Spatial signal amplification in cell biology: a lattice-gas model for self-tuned phase ordering
Teresa Ferraro, Antonio de Candia, Andrea Gamba, Antonio Coniglio
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] N ov Spatial signal amplification in cell biology: a lattice-gas model for self-tuned phase ordering
T. Ferraro, ∗ A. de Candia, A. Gamba,
2, 3 and A. Coniglio Dip. di Scienze Fisiche, Universit`a di Napoli “Federico II”, via Cintia, 80126, Napoli, Italia Politecnico di Torino and CNISM, Corso Duca degli Abruzzi 24, 10121 Torino, Italia INFN, via Pietro Giuria 1, 10125 Torino, Italia
Experiments show that the movement of eukaryotic cells is regulated by a process of phase separation oftwo competing enzymes on the cell membrane, that effectively amplifies shallow external gradients of chemicalattractant. Notably, the cell is able to self-tune the final enzyme concentrations to an equilibrium state of phasecoexistence, for a wide range of the average attractant concentration. We propose a simple lattice model inwhich, together with a short-range attraction between enzymes, a long-range repulsion naturally arises fromphysical considerations, that easily explains such observed behavior.
PACS numbers: 64.60.My, 64.60.Qb, 87.16.Xa, 87.17.Jj, 82.39.Rt, 82.40.Np
Specific moments of the life of a cell living in a multicel-lular community, such as migration, proliferation, organiza-tion in layers or complex tissues, imply spatial organizationalong some axis of direction. The original spatial symme-try of the cell must be broken to adapt to a highly structuredanisotropic environment. For instance, migrating cells mustorient towards sources of chemical attractants, mitotic cellsmust orient along the spindle-pole axis to bud daughter cells,epithelial cells must recognize the inner and outer part of tis-sues to define organ boundaries. From a physical point ofview, these spatial organization phenomena may be seen asself-organized phase ordering processes, where the cell state,spontaneously, or because driven by an external field, decaysinto a state of coexistence of two or more chemical phases,spatially localized in different regions in order to define a frontand a rear, a top and a bottom, an outer and an inner part.Local thermodynamic equilibrium requires precise tuning ofchemical potentials to nongeneric values to allow the coex-istence of different phases. To implement this requirementin a robust way, capable of giving a stable response over awide range of stimulation situations, biological systems mustbe endowed with self-organized tuning mechanisms leadingto phase coexistence and polarization.Directional sensing in chemotacting eukaryotic cells pro-vides a beautiful illustration of these principles [14]. At theheart of directional sensing lies a chemical phase separationprocess taking place on the inner surface of the cell mem-brane [3]. The main players of the process are the enzymesphosphatidylinositol 3-kinase (PI3K), and phosphatase andtensin homolog (PTEN), which catalyze the switch of thephospholipid phosphatidylinositol between the bisphosphate(PIP ) and the trisphosphate (PIP ) states. The phospholipidsare permanently bound to the inner face of the cell membrane,while the two enzymes diffuse in the cell volume and becomeactive when they are adsorbed on the membrane. PI3K ad-sorption takes place through binding to receptors of the ex-ternal attractant. PTEN adsorption takes place through bind-ing to the PTEN product, PIP , a process which introduces anamplification loop in the system dynamics [3, 5]. A secondamplification loop provided by PI3K binding to PIP [6] hasbeen recently observed. Although there are no relevant enzyme–enzyme orphospholipid–phospholipid interactions, the above describedcatalytic processes, together with phospholipid diffusion onthe cell membrane, mediate an effective short-range attractiveinteraction among enzymes of the same type. This interactiondrives the system towards phase separation in a PTEN-richand a PI3K-rich phase [3, 4]. Two different regimes of mem-brane polarization may be distinguished. In the presence ofan attractant gradient, anisotropy driven polarization is real-ized in a time of the order of a few minutes, and results in theformation of a PI3K-rich patch on the membrane side closer tothe attractant source and of a PTEN-rich patch in the comple-mentary region [5]. The process works as an efficient gradientamplifier: a few percent gradient is sufficient to completelypolarize the cell membrane. The orientation response is re-versible: by inverting the gradient direction the polarizationorientation is also inverted. On the other hand, cells exposedto uniform distributions of attractant polarize in random di-rections over a longer timescale. The average concentrationof attractant is of crucial importance, as shown by experimen-tally observed dose-response curves [7]: directional sensingdoes not take place neither at very low nor at very high attrac-tant levels, and there exists an optimal attractant concentrationsuch that the cell response is maximal.On the basis of a simple analogy with the physics of bi-nary mixtures, one would expect that the coexistence betweenthe PI3K-rich and the PTEN-rich phase would require a finetuning of the chemical potential difference between the twospecies. Surprisingly, phase separation takes place insteadfor a wide range of absolute concentration of the attractant,and therefore of absolute values of the chemical potential forPI3K adsorption. To explain this mechanism we propose herea simple lattice-gas model in which, together with the effec-tive short-range attraction between enzymes, a long-range re-pulsion naturally arises from the finiteness of the enzymaticreservoir, that easily explains the observed behavior. Model –
We represent the cell membrane by a square lat-tice of size L with N sites, using periodic boundary condi-tions. The sites i occupied by PI3K (PTEN) are describedby a S i = + −
1) spin [15]. We denote by N ± tot the totalnumber of ± N ± tot = N ± free + N ± . The probabil-ity that a PI3K enzyme binds to site i is proportional to thenumber of cytosolic PI3Ks and to the density of binding sites(activated receptors with local concentration c i and PIP ’s).As a first approximation, the PIP concentration can be as-sumed to be linearly dependent from the density of PI3Ks.This gives, on site i : P ( − → + ) (cid:181) " c i + a + c + + b + (cid:229) j ∈ ¶ i S j ! N + free (1)where a + , b + , c + are functions of the chemical reaction rates,and ¶ i are the nearest neighbors of i . Similarly, the probabilitythat a PTEN molecule binds to site i is proportional to thenumber of free PTENs, and to the concentration of PIP : P (+ → − ) (cid:181) a − c − − b − (cid:229) j ∈ ¶ i S j ! N − free (2)We interpret D H = ln [ P ( − → + ) / P (+ → − )] as an en-ergy difference (in units of k B T ) between states S i = + S i = −
1, depending both on the local field (cid:229) j ∈ ¶ i S j and on thenumber of cytosolic PI3Ks and PTENs. Since N + + N − = N ,we can express N + free and N − free as functions of the magnetiza-tion m = ( N + − N − ) / N . Linearizing D H around (cid:229) j ∈ ¶ i S j = m =
0, we obtain: D H = − J (cid:229) j ∈ ¶ i S j − h i + l m (3)where J = (cid:16) a + b + c i + a + c + + b − c − (cid:17) , h i = ln (cid:16) + c i a + c + (cid:17) − h , with h = ln (cid:16) a − c − m + a + c + m − (cid:17) , and l = (cid:0) m + + m − (cid:1) , with m ± = N ± tot / N −
1. If b + c + < b − c − we can neglect the dependence of J on the attractant concentration c i .Eq. (3) corresponds to the variation of the Hamiltonian H = − J (cid:229) h i j i S i S j − (cid:229) i h i S i + l N (cid:229) i < j S i S j . (4)The model (4) contains a short-range ferromagnetic interac-tion representing the effective attractive interaction betweenenzymes, a long-range antiferromagnetic interaction whichresults from the finiteness of the cytosolic enzymatic reser-voir, and an external site-dependent field representing the ef-fect of the attractant. The latter depends on the concentra-tion c i of activated receptors, which we take proportionalto the concentration of external attractant, in the form c i = c ( + e sin p x i L sin p y i L ) .When h i is independent of i , the second and third term ofEq. (4) can be written (apart from a constant) as N l ( h l − m ) ,so that energy minimization leads the system to self-tune tothe magnetization value h / l . Eq. (3) shows that S i is sub-ject to the action of an effective external field h eff , i = h i − l m .The value h eff , i measures the degree of metastability of the PTEN phase, and tends to zero during the self-tuning evo-lution of the system. To realize the phase separation, J hasto be greater than the critical value for the two-dimensionalIsing spin model ( J ≃ . h ≥
1. We set for definiteness J = h = l = a + c + = Simulations –
We study by Monte Carlo simulations thedynamics and the final state attained by the system, using asquare lattice of size L = e =
0, which corresponds to uniform stimulation. In the ab-sence of stimulation ( c =
0, implying h = − m = −
1) themembrane is uniformly populated by PTEN molecules. Set-ting c > h > − m tends asymptotically to h , while the effective field h eff tends to zero (Fig. 1), realizing the condition for phase co-existence. -1-0.500.51 1 10 10 e =0 h=1h=0h=-0.6h=0.6h=-1 time (MCS) m a gn e ti za ti on p e r s p i n m up down h=1h=0.6h=0h=−0.6h=−1 h=−1h=0h=0.6h=1h=−0.6 FIG. 1: Self-tuning dynamics in the presence of a uniform activa-tion field h . The magnetization m grows to compensate the externalactivation field h . On the right, equilibrium states corresponding todifferent values of h . After a rapid nucleation phase, a domain coarsening dy-namics follows: large domains grow and smaller onesshrink [8] (Fig. 2).
FIG. 2: Coarsening dynamics leading to random membrane polar-ization in the presence of a uniform activation field.
The final equilibrium state is characterized by the coexis-tence of the PI3K and the PTEN phase, localized in two com-plementary clusters. The equilibrium position of the PI3Kcluster, which determines the direction of cell movement,is random. This behavior is consistent with experiments inwhich cells exposed to a uniform attractant distribution orientrandomly (stochastic polarization) [7].In the presence of a gradient in the chemical attractant( e >
0) the PI3K cluster localizes around the maximum of theattractant density. To measure the polarization degree we de-fine the following order parameter: s = (cid:229) Ni ( c i − c ) S i (cid:229) Ni | c i − c | , (5)which is both a measure of the degree of order in the systemand of the correlation of the center of the PI3K cluster withthe maximum of the attractant density (Fig. 3). e =0.05 h = - 0.6h = 0h = 0.6h = 0.8 time (MCS) o r d e r p a r a m e t e r s up down h=0h=0.6h=−0.6h=0.8h=0.8h=−0.6h=0.6h=0 FIG. 3: Time evolution of the order parameter for different valuesof the activation field h , and for a fixed value e = .
05 of the gra-dient. At the end of the polarization process the PI3K cluster (grayin the panels on the right) is centered around the point of maximumattractant stimulation (crosses).
Dose-response curve –
Simulations reproduce the qual-itative behavior of experimentally observed dose-responsecurves [7, 9], showing no response for either very high orvery low values of the attractant concentration, and optimalresponse for intermediate values (Fig. 4). This effect can beexplained as follows. For very low c the critical radius forpatch nucleation is larger than the size of the cell, and no po-larization is possible. For very high c , such that h >
1, theequilibrium magnetization is 1, the whole system is uniformlypopulated by the PI3K phase, and again no polarization is pos-sible. Polarization is possible only for values which are inter-mediate between these two limit cases. -8 -7 -6 -5 -4 attractant concentration (M) % o f o r i e n t a ti on -1 attractant concentration c e qu ili b r i u m v a l u e o f s a) b) attractant concentration (M) attractant concentration c FIG. 4: a): Orientation degree of a population of cells as a func-tion of the attractant concentration, for a constant gradient (adaptedfrom [7]). b): Simulated equilibrium values of the order parameter s as a function of the attractant concentration c , for a constant gradient. Reversibility –
Polarization induced by the gradient is re-versible. By changing the gradient direction after the sys-tem has reached equilibrium, the position of the PI3K clus-ter adjusts to the new direction in a finite time (not shown).This effect reproduces the observed reorientation of eukary-otic cells under varying attractant gradients observed in theexperiments [10]. Interestingly, after changing the sign of therelative gradient we observed reorientation taking place by acollective movement of the PI3K cluster, and not by its evap-oration and successive recombination.
Gradient amplification and polarization time –
The tran-sient states are characterized by a coarsening dynamics withthe appearance of scaling laws in the process of domain for-mation [8, 11, 12]. Our simulations show that, for a conditionof uniform distribution of attractant, in the initial coarseningstage the average cluster radius h r i grows approximately as t / . In Fig. 5 the inverse length of the total cluster boundaryis plotted against time [16]. -6 -5 -4
10 10 e = 0 e = 0.001 e = 0.01 e = 0.1 t t time (MCS) ( bound a r y l e ngh t ) - FIG. 5: Time evolution of the inverse length of the total clusterboundary for different values of the gradient e . The dotted linesshow the slope of the power-law behaviors characterizing the growthregimes dominated, respectively, by the uniform component of theattractant ( ∼ t / ) and by the gradient ( ∼ t ). Arrows show the posi-tion of crossovers between the two scaling behaviors. We define the polarization time t p as the time for which theorder parameter s reaches 90% of its equilibrium value. Ifthe attractant is uniformly distributed the coarsening processstops when the average cluster radius becomes of the order ofthe cell size, r ∼ L , implying that the spontaneous cell polar-ization time scales as t p ∼ / L .In the case of an attractant gradient we observe instead adouble scaling behavior. For t < t e , where t e is a crossovertime depending on the amplitude of the gradient e , clustergrowth proceeds approximately as in the uniform case, while,for t > t e , the process of polarization becomes anisotropic,and the average cluster size grows approximately linearly in t (Fig. 5). The presence of this double scaling law implies thatthe polarization time behaves as t p ∼ a + b / e + c / e (Fig. 6).We can understand the double scaling law as follows. In thepresence of an attractant gradient polarization takes place intwo steps. In the initial (tuning) step the gradient of the attrac-tant is negligible with respect to the uniform component of theattractant and cluster growth is approximately unaffected byits presence. In the meanwhile, free enzymes shuttle from thecytosolic reservoir to the membrane, lowering the chemicalpotential for further cluster growth and effectively cancelingout the effect of the uniform component of the attractant. Thisprocess continues until times of order t e , when only the effectof the gradient component is left. At this point, fast polariza-tion in the direction of the gradient takes place. t p =a+b/ e +c/ e e po l a r i za ti on ti m e t p -3 -2 -1h=0 e =0.001 e =0.01 e =0.1 time (MCS) o r d e r p a r a m e t e r FIG. 6: Polarization time as a function of e , and time evolution of theorder parameter s for different values of e (inset). The anisotropic stage of cluster evolution leading to di-rected polarization occurs only if t e < t p . Otherwise, the pres-ence of a gradient of attractant becomes irrelevant and only thestage of isotropic patch growth actually occurs. The crossovertime t e increases with decreasing e until it becomes of the or-der of t p , implying the existence of a lower threshold e th ofdetectable gradients. For e > e th anisotropy-induced polar-ization is much faster than spontaneous polarization. Thisexplains the experimentally observed effect of gradient am-plification in chemotacting cells and the observation [9] of alower threshold of detectable gradients, below which there isno directional sensing. Our results also confirm the theoreticalpredictions of [12]. Discussion –
We have introduced a simple lattice-gas modelfor the process of eukaryotic directional sensing, which repro-duces important aspects of the observed phenomenology andsheds light on the underlying physical mechanism. The modelmaps signaling molecules and enzymes in spin variables, andthe effective interaction between enzymes on the membraneinto a ferromagnetic coupling. Enzymes shuttling from thecytosolic reservoir to the membrane is shown to provide afundamental self-tuning mechanism which drives the systemtowards phase coexistence and polarization, by counteractingthe effect of the external activation field. In the presence ofan attractant gradient this mechanism cancels out the isotropiccomponent of the attractant distribution in a first (tuning) stageof cluster growth, preparing the ground for fast directed polar-ization in the direction of the gradient in the next stage. Thecontrol provided by enzyme shuttling is encoded in the cou-pling of the effective magnetic field h eff with the local orderparameter m , thus realizing an effective long-range repulsionbetween enzymes and introducing in the model an element ofself-organization. The existence of two distinct stages in clus- ter evolution when an attractant gradient is present is signaledin the model simulations by the emergence of a double powerlaw for the time evolution of clusters of signaling molecules.This shows up in the dependence of directed polarization timefrom the gradient: for e ≪ t e scales as e − .Previous models of eukaryotic polarization postulated theexistence of a global inhibitor, which was needed to cancelout the average value of the attractant leaving only the gradi-ent component (see [13] for a review of previously proposedmodels of eukaryotic chemotaxis). The weak point of this ap-proach is the necessity to fine-tune the activity of the globalinhibitor in order to attain perfect cancellation of the averageattractant value. In our scheme instead, the exchange of chem-ical factors between the cell membrane and a finite cytoso-lic reservoir realizes a self-tuning mechanism which naturallyleads to an equilibrium state characterized by the coexistenceof two distinct phases, similarly to what happens in the caseof first-order phase transitions in a closed liquid-gas systemor in the precipitation of a supersaturated solution. The sens-ing mechanism encoded in the model is particularly robust,allowing the cell to respond over a wide range of attractantconcentrations.Our model explains the experimentally observed behaviorof chemotacting cells and reproduces several effects, suchas gradient amplification, typical dose-response curves, re-versibility of orientation. More generally, it shows that im-portant biological functions may be described at a physicallevel as self-organized phase transition processes.We gladly acknowledge useful discussions with S. Di Talia,I. Kolokolov, V. Lebedev and G. Serini. ∗ Electronic address: [email protected][1] A. Ridley et al. , Science , 5651 (2003).[2] D.A. Lauffenburger and A.F. Horwitz, Cell , 359 (1996).[3] A. Gamba, et al. Proc. Natl. Acad. Sci U.S.A. , 16927(2005).[4] A. de Candia et al. , Sci. STKE , pl1 (2007).[5] M. Iijima and P. Devreotes, Cell , 599 (2002).[6] M. Dance et al. , J. Biol. Chem. , 23285 (2006).[7] S.H. Zigmond, J. Cell. Biol. , 606 (1977).[8] E.M. Lifshitz, and L.P. Pitaevskii, Physical Kinetics (PergamonPress, 1981).[9] L. Song et al. , Eur. J. Cell Biol. , 981 (2006).[10] C. Janetopoulos et al. , Proc. Natl. Acad. Sci. U.S.A. , 16606(2004).[11] A. Bray, Adv. Phys. , 357 (1994).[12] A. Gamba et al. , Phys. Rev. Lett. (2007).[13] P. Devreotes and C. Janetopoulos, J. Biol. Chem. , 20445(2003).[14] The biological facts presented below are taken, where not oth-erwise specified, from the reviews [1] and [2].[15] One can imagine performing a coarse-graining of the system onan appropriate length scale and associating a + −−