Minimal Multi-Scale Model for Liquid-Liquid Phase Separation Regulated by Small Molecules
MMinimal Multi-Scale Model for Liquid-Liquid Phase Separation Regulated by SmallMolecules
Chun-Lai Ren, Pengfei Zhang, and Yu-qiang Ma ∗ National Laboratory of Solid State Microstructures and Department of Physics,Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China Center for Advanced Low-dimension Materials, Donghua University, Shanghai, China
Liquid-liquid phase separation (LLPS) of proteins regulated by small molecules, such as Adenosinetriphophate (ATP), ssDNA and RNA, has been observed. However, the physical mechanism of thisregulation remains largely unknown. Here we develop a minimal multi-scale model to quantitativelystudy the influence of ATP on LLPS of Fused in Sarcoma (FUS). Based on Flory-Huggins theory,we explicitly include multivalent interactions instead of treating all interactions via simple Floryparameters. We find the nonlinear shift of phase diagrams with the increase of ATP concentration,which agrees with the first enhancement and then inhibition of LLPS observed in the experiment.The reason lies in the positive and negative cooperation between ATP-FUS and FUS-FUS interac-tions, where the existence of the ATP bridging plays a key role. Furthermore, we systematicallystudy the phase behavior in a larger parameter space, and introduce a quantity to reflect differentregulation degrees. The present model provides a clear physical picture relating microscopic inter-actions to macroscopic phase diagrams, as well as a good example for adapting a classical theory inpolymer physics to understanding the new phenomena in intracellular phase separation.
Discoveries have shown that phase separation repre-sents an important and ubiquitous mechanism underly-ing many biological processes [1–6]. Intracellular liquid-liquid phase separation (LLPS) is generally believed todrive the formation of membrane-less compartments,such as stress granules in the cytoplasm [7], and Cajalbodies [8] and nucleoli [9] in the nucleus. Liquid droplets,with the typical size of micronmeter, are composed ofprotein or protein/RNA molecules via weak short-rangedmultivalent interactions [10] or intrinsically disorderedregions [11, 12] at the molecular level. Although theirorganizations are typically highly dynamic and accom-panied by the rapid exchange of components with theirsurroundings, their overall size and shape are usually sta-ble for minutes or hours [13, 14]. Thermodynamics hasbeen demonstrated applicable in studying P granule as-sembly [15], nuclear body assembly [16], the formationof nucleolus [17], and is able to provide a simple physicalpicture of phase separations in cell [18, 19]. However, itstill remains a great challenge to build classical thermo-dynamic models to study intracellular phase transition.The biggest difficulty lies in the lack of a quantitativeability of these models to characterize the mechanismat the molecular level. The relationship between mi-croscopic interactions and macroscopic phase behavioris still elusive.Thanks to the rapid development of experimental tech-nology, microscopic mechanisms of protein phase separa-tion have been continuously revealed. One type of theextensively investigated proteins for the phase separa-tion properties is Fused in Sarcoma (FUS) family pro-teins, about 30 proteins in the human genome. Theybelong to a class of intrinsically disordered scaffold pro-teins that share similar domain structures. FUS proteinsare composed of two modules: a low sequence complex- ity domain (PLD) and a RNA-binding domain (RBD)[20–22]. A recent study discovered that the moleculardriving forces for LLPS in FUS are multivalent inter-actions among tyrosine (TYR) residues from PLD andarginine (ARG) residues from RBD [23], where the satu-ration concentration as an indicator of LLPS was quanti-tatively measured. The valency of the multivalent inter-actions was found as a key factor affecting LLPS in FUS.Although the importance of multivalent interactions hasbeen shown [24], quantitative studies on how multivalentinteractions affect the general phase separation behaviorsare still rare.On the other hand, Adenosine triphosphate (ATP) isknown as an energy fuel for biological reactions typicallywith the concentration of micromolar [25–27]. In cells,surprisingly, the concentration of ATP is at millimolar,much higher than that as fuel. To disclose the reasonwhy there is a high concentration of ATP in cell, studieshave shown that the high concentration of ATP is relatedto the dissolution of protein condensers [28, 29]. More-over, RNA and ssDNA also exhibit qualitatively similareffect on intracellular LLPS [30, 31]: an enhancementof LLPS at low concentrations of small molecules andinhibiting LLPS at high concentrations. However, themechanism of such small molecules regulating LLPS isstill rather controversial. For example, it remains un-clear whether ATP modulates LLPS via nonspecific effect(like hydrotrope) or specific interactions [28, 31]. Mean-while, there is emerging evidence that abnormal proteinaggregation is associated with many human diseases, in-cluding cancer, neurodegeneration, and infectious disease[14]. To uncover the influence of small molecules on pro-tein condensates is of both fundamental and physiologicalsignificance.Here we aim to develop a multi-scale theory to build a r X i v : . [ c ond - m a t . s o f t ] F e b a direct relationship between microscopic driving forceand macroscopic phase separation, to offer a quantitativestudy compared with experiments, to clarify the mecha-nism of the regulation of LLPS by ATP, and to expose thegeneral factors regulating LLPS. We explicitly take mul-tivalent interactions into account within the frameworkof the mean field theory, following ideas from the gela-tion of associating polymers [32–34], hydrogen bondingin aqueous polymer solutions [35, 36], and cross-linkingin salt-doped polymer blends [37].For FUS solutions, multivalent interactions are specificbinding between TYR and ARG residues. Each proteinoffers several TYR and ARG residues [23]. The num-ber of TYR or ARG residues is named as the valencyof the multivalent interaction. According to experimen-tal data by Wang et al [23], we model each FUS pro-tein as a chain with length N = 526. Each FUS pro-tein has m TYR residues and m ARG residues, with m = m = 34. Solvents are modeled as chains withthe length of N = 50. The incompressible solution con-sists of n p FUS proteins and n s solvent molecules. Thedimensionless free energy of the reference state, a solu-tion without nonspecific interactions between FUS pro-teins and solvents, βF ref , has purely entropic characterand is written as: βF ref = φN ln φ + (1 − φ ) N ln(1 − φ ),where φ = n p N vV is the volume fraction of FUS. v isthe volume of one residue of protein, which is taken asthe unit volume, and V = n p N v + n s N v is the volumeof system. Next, we take the specific binding betweenTYR and ARG residues into account, assuming that eachTYR/ARG has only one binding site. The partition func-tion aroused by the formation of k TYR-ARG bonds [32–37] is Z binding = C kn p m C kn p m k !( vV ) k exp( kβ(cid:15) ). − β(cid:15) isthe free energy associated with the binding between aTYR and an ARG, which includes the affinity betweenTYR and ARG residues as well as the entropic loss duringthe formation of TYR-ARG bond [33, 35]. More specifi-cally, the specific chemical structures of TYR and ARGside chains appear to be important determinants of theformation of TYR-ARG bond [23]. Such reduction ofconformational entropy is a common character for spe-cific interactions, such as hydrogen bonds and cation- π interactions. According to βF binding ≡ − vV ln Z binding ,the dimensionless binding free energy can be written as βF binding = m φN [ p ln p + (1 − p ) ln(1 − p )]+ m φN (cid:18) − p m m (cid:19) ln (cid:18) − p m m (cid:19) − p m φN (cid:18) ln m φN e + β(cid:15) (cid:19) , (1)where p = kn p m is the fraction of TYR in the formationof TYR-ARG bonds. It reflects the extent of protein net-work crosslinked by TYR-ARG bonds. Thus, the totalfree energy of FUS solution is F tot = F ref + F binding . FIG. 1. Phase separation in FUS solutions without ATP.Phase diagrams as a function of β(cid:15) (a), the faction of TYR-ARG binding (b), and effective interaction for coexistingphases (c). The black curves represent the dilute phase, andthe red ones are the condensed phase. Dash lines are tie-lines relating two coexisting phases. Blue dots are chosen astwo coexisting phases of FUS solutions driven by TYR-ARGbinding. After the minimization with respect to p and φ , we canconstruct the phase diagram. More details are given bySection I in the Supplemental Material (SM) [38].It should be stressed that all the input parameters areobtained from experimental data except the free energyloss of TYR-ARG binding ( β(cid:15) ). Theoretically predictedphase diagram as a function of β(cid:15) is shown in Fig. 1(a). We confirm that although each microscopic spe-cific interaction is in the order of k B T , it is enough tocause macroscopic phase separation. The theoretical pre-diction that the binding free energy is of the order ofthermal energy accords with the experimental definitionof weak multivalent interaction. At β(cid:15) = 2, we findthat the concentration of the condensed phase is about100 times higher than that of the dilute phase, whichagrees with experimental observations on condensers[12].Therefore, − k B T is chosen as TYR-ARG binding freeenergy for FUS solutions. The saturation concentration,which is the threshold of the phase separation, is de-fined as the the concentration of dilute phase when twophases coexist. Here we obtain the saturation concen-tration φ sat = 0 .
006 for FUS. If average protein radiusin solvent is around 10 nm , the saturation concentrationof FUS solution is several µM , quantitatively consistentwith the experiments by Wang et al [23]. At the molecu-lar level, two coexisting phases show distinct TYP-ARGbinding fractions, as shown in Fig. 1 (b). In the di-lute phase, the fraction of TYR-ARG binding is almostzero, implying that FUS proteins do not interact eachother. On the other hand, in the dense case, finite valuesof TYR-ARG binding fractions indicate that multivalentinteractions between TYR and ARG residues are drivingforces leading to the aggregation of FUS proteins via theformation of protein network. From the second derivativeof the free energy (see Section I in SM [38]), we are able toextract the effective interaction χ (cid:48) between proteins andsolvent molecules. For the case of FUS with m = m , χ (cid:48) N = m pφ (1+ p ) > φ and the extent of TYR-ARG binding p ,clearly shown in Fig. 1 (c). In polymer physics, the Floryparameter χ usually is assumed to be independent of theconcentration, which typically represents the nonspecificinteraction between different species. However, here weget the effective interaction is a complex function of theprotein concentration. This result, to some extent, re-flects the difference between nonspecific interaction andspecific interaction at the mean-field level.Next, we investigate the influence of ATP on the phasetransition of FUS solutions. Now the reference system isa FUS solution with freely moving ATP molecules with-out any nonspecific interactions. For the purpose of de-termining the phase behavior, it is more convenient tointroduce a free energy corresponding to a semiclosedsystem [37]. It means that ATP molecules are free toexchange with a reservoir at the chemical potential of µ AT P , which is determined by ATP concentration ( ϕ ) inthe reservoir (See Section II in SM [38]). Thus, the di-mensionless free energy for the reference system becomes: β F ref = φN ln φ + (1 − λφ − φ ) N ln(1 − λφ − φ ) + λφ ln λφ − λφµ AT P , where λ = n ATP n p N is the ratio of the number ofATP molecules and that of FUS residues. According tothe latest study by Kang et al [31], the main interactionbetween ATP and FUS is the binding between ATP andARG residues. Therefore, there are two types of bondsin the solution: TRY-ARG and ATP-ARG bonds, whichcharacterize the FUS-FUS and ATP-FUS interactions,respectively. Given the fact that ATPs are amphiphilicmolecules, each ATP molecule can provide up to two sitesfor binding with ARG residues (i.e., L = 2). The parti-tion function due to the formation of k TRY-ARG bondsand b ATP-ARG bonds can be written as: Z binding = C kn p m C bn ATP L C k + bn p m ( k + b )!( vV ) ( k + b ) exp( kβ(cid:15) + bβ(cid:15) ). n AT P is the number of ATP molecules in the solution. − β(cid:15) is the binding free energy aroused by the formationof a ATP-ARG bond and − β(cid:15) is again that for TRY-ARG binding. The associated dimensionless free energyis given by: β F binding = m φN [ p ln p + (1 − p ) ln(1 − p )]+ Lλφ [ q ln q + (1 − q ) ln(1 − q )]+ m φN (cid:18) − pm m − qLλN m (cid:19) ln (cid:18) − pm m − qLλN m (cid:19) − pφ m N (cid:18) ln m φN e + β(cid:15) (cid:19) − qLλφ (cid:18) ln m φN e + β(cid:15) (cid:19) , (2)where q = bLλn p N is the fraction of ATP participatingin ATP-ARG binding. Therefore, the total free energybecomes F tot = F ref + F binding . It can be seen that p and q are coupled in Eq.(2), indicating the correlation FIG. 2. The influence of ATP on LLPS of FUS solutions. (a)The saturation concentration φ and FUS volume fraction ofdense phase φ (inset); (b) The fraction of TRY-ARG binding p = kn p m and the ratio of the number of ATP-ARG bondsto that of ATP molecules g = bn ATP (inset) as functions ofATP concentration. Subscripts 1 and 2 represent conditionsof dilute and dense phase, respectively. Black curves are thosein dilute phases, and red ones are for dense phases. Solid lineswith solid symbols correspond to the case of ATP with L = 2,and dash lines with hollow symbols reflect the condition of L = 1. Fixed parameters are β(cid:15) = 2 and β(cid:15) = 3 . between ATP-FUS and FUS-FUS interactions. To obtainphase diagrams, more details can be found from SectionII in SM [38].When ATPs are introduced into the FUS solution, theinhomogeneous distribution of ATP happens when phaseseparation occurs. Theoretical calculation shows thatmore ATP molecules prefer to stay in the dense phase (seeFigure S1 [38]), in agreement with experimental observa-tions [28]. Figure 2 (a) shows the change of the saturationconcentration φ with the increase of ATP concentration.The solid black curve shows that φ decreases first andthen increases. This indicates that a small amount ofATP leads to the occurrence of phase transition in thesolution with less FUS concentration; but a large amountof ATP results in the phase separation in the solutionwith more FUS proteins. The inset of Fig. 2 (a) presentsthe volume fraction of dense phase φ , which shows theincrease followed by the decrease. The phase-separatedregion represented by ∆ φ = φ − φ increases first andthen decreases, indicating less ATP enhancing LLPS andmore ATP inhibiting LLPS. To explore the mechanism,we compare ATP with the other case, in which all pa-rameters are the same to ATP except for L = 1. Underthe condition of L = 1, φ increases and φ decreasesmonotonously in Fig. 2 (a). It is obvious that LLPS willbe always inhibited when small molecules with L = 1 areadded. This result implies the importance of two bindingsites provided by ATP molecules. More direct evidencebased on the molecular level is given in Fig. 2 (b). Theinset of Fig. 2 (b) shows the variation of the ratio ofthe number of ATP-ARG bonds to that of ATP with theincrease of ATP concentration. By comparing the twocases, we find that the ability of ATP to form bonds withARG residues is almost twice that of small molecules with L = 1. It implies that a reasonable amount of ATP must FIG. 3. Schematic representation of the influence of ATP onLLPS. (a) Specific binding between TYR and ARG residuesleading to LLPS in FUS solutions. (b) Enhancement of LLPSdue to the ATP bridging effect at a small amount of ATP. (c)Inhibiting of LLPS with the addition of large amount of ATP.Red chains are FUS proteins. Green ‘v’ symbols are ARGresidues. Yellow triangles are TYR residues. Blue diamondsare ATP molecules. bind with two ARG residues in the formation of densephase, which is named as the bridge effect of ATP. Sucheffect, in turn, effectively decreases the TYR-ARG bind-ing as a driving force leading to the phase separation, asshown in the main graph of Fig. 2 (b). Thus, the behav-ior of ATP binding with two ARG residues provides thepositive cooperation with TYR-ARG binding in the for-mation of protein network. However, the bridge effect de-pends on the ATP concentration, which disappears withthe inclusion of large amount of ATP due to the limitednumber of ARG residues. Hence, in the case of largeATP concentration, ATP molecules start to bind withone ARG residue or stay free, similar to small moleculeswith L = 1. Under this circumstance, the formation ofthe protein network via TYR-ARG binding is inhibited,in which the influence of large amount of ATP is calledas the negative cooperation with TYR-ARG binding. Tovividly reflect the influence of ATP depending on its con-centration, a schematic representation is given in Fig. 3.We note that not only ATP, but also RNA and ssDNAhave the similar effect on LLPS [30, 31]. It is thus usefulto study a larger parameter space to explore more gen-eral conclusions. First, from the perspective of protein-protein interaction, a phase diagram in the φ - β(cid:15) planeat various ϕ is given in Fig. 4 (a). The larger bindingenergy between protein residues, the wider the phase-separated regime occurs. Thus, the phase diagram hasa lower critical point. When a small amount of smallmolecules is added, the binodal curve moves downward,implying the enlarged phase-separated region. With theincrease of small molecules, the binodal curve goes up,and the phase-separated region narrows down. The non-linear shift of the binodal curve at different ϕ clearly in-dicates that the enhancement and inhibition of LLPS de-pend on the concentration of small molecules. To furtherquantify such an influence, we introduce S = ∆ φ − ∆ φ , FIG. 4. (a)-(c) Phase diagrams as functions of bindingfree energy of protein residues β(cid:15) , the concentration ofsmall molecule ϕ , and the binding free energy between smallmolecule and protein residue β(cid:15) , respectively. (d)-(f) thequantity S to reflect different extent of the influence of smallmolecules on LLPS corresponding to different cases in (a)-(c),respectively. Fixed parameters in (a) and (b) are L = 2 and β(cid:15) = 3 .
5. Fixed parameters in (c) are β(cid:15) = 2 and ϕ = 0 . where ∆ φ and ∆ φ are the volume fraction differencebetween dense and dilute phases for conditions with andwithout small molecules respectively. S >
S < S reflects the extent of enhancement or inhi-bition. Figure 4 (d) shows the change of S in differ-ent conditions corresponding to Fig. 4 (a). For twocases of ϕ = 0 .
002 and 0.02, S shows an increase be-fore β(cid:15) = 1 . β(cid:15) < .
6, ∆ φ = 0 (i.e., there is no phase sepa-ration) in the pure protein solution. However, when β(cid:15) is larger than 1.6, the value of ∆ φ will increase withincreasing β(cid:15) , and there may exist a change from theenhancement ( S >
0) to inhibition (
S < β(cid:15) = 1 .
2, a closed-loop comes into being, since nophase separation occurs at ϕ = 0. With the increaseof small molecules, the phase-separated region first be-comes larger and then shrinks in cases of β(cid:15) = 1 .
2, 1.6and 2.0. For β(cid:15) = 2 .
4, the phase-separated regions al-ways reduce with increasing ϕ . Accordingly, Fig. 4 (e)exhibits a nonmonotonic variation with the ATP con-centration in the former three cases, namely S first in-creases and then decreases. The maximum S implies theexistence of an optimal concentration of small moleculescharacterizing the strongest enhancement. It can be seenthat the effect of small molecule on LLPS is always en-hancement for small β(cid:15) , such as β(cid:15) = 1 .
2, while in-hibition for large β(cid:15) , like β(cid:15) = 2 .
4. Only when β(cid:15) is medium, its influence includes both enhancement andinhibition. Last, from the view of the interaction be-tween small molecule and protein, Fig. 4 (c) shows thediagram as a function of β(cid:15) . At L = 1, the phase separa-tion is inhibited with the increase of β(cid:15) . This is becausesmall molecule-protein binding from the beginning com-petes with protein-protein binding to become the domi-nate interaction due to the increase of β(cid:15) , causing thatprotein network becomes more and more difficult to formsince ARG residues are occupied by small molecules with L = 1. Once small molecules can bind with two or moreprotein residues, the case is completely opposite. As thecases of L = 2 and 3 show, the phase separation is en-hanced with the increase of β(cid:15) . Under those cases, smallmolecule can bind with more than one ARG residues,which plays the bridge effect and works as the additionaldriving force leading to the formation of protein network.Figure 4 (f) further verifies the complete different influ-ences depending on whether the bridging effect of smallmolecules exists or not.In conclusion, we develop a minimal multi-scale theory,which links the molecular interaction and macroscopicphase diagram directly. We confirm a feasibility schemeof regulating LLPS by introducing small molecules. Theregulation can be achieved by the introduction of addi-tional specific interaction comparable to protein-proteininteraction, where the positive or negative cooperationmay lead to the enhancement or inhibition of LLPS. Thebridging effect resulting from small molecules bindingwith several protein residues is the key to the regula-tion. Although it is a classic mean field theory in polymerphysics, it still can be used to explain the complex mech-anism resulting from the competition between differentmultivalent interactions, to obtain phase diagrams in asimple and clear way, and to show its strong vitality inthe field of intercellular phase separation. By the virtueof the simplicity and instructiveness, it is an effective wayto adapt classical theories to new fields. Acknowledgment.
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