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Novel physics arising from phase transitions in biology

AbstractPhase 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.

Topics: Phase transition (51%)

Summary (4 min read)

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.

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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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Figures (15)
Citations
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Journal Article
TL;DR: A minimal model for an active colloidal fluid in the form of self-propelled Brownian spheres that interact purely through excluded volume with no aligning interaction undergoes an analog of an equilibrium continuous phase transition, with a binodal curve beneath which the system separates into dense and dilute phases whose concentrations depend only on activity.
Abstract: We examine a minimal model for an active colloidal fluid in the form of self-propelled Brownian spheres that interact purely through excluded volume with no aligning interaction. Using simulations and analytic modeling, we quantify the phase diagram and separation kinetics. We show that this nonequilibrium active system undergoes an analog of an equilibrium continuous phase transition, with a binodal curve beneath which the system separates into dense and dilute phases whose concentrations depend only on activity. The dense phase is a unique material that we call an active solid, which exhibits the structural signatures of a crystalline solid near the crystal-hexatic transition point, and anomalous dynamics including superdiffusive motion on intermediate time scales.

214 citations


Journal ArticleDOI
TL;DR: It is found that TAF15 has a unique charge distribution among the FET family members that enhances its interactions with the C-terminal domain of RNA polymerase II, suggesting that positive feedback between interacting transcriptional components drives localized phase separation to amplify gene expression.
Abstract: Membraneless organelles or condensates form through liquid–liquid phase separation1–4, which is thought to underlie gene transcription through condensation of the large-scale nucleolus5–7 or in smaller assemblies known as transcriptional condensates8–11. Transcriptional condensates have been hypothesized to phase separate at particular genomic loci and locally promote the biomolecular interactions underlying gene expression. However, there have been few quantitative biophysical tests of this model in living cells, and phase separation has not yet been directly linked with dynamic transcriptional outputs12,13. Here, we apply an optogenetic approach to show that FET-family transcriptional regulators exhibit a strong tendency to phase separate within living cells, a process that can drive localized RNA transcription. We find that TAF15 has a unique charge distribution among the FET family members that enhances its interactions with the C-terminal domain of RNA polymerase II. Nascent C-terminal domain clusters at primed genomic loci lower the energetic barrier for nucleation of TAF15 condensates, which in turn further recruit RNA polymerase II to drive transcriptional output. These results suggest that positive feedback between interacting transcriptional components drives localized phase separation to amplify gene expression. Wei et al. show that clusters of unphosphorylated RNA polymerase II seed the nucleation of phase-separated condensates of TAF15, which further recruit RNA polymerase II to amplify transcriptional activation.

65 citations


Journal ArticleDOI
Abstract: Endeavoring toward a transferable, predictive coarse-grained explicit-chain model for biomolecular condensates underlain by liquid-liquid phase separation (LLPS) of proteins, we conducted multiple-chain simulations of the N-terminal intrinsically disordered region (IDR) of DEAD-box helicase Ddx4, as a test case, to assess roles of electrostatic, hydrophobic, cation-π, and aromatic interactions in amino acid sequence-dependent LLPS. We evaluated three different residue-residue interaction schemes with a shared electrostatic potential. Neither a common hydrophobicity scheme nor one augmented with arginine/lysine-aromatic cation-π interactions consistently accounted for available experimental LLPS data on the wild-type, a charge-scrambled, a phenylalanine-to-alanine (FtoA), and an arginine-to-lysine (RtoK) mutant of Ddx4 IDR. In contrast, interactions based on contact statistics among folded globular protein structures reproduce the overall experimental trend, including that the RtoK mutant has a much diminished LLPS propensity. Consistency between simulation and experiment was also found for RtoK mutants of P-granule protein LAF-1, underscoring that, to a degree, important LLPS-driving π-related interactions are embodied in classical statistical potentials. Further elucidation is necessary, however, especially of phenylalanine's role in condensate assembly because experiments on FtoA and tyrosine-to-phenylalanine mutants suggest that LLPS-driving phenylalanine interactions are significantly weaker than posited by common statistical potentials. Protein-protein electrostatic interactions are modulated by relative permittivity, which in general depends on aqueous protein concentration. Analytical theory suggests that this dependence entails enhanced interprotein interactions in the condensed phase but more favorable protein-solvent interactions in the dilute phase. The opposing trends lead to only a modest overall impact on LLPS.

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Journal Article
TL;DR: The phenomena of cytoplasmic streaming, elastotaxis, and active mechanosensing find natural explanations within the model of hydrodynamic velocity, concentration, and stress fields in a suspension of active, energy-dissipating particles.
Abstract: We study the interplay of activity, order, and flow through a set of coarse-grained equations governing the hydrodynamic velocity, concentration, and stress fields in a suspension of active, energy-dissipating particles. We make several predictions for the rheology of such systems, which can be tested on bacterial suspensions, cell extracts with motors and filaments, or artificial machines in a fluid. The phenomena of cytoplasmic streaming, elastotaxis, and active mechanosensing find natural explanations within our model.

27 citations


Posted Content
18 Aug 2017
TL;DR: It is indicated that ATP is continuously hydrolysed to deter SG formation under normal conditions, and specific predictions that can be tested experimentally are provided.
Abstract: Stress granules (SG) are droplets of proteins and RNA that form in the cell cytoplasm during stress conditions. We consider minimal models of stress granule formation based on the mechanism of phase separation regulated by ATP-driven chemical reactions. Motivated by experimental observations, we identify a minimal model of SG formation triggered by ATP depletion. Our analysis indicates that ATP is continuously hydrolysed to deter SG formation under normal conditions, and we provide specific predictions that can be tested experimentally.

15 citations


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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 [28]....

    [...]

  • ...The diffusive profile leads to a flux Jout→in = D∇Pout|r=R of molecules P entering the drop at the interface [27]:...

    [...]

  • ...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 [27]:...

    [...]


Journal ArticleDOI
Abstract: The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

4,818 citations


"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 [32]....

    [...]


Journal ArticleDOI
TL;DR: A model is proposed for the evolution of the profile of a growing interface that exhibits nontrivial relaxation patterns, and the exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations.
Abstract: A model is proposed for the evolution of the profile of a growing interface. The deterministic growth is solved exactly, and exhibits nontrivial relaxation patterns. The stochastic version is studied by dynamic renormalization-group techniques and by mappings to Burgers's equation and to a random directed-polymer problem. The exact dynamic scaling form obtained for a one-dimensional interface is in excellent agreement with previous numerical simulations. Predictions are made for more dimensions.

3,929 citations


"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 [62]....

    [...]

  • ...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 [62]....

    [...]


Book
01 Jan 1995
Abstract: This book brings together two of the most exciting and widely studied subjects in modern physics: namely fractals and surfaces. To the community interested in the study of surfaces and interfaces, it brings the concept of fractals. To the community interested in the exciting field of fractals and their application, it demonstrates how these concepts may be used in the study of surfaces. The authors cover, in simple terms, the various methods and theories developed over the past ten years to study surface growth. They describe how one can use fractal concepts successfully to describe and predict the morphology resulting from various growth processes. Consequently, this book will appeal to physicists working in condensed matter physics and statistical mechanics, with an interest in fractals and their application. The first chapter of this important new text is available on the Cambridge Worldwide Web server: http://www.cup.cam.ac.uk/onlinepubs/Textbooks/textbookstop.html

3,887 citations


Journal ArticleDOI
TL;DR: This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments, and highlights the experimental relevance of various semimicroscopic derivations of the continuum theory for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material.
Abstract: This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments. This approach offers a unified framework for the mechanical and statistical properties of living matter: biofilaments and molecular motors in vitro or in vivo, collections of motile microorganisms, animal flocks, and chemical or mechanical imitations. A major goal of this review is to integrate several approaches proposed in the literature, from semimicroscopic to phenomenological. In particular, first considered are ``dry'' systems, defined as those where momentum is not conserved due to friction with a substrate or an embedding porous medium. The differences and similarities between two types of orientationally ordered states, the nematic and the polar, are clarified. Next, the active hydrodynamics of suspensions or ``wet'' systems is discussed and the relation with and difference from the dry case, as well as various large-scale instabilities of these nonequilibrium states of matter, are highlighted. Further highlighted are various large-scale instabilities of these nonequilibrium states of matter. Various semimicroscopic derivations of the continuum theory are discussed and connected, highlighting the unifying and generic nature of the continuum model. Throughout the review, the experimental relevance of these theories for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material is discussed. Promising extensions toward greater realism in specific contexts from cell biology to animal behavior are suggested, and remarks are given on some exotic active-matter analogs. Last, the outlook for a quantitative understanding of active matter, through the interplay of detailed theory with controlled experiments on simplified systems, with living or artificial constituents, is summarized.

2,573 citations


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. 

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.