 2018], licensed under CC BY 4.0. Figure b): c©Nikola Bilic, Dreamstime.com. Figures c) and d) reprinted from Biophysical Journal, vol. 98, Petitjean et al., Velocity fields in a collectively migrating epithelium, pp. 1790-1800, Copyright (2010), with permission from Elsevier.

Figure 14. A smectic liquid crystal is a layered structure of liquid crystals (red ellipsoid) and the function h(x, y) quantifies the deviation of the layer at position (x, y) from the expected location.

Figure 4. Model of non-equilibrium phase separation in the cell cytoplasm. We consider the ternary mixture from Sec. 2.2 with the addition of chemical reactions. Molecules P can convert into S at rate k and vice versa at rate h (Eq. (6)). S do not participate to phase separation i.e. its concentration remains homogeneous (Ŝin = Ŝout). Figure reprinted from [Wurtz J.D. and Lee C.F., New Journal of Physics, vol. 20, no. 4, 045008, 2018], licensed under CC BY 3.0.

Figure 8. Molecular fluxes at a drop interface for varying drop radius R and fixed supersaturation ∆. The efflux Jin→out (red curve, Eq. (13)) driven by chemical reactions competes with the influx (brown curve) which is identical to that at the equilibrium condition (i.e without chemical reactions, see Eq. (5) and Fig. 3 a)). Drops grow if Jout→in > Jin→out and shrink otherwise. Two steady states radii R∗− < R ∗ + exist. R ∗ − (purple circle), reminiscent from the equilibrium case, is unstable against Ostwald ripening: smaller drops shrink (left arrow) and larger drops grow (right arrows). At large radius the efflux Jin→out dominates so drops shrink. There exist therefore a second steady state R∗+ (green disk) that is stable against Ostwald ripening: larger drops shrink and smaller drops grow. The insert shows the effect of decreasing the supersaturation ∆. The smallest unstable steady state is labelled Rc, the largest stable steady state is labelled Ru, and the critical radius Rl (black square) is the boundary between the unstable and the stable region (Eq. (14)).

Figure 13. The flow of the two coupling coefficients g (b) ` and g (λ) ` under a DRG transformation for d = 3 to order . There are one stable and two unstable fixed points. The stable fixed point (red square) corresponds to the incompressible active fluid universality class found recently [57]. The unstable fixed points lying on the g(b)-axis (purple circle) corresponds to the ferromagnets with dipolar interactions universality class found in 1973 [58]; while the unstable fixed point lying on the g(λ)-axis corresponds to the randomly stirred fluid universality class found in 1977 (model B in [41]). Figure reprinted from [Chen L., J. Toner and Lee C.F., New Journal of Physics, vol. 17, 042002, 2018], licensed under CC BY 3.0.

Figure 5. Non-driven and driven chemical reactions in phase separating systems. We consider here for clarity a particular example from our ternary mixture, where P molecules (red disks) form bonds (black lines) with neighbouring P molecules while molecules S (blue disks) do not interact. The chemical conversion of a given P into a S requires an activation energy ∆U to break these bounds. Since drops are enriched in P , many bonds need to be broken and ∆U is high (upper graph). Conversely the cytoplasm is poor in P and ∆U is thereby small (lower graph). At thermal equilibrium the chemical conversions are non-driven and energy required to overcome the barrier ∆U is provided by thermal fluctuations alone, and therefore the reaction rate constant k decreases exponentially with ∆U [31]. As a result, k is thus concentration-dependent: k is small inside drops where ∆U is high, and large in the cytoplasm. In the case of driven chemical reactions, such as ATP-dependent phosphorylation, an external source of energy is provided [18] (yellow arrows). Hence k depends on the specificities of the reaction and can potentially be independent of ∆U i.e. concentration-independent. The rightmost part of the figure is adapted from [J.R. Wheeler et al., eLife vol. 5, e18413, 2018], licensed under CC BY 4.0.

Figure 12. In a generic incompressible active fluid, two distinct phases are possible. At the mean-field level, a disordered phase exists when the parameter a is negative, and an ordered phase (characterised by a non-zero mean speed of the system) emerges when a is positive. The transition between these two phases is continuous, or critical (region depicted in red). The surface plots depicts the “potential energy landscape” at the mean-field level for an active fluid in two dimensions. In the disordered phase, the energy landscape is like a parabolic bowl, while at the transition, the global minimum of the landscape becomes very flat. The landscape transitions further into the shape of a Mexican hat in the ordered phase.

Figure 2. Equilibrium phase diagram of a ternary mixture composed of molecules P (red discs), S (blue discs) and C (not shown). Outside the phase boundary (green line) the system is homogeneous (“ ” symbol). Inside the phase boundary (“♦” symbol) the system phase separates into two phases “in” and “out” of distinct concentrations. The coexistence concentrations P̂in,out, Ŝin,out are given by the intersections between the tie-lines (straight lines) and the phase boundary.

Figure 11. Drop spatial organisation. a) Snapshot of Monte Carlo simulations of a multi-drop system at steady state (no momentum transfer between molecules, see main text for details). Dro ps spontaneously distribute homogeneously in space. b) Schematic of the solute profile across three adjacent drops. The profile gradient between drops far from each other is stronger than between close drops. The solute influx is therefore larger on one side of the drop and weaker on the opposite side, causing the drop to grow on one side and dissolve on the other side. This leads to effective drop displacement tending to align drops on a lattice.

Full text

Novel physics arising from phase transitions in
biology
Chiu Fan Lee
Department of Bioengineering, Imperial College London, South Kensington
Campus, London SW7 2AZ, U.K.
E-mail: c.lee@imperial.ac.uk
Jean David Wurtz
Department of Bioengineering, Imperial College London, South Kensington
Campus, London SW7 2AZ, U.K.
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Novel physics arising from phase transitions in biology 2
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 non-
equilibrium 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.
1. Introduction
Collective phenomena are intimately linked to the phenomenon of phase transitions
in physics. At a typical phase transition, a many-body system with constituents that
interact only locally with their neighbours, be they molecules or living organisms,
can collectively change their behaviour upon a subtle change of a single parameter,
to the extent that the qualitative behaviour of the whole system is modified. Phase
transitions encompass many everyday phenomena such as oil drop formation in a
salad dressing and magnetisation in some metals. The study of phase transitions is of
fundamental interest to physicists because of the emergence of universal behaviours
at a phase transition. By a universal behaviour, we 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. Indeed, the generic non-equilibrium nature
of biological systems have given rise to novel universal behaviours not seen before.
In this review, we will focus on two such examples: phase separation with driven
chemical reactions, motivated by the mechanism underlying the formation of some
non-membrane bound organelles in cells [1, 2], and spontaneous symmetry breaking
in incompressible active matter, motivated by its relevance to biological tissues [3, 4, 5]
(Fig. 1).
In Sect. 2, we will first describe the relevance of phase separation in cytoplasmic
organisation and then review the latest findings on how driven chemical reactions
(e.g., adenosine triphosphate (ATP)-driven phosphorylation) can lead to co-existing
phase-separated protein drops in the cytoplasm, contrary to the universal coarsening
behaviour expected in its equilibrium counter part. In Sect. 3, motivated by the
collective behaviour found in motile organisms, we will introduce a generic model of
incompressible active fluids from a symmetry consideration. We will then elucidate
how a novel critical behaviour emerges at the onset of collective motion, and discuss
the universal behaviour of a two dimensional incompressible active fluid in the ordered
phase. Finally, we will end with Conclusion & Outlook.
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Novel physics arising from phase transitions in biology 3
Figure 1. Cytoplasmic phase separation and tissue dynamics as active matter.
a) In many distinct types of cells, certain proteins can phase separate from the
cytosol to assemble membrane-free organelles, such as the stress granules (yellow
drops) shown here in human epithelial cells (HeLa) [6], akin to oil drop formation
in an oil-water mixture (b). c) In a monolayer of Madin-Darby Canine Kidney
(MDCK) cells, the cells in the tissue can undergo dynamical rearrangement as
shown by the snapshot of the velocity field shown in (d) [7]. Figure a) is adapted
from [J.R. Wheeler et al., eLife vol. 5, pp. e18413, 2018], licensed under CC
BY 4.0. Figure b):
c
Nikola Bilic, Dreamstime.com. Figures c) and d) reprinted
from Biophysical Journal, vol. 98, Petitjean et al., Velocity fields in a collectively
migrating epithelium, pp. 1790-1800, Copyright (2010), with permission from
Elsevier.
2. Non-equilibrium phase separation: a mechanism for cytoplasmic
organisation
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. Recently an intriguing class
of organelles lacking a membrane is being studied intensely [8]. 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. For example P granules are involved in the asymmetric
division of the Caenorhabditis elegans embryo [9], and stress granules assemble during
environmental stress and protect cytoplasmic RNA from degradation [10] (Fig. 1
a)). Membrane-less organelles are generally spherical, fuse together upon contact
[11, 12], and their components quickly shuttle in and out [13, 14], thus resembling
liquid drops. Indeed, strong experimental evidence indicates that membrane-less
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Novel physics arising from phase transitions in biology 4
organelles are assembled via liquid-liquid phaseLiving and engineered systems rely
on the stable coexistence of two interspersed liquid phases. Yet, surface tension
drives their complete separation. Here, we show that stable drops of uniform and
tunable size can be produced through arrested phase separation in an elastic matrix.
Starting with a cross-linked, elastic polymer network swollen by a solvent mixture,
we change the temperature or composition to drive demixing. Droplets nucleate and
grow to a stable size that is tunable by the network cross-linking density, the cooling
rate, and the composition of the solvent mixture. We discuss thermodynamic and
mechanical constraints on the process. In particular, we show that the threshold for
macroscopic phase separation is altered by the elasticity of the polymer network, and
we highlight the role of correlations between nuclei positions in determining the drop
size and polydispersity. This phenomenon has potential applications ranging from
colloid synthesis and structural color to phase separation in biological cells. separation
[2, 15, 16], a common phenomenon in every day life responsible for example for oil
drop formation in water (Fig. 1 b)). Under the equilibrium condition phase separation
is well understood [17]. However cells are driven away from equilibrium by multiple
energy-consuming processes such as ATP-driven protein phosphorylation [18], which
can potentially affect the phase-separating behavior of membrane-less constituents.
For example P granules do not distribute homogeneously in the cytoplasm but
preferentially to the posterior side of the cell [19], and stress granules form and dissolve
according to environmental cues [20]. The fascinating physics associated to membrane-
less organelles are only beginning to be investigated [12, 21, 19, 11, 22, 23, 24].
In this section, we will start with a brief summary of relevant principles of
equilibrium phase separation in Sec. 2.2. We will then review the latest progress
on phase separation driven out of equilibrium by energy-driven chemical reactions in
Sec. 2.3. Specifically we will focus on a ternary fluid model of the cell cytoplasm where
chemical reactions can convert phase-separating molecules into soluble molecules and
vice versa. We will show how such reactions can control drops assembly and size,
and suppress Ostwald ripening, allowing a collection of organelles to coexist in the
cytoplasm.
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 [17]. The transition from the homogeneous
state to the phase-separated state is controlled by parameters such as temperature,
pressure or concentrations. The set of parameters leading to phase separation are
represented in a phase diagram as shown in Fig. 2, for a ternary mixture composed
of molecules P (red disks), S (blue disks) and C (not shown). The molecular
concentrations are labelled by the same symbols P, S, C. We assume incompressibility
and that all three types of molecules occupy the same volume, so the combined
concentration ψ ≡ P + S + C is homogeneous. The concentration C at any
point in the phase diagram is therefore given by ψ − P − S. Outside the phase
boundary (green curve) the system is homogeneous (“” symbol). Inside the phase
boundary (“♦” symbol) the system phase separates into two phases (“in” and “out”)
of distinct concentrations (
ˆ
P
in,out
,
ˆ
S
in,out
), given by the intersections between the tie-
lines (straight lines) and the phase boundary.
At the equilibrium condition a multi-drop system is unstable due to Ostwald
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Novel physics arising from phase transitions in biology 5
Figure 2. Equilibrium phase diagram of a ternary mixture composed of
molecules P (red discs), S (blue discs) and C (not shown). Outside the phase
boundary (green line) the system is homogeneous (“” symbol). Inside the phase
boundary (“♦” symbol) the system phase separates into two phases “in” and
“out” of distinct concentrations. The coexistence concentrations
ˆ
P
in,out
,
ˆ
S
in,out
are given by the intersections between the tie-lines (straight lines) and the phase
boundary.
ripening that causes large drops to grow and small drops to evaporate [25, 26], and
coalescence caused by the fusion of drops upon contact [27]. Eventually a unique drop
remains in a finite system. Since the crowded environment of the cytoplasm inhibits
the diffusion of macromolecular aggregates [28] we will ignore drop coalescence in this
review and focus on Ostwald ripening.
Ostwald ripening is caused by two ingredients. One is the Gibbs-Thomson relation
that relates the coexistence concentration to the drop radius. For example for the P
concentration we have:
P
in
(R) =
ˆ
P
in
(1)
P
out
(R) =
ˆ
P
out
1 −
ˆ
P
out
l
c
R
!
, (2)
were l
c
is a capillary length and
ˆ
P
in,out
are the coexistence concentrations for a flat
interface (R → ∞, Fig. 2). The smaller the drop, the larger the concentration outside
which is a consequence of the Laplace pressure [17].
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 [26]:
∂P
out
(r, t)
∂t
= D∇
2
P
out
(r, t) , (3)
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Frequently asked questions

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.