Critical Phenomena of Liquid-Liquid Phase Separation in Biomolecular Condensates

Junlang Liu Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA, 02138 [email protected] (Dated: May 21, 2021) Biomolecular condensates, an important class of intracellular organelles, is considered to be formed by Liquid-Liquid Phase Separation (LLPS) in vivo. Here, in this report, we focus on the critical region of LLPS, where we believe is the state that most cells live for spatiotemporally-resolved physiological activities. First, a brief introduction of LLPS and recent progress is presented. We then discuss critical phenomena of passive LLPS and active LLPS respectively from mean-field perspective with the help of some reported results. Current descriptions of passive LLPS are mainly based on Flory-Huggins model, which is still within the framework of mean-field Landau-Ginzburg model. For active LLPS, the application of external fields do affect the critical behaviour. This not only implies the importance of taking external energy/biochemical reactions into account for more precise description of in vivo LLPS, but also could be the origins of such rich intracellular activities.

I. INTRODUCTION (such as ATP or post-translational modifications) influence LLPS (i.e., out-of-equilibrium active process)? [4] Intracellular compartmentalization enables dynamical 3) How does heterogeneous cellular environments inter- and accurate information processes during the growth act with condensates? [5] and development of lives. Compared with well-studied membrane-bound organelles, researches on membrane- Although huge amounts of researches have been done, less organelles are still in its infancy. It has been well very few of them took a serious investigation on criticality established during the past decade that membrane-less or universality of LLPS/liquid-gel-solid transition. It’s organelles are biomolecular condensates, comprised of highly possible that cells live in the vicinity of the critical proteins and/or RNAs, with significantly higher den- point of LLPS such that they can function in a timely sity compared with their surroundings but also internal manner as responses to any stimuli or biological signals. mobility and rapid component exchange with surround- [6] In this report, we will focus on the critical phenomena ings, which highlights its liquid-like properties. Exam- of LLPS. Standard and extended Flory-Huggins model ples include protein condensates and ribonucleoprotein will be presented with related analysis of critical points (RNP) condensates such as the nucleolus, Cajal bod- for passive LLPS. A dynamic percolation-like model will ies, stress granules, P-bodies, and so on. Therefore, be discussed for criticality of out-of-equilibrium active Liquid-Liquid Phase Separation (LLPS) is widely recog- LLPS. Future research directions will be concluded in nized as the mechanism underlying formation and func- the final part of this report. tions of those membrane-less organelles [1]. Controlled by external stimuli or biological signals, proteins and/or RNAs can form into biomolecular condensates or dissolve II. PASSIVE LIQUID-LIQUID PHASE into solutions so as to fulfill the biological functions of SEPARATION membrane-less organelles in a spatiotemporally resolved manner.Recent researches also suggest that gelation (i.e., All different kinds of LLPS and liquid-gel-solid transi- liquid-gel transition) could be the intermediate step be- tion in biological context can be roughly categorized into tween the transition from liquid-like biomolecular con- two types: passive process and active process [7]. There densates to solid-like biomolecular condensates, the latter is no obvious deviation from equilibrium thermodynam- of which are usually observed in diseases, such as Amy- ics for passive processes, while due to external work or otrophic lateral sclerosis (ALS) and Alzheimer’s Disease internal energy sources, thermodynamics of active pro- (AD) [2]. cess is fundamentally different (see next section).

All of those exciting experimental discoveries denote One of the simplest models to describe passive LLPS the importance of LLPS and related liquid-gel-solid tran- is Flory-Huggins model. Protein chains (i.e., polypep- sition on better understanding, modulation of biological tides) can be considered as a chain of dipoles. Taken sol- processes and disease treatments. Several topics listed as vent molecules into account, there exists three types of follows are of great interests to the whole community: 1) interactions: peptide-peptide, solvent-solvent, peptide- As a single condensate could contain hundreds of differ- solvent. Strong peptide-peptide interactions enable con- ent kinds of proteins, what are mechanisms that guide densation of protein chains and expel solvent molecules the formation of multicomponent mixtures of proteins into the bulk solution. Through introducing Flory in- and RNAs? [3] 2) How do external energy/chemicals teraction parameter, Flory-Huggins free energy can be 2 expressed as follows [8, 9]: interactions. Following the steps as Eq. (3-4) and expanding near the critical point, we can obtain the crit- φ βf = ln φ + (1 − φ) ln(1 − φ) + χφ(1 − φ), (1) ical point σ3r ≈ 0.5 of extended Flory-Huggins free en- N 1 ergy equation and the same scaling exponent β = 2 (i.e., 1 where φ is the volume fraction of peptides, β equals to |φ − φc| ∝ |σ − σc| 2 ) as the standard Flory-Huggins the- 1 , χ is the Flory interaction parameter, which quan- kBT ory. Therefore, although taking electrostatic interactions tities interactions between peptides and solvent per site into consideration makes Flory-Huggins model more re- and can be expressed as follows: alistic and useful, such modification does not substan- 1 tially change the description of critical behavior. Those χ = zβ ups − (upp + uss) , (2) two equations are in same universal class as mean-field 2 Landau-Ginzburg model. where upp, uss and ups are interaction energy per site for peptide-peptide, solvent-solvent, peptide-solvent interactions, respectively. III. ACTIVE LIQUID-LIQUID PHASE SEPARATION Eq. (1) is a lattice model based on mean-field assumption. The first and second terms of are mixing entropy Most of the LLPS phenomena observed in biological per lattice site and third term is mixing energy per lattice context are influenced or driven by external energy (e.g., site (similar to question 1 of chapter 1 [10]). According mechanical forces by cytoskeleton) or internal energy to sources (e.g., ATP), the active (i.e., out-of-equilibrium) ∂2(βf) 1 1 essence of which inevitably alters boundaries and kinetics = + − 2χ = 0, (3) of phase separation [7]. Here, we will use directed perco- ∂φ2 Nφ 1 − φ lation model, one of the simplest possible models exhibit- ∂3(βf) 1 1 = − + = 0, (4) ing the out-of-equilibrium phase separation, to describe ∂φ3 Nφ2 (1 − φ)2 the dynamic interactions between different protein/RNA regions. For simplicity, directed percolation model can it’s easy to find the Flory interaction parameter at crit- be considered as a network with N nodes, in which each ical point approximates 0.5 (the exact solution is χ = √ c node can be in either active or inactive state. Any ac- N+2 N+1 2N ), which means phase separation will be facil- tive nodes can be inactivated stochastically at the rate itated for Flory interaction parameter larger than 0.5. of µ, while only inactive nodes at the nearest sites of ac- Expanding at the vicinity of critical points, we can have tive nodes can be activated randomly at the rate of λ. 1 There exists sustained activity and stable formation of |φ − φ | ∝ |χ − χ | 2 , (5) c c biomolecular condensates in the ’active phase’ above the 1 critical point. To the contrary, below the critical point which denotes that the scaling exponent β equals to 2 . Although its simplicity, it has been adopted as a frame- is the ’absorbing phase’, where activity will decay into work to understand LLPS [11]. quiescence and no stable biomolecular condensates can be observed. Thus, in this region, stochastic fluctuations Besides short-range dipole-dipole interactions, long- will greatly influence dynamics and is non-negligible. range electrostatic interactions between charged peptides also play an important role in LLPS [12]. Taken that In the large N limit (i.e., continuum limit) and taken into consideration, Overbeek and Voorn extended Flory- stochastic noise into account, directed percolation model Huggins free energy equation as follows: can be described by the following Langevin equation [14]: φ φ p βf = ln + (1 − φ) ln(1 − φ) − α(σφ)3/2, (6) ρ˙(r, t) = aρ(r, t) − bρ2(r, t) + D∇2ρ(r, t) + ρ(r, t)η(r, t), r 2 (7) where r is molecular weight of peptides, σ is charge density, which is deduced according to Debye–Hückel theory where ρ(r, t) is the density of activity at r and time t, a and based on the assumption that the influence of poly- is the distance of order parameter, active rate, towards electrolytes is similar to that of monovalent ions with the critical point, b, D are constant and η(r, t) is Gaus- equivalent charges [13]. α is a solvent constant deter- sian noise with mean zero and variance σ2. Such prob- mined by thermal energy kBT and molar volume of sol- abilistic description of noise distinguishes directed per- vent, which equals to 3.5 for water under room tempera- colation model from other deterministic model, such as ture. Ising model.

We can ﬁnd that ﬁrstly, the Flory interaction term Fluctuations mentioned above normally lead to the so- is neglected here compared with long-range electrostatic called ’avalanching behavior’, in which absorbing phase interactions. Secondly, the mixing entropy of peptides will be perturbed by an activation seed and a series of φ φ (i.e., r ln 2 ) is reduced due to the strong electrostatic events will be triggered before the whole system returning 3 back to absorbing state again. The PDF of avalanche-size According to Ref. [19], we have

S and avalanche-time T in the vicinity of critical point 2v x2 4µ −ν−1 − 4µT are scale-invariant and expressed as follows [15]: F (T ) = Γ(v−1) (1 + β)(4µT ) e (18) 3+β −v−1 − 2 −τ ∼ T = T , (19) P (S) ∼ S , (8) which indicates that α equals to 2 − h . The avalanche- −α 2µ F (T ) ∼ T . (9) size can be obtained through the following integral [18]: Z T Z 1 t t Similarly, the averaged avalanche-size scales with S = v(t)dt ∼ T 2 v˜( )d( ) (20) avalanche-time in the following manner [16]: 0 0 T T where homogeneity assumption is applied and v(t) is per- hSi ∼ T γ , (10) colation displacement at time t. Therefore, γ = 2 and α − 1 τ = 3 − h according to Eq. (11). γ = . (11) 2 4µ τ − 1 Based on the review and discussion above, we have two A more general expression of Langevin equation can interesting findings as follows. Firstly, if external fields 3 be deduced from Eq. (7) as follows: do not exist, we will have α = 2, τ = 2 and γ = 2, which coincide with results of standard random walk as √ ρ˙ = ρξ(t), (12) reported by Ref. [20]. External fields decrease the scaling exponent α and τ, which means due to the random acti- where ξ(t) is the Gaussian noise with mean zero and vation of inactive nodes, it takes longer time for systems variance 2σ2, D∇2ρ(r, t) becomes zero in the mean-field in absorbing phase return back to quiescence after pertur- limit, aρ(r, t) is neglected near the critical point, bρ2(r, t) bation. Secondly, as external fields vary from each other, vanishes through re-scaling. Eq. (12) can be further there is no universal class for such out-of-equilibrium be- rewritten to an equivalent expression [17] as follows in havior. order to guarantee ρ = 0 is the absorbing state:

σ2 √ IV. CONCLUSION ρ˙ = − + ρη(t), (13) 2 In conclusion, we have discussed two simplest possible where η(t) is the Gaussian noise with mean zero and vari- models from mean-field perspective for passive and ac- 2 √ ance σ . Through applying x = ρ and U(x) = λlogx, tive LLPS, respectively and particularly focused on crit- 2 ical phenomena. Flory-Huggins model is widely used to explore passive LLPS. Although extended Flory-Huggins dx σ2 = − + η(t) (14) model may give out a better description through con- dt 4x sidering electrostatic interactions, there still exist sev- dU(x) = − + η(t) (15) eral important factors needed to be taken into account, dx such as charge pattern of protein/RNA chains, hydropho- λ = − + η(t). (16) bic interactions, rigidity of existed condensates. Besides, x there is still the lack of experimental exploration of these critical phenomena. For active LLPS, we find that dif- In the case that we denote variance of the Gaussian noise σ2 ferent external fields may lead to totally different criti- as 2µ, it is easy to find that µ = λ = 4 . Thus, β ≡ cal behaviours, which highlights the importance of tak- λ/µ = 1, which quantities the relationship between the ing external fields (e.g., posttranslational modifications, logarithmic potential and the noise variance and it is well- ATP, mechanical forces, etc.) into consideration due to balanced here. the complexity and heterogeneity of intracellular environments. In the future, more experimental researches can In order to further mimic dynamic interactions be- be conducted at the vicinity of critical point for the bet- tween different protein/RNA regions, we can introduce ter understanding of in vivo LLPS, while in the theory an external field allowing the spontaneous activation of side, more universal descriptions of this intricate process any inactive sites at a fixed rate h [18]. In this case, is highly demanded. Eq. (14) becomes

dx σ2 − 4h ACKNOWLEDGMENTS = − + η(t), (17) dt 4x I would like to thank Professor Mehran Kardar for his the balance between the logarithmic potential and the great eﬀorts and wonderful teaching during the whole h noise variance is broken down as β becomes 1 − µ . semester. This course leads me, a completely layman before taking it, into this exciting ’ﬁeld’ and endows me a whole new perspective for my future researches. I also would like to thank Mr. Alex Siegenfeld for his helpful recitation. 4

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