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Self-Assembly of a Filamen by Curvaure-Inducing Proeins James Kwiecinski, S. Jonahan Chapman, Alain Goriely Oxford Mahemaical Insiue, Andrew Wiles Building, Woodsock Road, Oxford, OX 6GG, Unied Kingdom

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Self-Assembly of a Filamen by Curvaure-Inducing Proeins James Kwiecinski, S. Jonahan Chapman, Alain Goriely Oxford Mahemaical Insiue, Andrew Wiles Building, Woodsock Road, Oxford, OX 6GG, Unied Kingdom Absrac We explore a simplified macroscopic model of membrane shaping by means of curvaure-sensing BAR proeins. Equaions describing he inerplay beween he shape of a freely floaing filamen in a fluid and he adhesion kineics of proeins are derived from mechanical principles. The consan curvaure soluions ha arise from his sysem are sudied using weakly nonlinear analysis. We show ha he sabiliy of he filamen s shape is compleely characerized by he parameers associaed wih proein recruimen and esablish ha in he bisable regime, proeins aggregae on he filamen forming regions of high and low curvaure. This paern formaion is hen followed by phase-coarsening ha resolves on a ime-scale dependen on proein diffusion and drif across he filamen, which conend o smooh and mainain he paern respecively. The model is generalized for muliple species of BAR proeins and we show ha he sabiliy of he assembled shape is deermined by a compeiion beween proeins aaching on opposing sides. Keywords: Self-Assembly, Membrane Remodeling, BAR Proeins. Inroducion Self-assembly is a ubiquious phenomenon ha exiss over a large range of lengh scales in sysems boh physical and biological in naure. Examples range from he asronomically large, such as he formaion of galaxies and planeary sysems [], o he nano-scale, such as he echnology of DNA origami [37]. In is mos basic definiion, such self-assembling sysems are: i) comprised of pars or componens ha exhibi ineracion; ii) a hermodynamic nonequilibrium iniially, bu end o equilibrium; iii) hermodynamically closed [, 38]. Self-assembly is a process of energy minimizaion ha ends in a final, well-defined srucure ha is uniquely deermined by he properies of he ineracing componens which remain unchanged during he ransiion o hermodynamic equilibrium. Global order in he sysem is encoded in he iniial se-up and he specific relaionships ha exis beween componens; no addiional energy is necessary o drive he process [5]. An ineresing example of self-assembly is found in he shaping of biological membranes, in paricular lipid bilayers, ha occur a he cellular level. Such obecs are imporan building blocks which no only coa pars of he cell, such as he nucleus and he endoplasmic reiculum [4], bu also form independen biological obecs wihin he cell, such as vesicles and ubules which are necessary for he inra-cellular ranspor of wases, nuriens, and proeins [8, 46]. The primary mechanism believed o be responsible for he highly curved geomeries observed involves he recruimen of membrane-shaping proeins from he cellular fluid, such as he BAR Bin/Amphiphysin Rvs) and ENTH Epsin N-Terminal Homology) proein families [3, 9, 35, 5]. Such proeins bind direcly ono he bilayer by means of elecrosaic ineracions and bend i by he inserion of amphipahic helix funcional groups [3, 3], wih he magniude of he induced curvaure dependen on he deph of inserion ino he lipid monolayer [] and he number of aaching proeins [44]. Moreover, he BAR and ENTH families ac as sensors of curvaure [6, 34], meaning ha he shape of he membrane deermines he adhesion kineics of he proeins. In oher words, we have a sysem where he curvaure of he membrane regulaes he concenraion of aached proeins, and vice-versa, in an ineracing process [4]. Corresponding auhor addresses: James Kwiecinski), S. Jonahan Chapman), Alain Goriely) Preprin submied o Elsevier December 5, 6 Previous invesigaions on he subec have been from wo perspecives; namely, compuaional, involving large-scale coarse-grain simulaions [, 4, 9], and heoreical, which consider he minimizaion of membrane free energies or elecrical poenials o deermine he naure of equilibrium configuraions. In he former, many ineresing aspecs of BAR proeins have been found, including he linear aggregaion of proeins, leading o membrane ubulaion [43, 49], and membrane fissioning, leading o closed ubular neworks [4]. From a heoreical poin of view, he effecs of single BAR proeins, in he conex of elecric fields and poenials [5], and a coninuum of hem aaching o he lipid bilayer have also been sudied, bu no o he same exen as compuaional models. In paricular, equilibrium configuraions of he self-assembled sysem, and he role ha BAR proeins play in he sabiliy of hese final shapes, are considered. Early research on he subec sudied no he formaion of vesicles or oher independen biological obecs, bu raher he oscillaions ha can exis on he cell-membrane known as circular dorsal ruffles [4, 33]. I is found ha BAR proeins provide a sabilizaion of he geomery; a poin which has been furher invesigaed in he conex of fla membrane geomeries [39] and pearling insabiliies in cylindrical geomeries [3]. As a saring poin o undersand his phenomenon, we derive he simples, non-rivial sysem ha allows us o explore he inerplay beween an underlying geomery and curvaure-sensing proeins. We focus on he shaping of a filamen and derive a macroscopic model for ime-dependen self-assembly using conceps from coninuum and saisical mechanics. The resul is a hermodynamically consisen sysem of equaions in erms of experimenal parameers ha allows us o furher explore he role of he filamen mechanics, he adhesion kineics of he aaching proeins, and he ineracions beween hese componens.. Mahemaical Model Our mahemaical model is based on he following assumpions: i) We consider a hermodynamically closed sysem in D space which only conains he main coninuum, he curvaure-inducing proeins, and he ineracions beween hese. There is no forcing or energy inpu from he ouside environmen; ii) We ake he coninuum o be a D filamen; explicily, an elasic rod which is inexensible and unshearable wih consan lengh L ha is parameerized wih an arc-lengh coordinae s [, L]. This geomery is a simplificaion of he D lipid bilayer wihou ransverse mechanical effecs and area dilaion, despie he membrane being able o endure srains of -3% [36]. The dependen variable of ineres is he filamen curvaure κ, s) a ime ; iii) The coninuum freely floas in a fluid which is populaed wih a single ype of BAR proein modeled as a hin filamen wih consan curvaure. The proeins induce curvaure along one principal direcion which is always aligned wih ha of he main filamen and hey have a hermodynamically favorable arge curvaure κ. The number of bound proeins per uni lengh is given by c, s). Addiionally, we make wo imporan hermodynamic assumpions. iv) We suppose ha proeinmembrane ineracions are a hermodynamic equilibrium wih respec o energy exchange beween hese componens. A number of heoreical models have made his same assumpion [3, 33] as well as compuaional models [45]. In he laer case, resuls prediced by simulaions have been experimenally verified [4], which suggess ha his assumpion is valid when sudying qualiaive aspecs of he mahemaical model. v) There are hree sages in he proein aachmen corresponding o he unbound, ransiion, and bound saes. An unaached proein floaing in he cellular fluid is assumed o have zero bending energy unbound sae), bu when he proein aaches o he main filamen, i deforms iself o mach he curvaure of he filamen and acquires bending energy dicaed by bending siffness B BAR bound sae). We suppose here is an inermediary sep o aachmen whereby he proein acquires bending energy deermined by bending siffness B s, which is no necessarily he same as B BAR, and we assume ha his ransiion sae is sufficienly long-lived such ha hermodynamic equilibrium is reached. We now derive equaions describing he ime-evoluion of κ, s) and c, s)... Equaions for Filamenary Mechanics We model he membrane cross-secion as a Kirchhoff elasic filamen whose moion is confined o a plane [, 9]. The posiion of he filamen s cenerline is denoed by r, s) and he curve is geomerically described by a local direcor basis {d, d, d 3 } which is righ-handed and orhonormal. The vecors d 3 and d are he angen and normal Frene vecors respecively, whils d poins perpendicularly ouwards from he plane and is consan wih respec o boh and s see Fig. ). To complee he geomeric descripion, we inroduce ,s) c,s) d,s ) d 3,s ) d,s ) d 3,s ) BAR BAR c,s),s) d 3,s ) d,s ) d,s ) P d 3,s ) P Figure : The mahemaical model of D self-assembly as i evolves from = o =. A filamen of curvaure κ, s), ha is described by local normal and angenial vecors d, s) and d 3, s), is submersed in a fluid populaed wih a single ype of BAR proein which has consan inrinsic curvaure κ. Unbound proeins in he layer of hickness τ BAR around he filamen shown as black) have consan concenraion P, and can aach ono he filamen resuling in a number per uni lengh c, s) shown as red). The aached proeins give he filamen an inrinsic shape ha evolves in ime due o hermodynamic consrains on he proein adhesion kineics. Color online) a srain vecor u = κd and spin vecor w = w d, wih w being a measure of he angular velociy of he direcor basis, so ha we have he following kinemaic relaions: r s d i s d i = d 3, ) = u d i, ) = w d i. 3) Defining he velociy of he rod in he local basis v = v d v 3 d 3 = r/, we use )-3) o obain geomeric consrains for v and v 3 : as well as a compaibiliy relaion, given ha d i / s = d i / s: = v s κv 3 w, 4) = v 3 s κv, 5) κ = w s. 6) Considering he mechanics of he coninuum, we suppose ha he only conribuion o he applied force comes from fluid drag, which is proporional o he velociy of he filamen v. More specifically, he applied force per uni lengh is f = f d f 3 d 3 = η v d η 3 v 3 d 3, where η and η 3 are he drag coefficiens per uni lengh in he normal and angenial direcions. Given ha we are ineresed in moions ha occur a low Reynolds numbers, i is reasonable o assume ha he inerial erms can be negleced so ha he resuling dynamics are firs order in ime [8]. Furhermore, we use slender body heory o simplify he applied force so ha η = η 3, where η 3 = πµ/a ln L/r) for a filamen of lengh L, cross-secional area A, and radius r L which is submersed in a fluid of dynamic viscosiy µ [7]. Inroducing he resulan force n = n d n 3 d 3 and momen m, we balance linear and angular momena o obain wo equaions of moion 3 [3]: n s f = ρa r m s d 3 n = ρi, 7) ) d d, 8) for a filamen of consan densiy ρ and second momen of area I. Lasly, o relae mechanical sresses o srains, we use a linear consiuive relaion of he form: m = EI κ γc) d, 9) where E is he Young s modulus of he rod, he number of proeins per uni lengh c akes on a role ha is analogous o he inrinsic curvaure, and γ is a dimensionless measure of he curvaure he proeins induce. The laer is included o accoun for he observaion ha he locally induced curvaure is proporional o he number of proeins ha have aached o he sie [44] as well as he deph of amphipahic helix inserion []. Relaion 9) is derived from an elasic energy which is quadraic in he curvaure of he filamen, however i has he same form as he Helfrich energy of a lipid bilayer reduced o a D geomery []. Furhermore, using 9) incurs an error O ω ) where [7]: ω = max s [,L] { κ, s) r, c, s) r, r/l}. ) Combining )-3) wih 7)-9) and he aforemenioned simplificaions, we include 4)-6) so ha we have he following sysem of 6 equaions for 6 dependen variables {v, v 3, w, κ, n, n 3 }: ρa v ρa v 3 ρi w κ = n s κn 3 ρaw v 3 η v, ) = n 3 s κn ρaw v η v 3, ) = EI s κ γc) n, 3) = w s, 4) = v s κv 3 w, 5) = v 3 s κv. 6) To close he sysem, we impose free-ends boundary condiions, which implies ha n = m = a s = {, L}. In erms of he dependen variables, his corresponds o n = n 3 = and κ = γc applying a boh ends of he filamen... Equaion for Proein Adhesion Kineics We suppose here are freely floaing proeins in he cellular fluid which is aken o occupy an area much larger han ha of he filamen. The soluion is assumed o be of sufficienly high concenraion and wellmixed such ha he proeins are uniformly disribued. We define a region of consan widh around he filamen ha is one BAR proein hick τ BAR and presume ha proeins can aach o he filamen or deach from i in his region. The number of freely floaing proeins per uni area of he region is P whils he number per uni lengh of proeins bound o he filamen is c see Fig. ). We summarize he kineics as a chemical reacion of he form: P K f κ)τ BAR c, 7) K rκ) where K f κ) and K r κ) are he forward aachmen) and reverse deachmen) raes of reacion, boh of which depend on he curvaure of he filamen. 4 We apply he principle of mass acion o 7) and wrie he ime-evoluion of c as: [ ] c = K Kf κ) r κ) K r κ) τ BARP c. 8) If he assumpions regarding he cellular fluid hold in regards o is spaial uniformiy and high concenraion, and we addiionally suppose ha τ BAR is much smaller han he lengh of he filamen L, hen changes o P will be insanly replenished by he surrounding soluion. The laer assumpion is biologically feasible given ha τ BAR 9 m, wih he upper bound corresponding o he hickness of DNA [6], and L 7 from discussions on ube radii [5]. As such, P is effecively consan and is reaed as such in he forhcoming analysis. To find explici expressions for he raes of reacion, we use assumpion iv) and assign a Bolzmann facor o he raio of reacion raes: K f κ) K r κ) = exp E k B T ), 9) where k B is Bolzmann s consan, T is he absolue emperaure in he sysem, and E is he energy difference of he proein as i ransiions from he unbound o he bound sae. Since he BAR proein is modeled as a hin filamen of consan curvaure, i has a microscopic bending energy given by [7]: E = E b E u B BARL BAR κ κ ), ) where E u and E b are he free energies of he proein in he unbound and bound sae respecively, B BAR is he bending siffness of he proein in he bound sae, L BAR is he lengh of he proein, and κ is he hermodynamically favorable arge curvaure ha is equivalen o he inrinsic curvaure of he proein. Combining 9) and ), we have: K f κ) K r κ) = K eqmexp ξl ) BAR κ κ ), ) where ξ = B BAR /k B T is he persisence lengh of he proein, being a measure of is mechanical siffness o hermal flucuaions, and K eqm = exp E u E b ) /k B T ) is a quaniy ha measures he exen of he aachmen reacion; ha is, if K eqm, hen mos of he proeins will aach ono he filamen, whils if K eqm, very few will aach. To deermine expressions for K f κ) and K r κ), we use assumpion v) and inroduce an inermediae sae so ha he kineics of 7) now read: K in K rκ) c in K in P K f κ)τ BAR c, ) where K in is he rae of he inermediae reacion and is assumed o be independen of κ, and c in is he number per uni lengh of proeins aached o he filamen ha are in he ransiion sae. Using ), we can find K f κ) by assigning a Bolzmann facor in a similar manner as before, wih he energy difference now being beween he unbound sae and he ransiion sae of he proein []: K f κ) = K f K in exp ξ ) sl BAR κ κ ), 3) where ξ s = B s /k B T , K f = exp E u E s ) /k b T ), and E s is he free energy of he proein in he ransiion sae. By considering he energy difference beween he bound and he ransiion sae, we find K r κ): ) LBAR K r κ) = K r K in exp ξ ξ s ) κ κ ), 4) where K r = exp E b E s ) /k b T ) and is relaed o K eqm and K f by K eqm = K f /K r. 5 Some commens regarding 3) and 4) are necessary. We noe he former predics ha he highes aachmen rae occurs when he curvaure of he filamen κ maches he arge curvaure of he proein κ. This resul is expeced given ha he aaching proein does no need o bend o mach is curvaure wih ha of he filamen, meaning no energy cos is incurred during he aachmen process. Similar reasoning holds for 4), provided ha ξ ξ s, which hen implies ha he deachmen reacion for a filamen curvaure κ κ occurs a a higher rae han when κ = κ i.e. K r κ) K r κ )). Proeins aaching o he filamen for κ κ incur an energy cos and are hermodynamically undesirable, so hey do no bind o he same exen as when κ = κ. However, here is nohing o sugges ha ξ s ξ canno also occur. In his case, he opposie holds rue; namely, if he filamen curvaure is κ κ hen he proein will deach a a lower rae han when κ = κ. This is couner-inuiive given he previous reasoning, however his is a physically realizable sysem in he form of a simple oy: he Chinese finger rap. The idea of he oy is simple: i is easy o inser one s fingers ino he rap, bu i is difficul o pull hem ou using force. The reason is ha when he fingers are forced ou, sresses are creaed in he sysem ha rap he fingers. In a similar manner, sresses ha are generaed by he mismach of curvaures beween he filamen and he proein cause he aached proein o be rapped. In eiher case, his mahemaical model is able o emulae boh physically realizable scenarios. Lasly, we suppose ha proeins ha have aached ono he filamen can move in wo ways: he diffusion of proeins from regions of high concenraion o low concenraion and he drifing of proeins which seek o minimize he quaniy κ κ ) for he hermodynamic reasons oulined previously. Combining he effecs of proein movemen wih 8), ), and 4), we obain: ) c = K LBAR rk in exp ξ ξ s ) κ κ ) {K eqm τ BAR P exp ξl ) } BAR κ κ ) c D c s χ s [c s κ κ ) ], 5) where D is he diffusion consan of proeins on he filamen and χ is he sensiiviy of an individual proein o move owards regions wih low κ κ ). The drif erm shares a similar form o he Nerns- Planck equaion for charged paricles minimizing heir elecric poenial [5] and he Keller-Segel model for he movemen of bacerial populaions due o gradiens in chemoaracans [47]. To close 5), we suppose ha no aached proeins flow off he edges of he filamen, so ha c/ s = a s = {, L}..3. D Self-Assembly Equaions Equaions )-6) and 5) form he D self-assembly sysem. We inroduce he following nondimensionalizaions and rescaling o simplify: ŝ = s/l, ˆ = EI/η L 4, ˆκ = κl, ĉ = γcl, ˆn = n L /EI, ˆn 3 = n 3 L /EI, ˆv = v η L 3 /EI, ˆv 3 = v 3 η L 3 /EI, ŵ = w η L 4 /EI, ˆκ = κ L, ˆP = γkeqm τ BAR P L, ˆξ = ξl BAR /L, ˆξ s = ξ s L BAR /L, ˆKr = K r K in η L 4 /EI, ˆD = Dη L /EI, ˆχ = χη /EI, so ha he equaions of ineres become upon dropping he ha symbol): α v α v 3 β w κ = n s κn 3 αw v 3 v, 6) = n 3 s κn αw v v 3, 7) = s κ c) n, 8) = w s, 9) = v s κv 3 w, 3) = v 3 s κv, 3) 6 Quaniy Esimaed Value κ, κ, c, P ξ, ξ s 3 K r D, χ α, β 5 36 Table : Esimaes of imporan variables and quaniies ha appear in 6)-3). where α = ρaei/η L 4 r /L 4 and β = ρei /η L 6 r /L 6, boh of which are small given ha r L as seen in Secion., and ) { c = K rexp ξ ξ s) κ κ ) P exp ξ ) } κ κ ) c D c s χ [c s ] s κ κ ), 3) wih boundary condiions given by n = n 3 =, κ = c, and c/ s = a s = {, }. For he curren analysis and following numerical sudies, we include esimaes of some of he imporan quaniies for a ypical biological sysem in Table. In paricular, we have used esimaes for curvaure based on discussions in [5], which propose he radius of remodeled ubes o be of he order 8 7 m, and have assumed ha he lengh of he filamen is such ha he self-assembled sysem forms complee consan curvaure soluions wih lile o no overlap. We esimae η 3 4, aking he dynamic viscosiy of fluid o be approximaely ha of waer µ 4 m s and he radius of he filamen r 9 m o coincide wih he hickness of a lipid bilayer. We consider he bilayer o be made of packed hydrocarbon chains and use a Young s modulus of E 7 8 Nm, which provides a reasonably good esimae of he maerial properies of a biological membrane according o [], and ake he densiy of he filamen o be ρ 3 kg.m 3 o approximae he fluid naure of he bilayer. Furhermore, we esimae he proein persisence lengh o be ξ 8 m, wih he larger bound coinciding wih ha of DNA [3], and he diffusion of proeins on he filamen o be D m s, according o sudies of membrane proein movemen on lipid bilayers found in [48]. Lasly, we assume K r varies subsanially based upon chemical riggers which can affec proein adhesion. Given he small esimaed values of α and β, we consider he limi α, β in he forhcoming analysis. 3. Sabiliy Analysis of Consan Curvaure Soluions 3.. Seady Saes To deermine he ime-independen soluions of 6)-3), we firs consider when such soluions occur in he mechanics of he filamen. We denoe he seady sae of a varia

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