Hydrodynamics of transient cell-cell contact: The role of membrane permeability and active protrusion length
[['In many biological processes , two or more cells come into physical contact to form a cell-cell interface .', 'These include cell-cell contacts like those in the epithelium [1 , 2] that change on timescales of hours , and also transient contacts that form on seconds timescales , including those formed by lymphocytes and other immune cells that must interrogate many cells rapidly [3 , 4] .', 'A fundamental question for all cell-cell interfaces is how receptors and ligands come into contact , despite being separated by extracellular fluid , various large surface molecules like ectodomains of membrane proteins , and other structures in the negatively-charged glycocalyx .', 'The contribution of large surface molecules has received most attention , for example producing spatial pattern formation based on molecular size [5–9] of the T cell receptor ( TCR ) and the immunotherapy target PD-1 [10] .', 'In this work , we focus on the role of the fluid [11–14] .', 'To highlight the potential importance of the hydrodynamics of extracellular fluid at an interface , we perform a preliminary calculation ( unrealistically ) assuming cells are rigid , impermeable spheres of radius rcell .', 'In order to bring these cells into close contact , a force F pushes them together , as shown in Fig 1A .', 'This fluid dynamics problem can be solved analytically for the separation distance z , yielding [15 , 16]', 'd z d t = - 1 6 π η r cell ( z r cell ) F ( 1 )', 'where η is the extracellular fluid viscosity .', 'This equation is reminiscent of the Stokes drag formula for a sphere in free fluid , but modified by a factor ( z/rcell ) ∼ ( 10 μm/10 nm ) ∼ 103 .', 'In other words , the force required to move two cells together is increased by a thousand-fold , a strikingly large correction .', 'This observation , known as the “lubrication limit” , “confinement effect” or “thin-layer effect” [11 , 15 , 16] , heuristically arises because a small change in z requires incompressible fluid to move a large distance to outside the interface .', 'The cell surface is not a rigid sphere , but a deformable membrane subject to thermal undulations , active forces , and hydraulic permeability due largely to membrane inclusions like aquaporins .', 'In this context , we ask , what is the role of the fluid in close-contact formation ?', 'Are thermal undulations sufficient for receptor proximity ?', 'Are typical F-actin filopodial forces , ∼10 picoNewtons [17 , 18] , sufficient for receptor proximity ?', 'And how much force is required for rapid proximity ( <1 second ) ?', 'If there is a significant thin-layer effect , the force required will increase for smaller cell-cell distances , but larger distances require longer protrusions , suggesting the possibility of an optimal “attack range” which might explain the biological benefit of filopodia .', 'If the membrane is permeable to extracellular fluid [19] , how much permeability is required for rapid proximity ?', 'Factors that influence permeability , such as aquaporins , are under regulation [20] , differentially localized , and impact cell processes including cancer angiogenesis [21] , raising the possibility that cell-cell contact can be regulated in this way .', 'In contrast to previous theoretical studies of cell-cell interfaces , many of which capture membrane and molecular dynamics but exclude hydrodynamics , or exploit equilibrium statistical physics and therefore omit dynamics , studying the influence of active forces requires a full fluid dynamics model .', 'Such models have been studied using both analytical methods [22 , 23] and computational methods [24] , reviewed in [25] .', 'We have developed a computational fluid dynamics algorithm capable of simulating fluid-structure interactions with thermal fluctuations on seconds- and microns-scales [26] based on the Stochastic Immersed Boundary Method [24 , 27–30] .', 'Here , we use this to simulate two opposing membranes , variously including thermal fluctuations , active forces , and membrane permeability .', 'We find that the thermal fluctuations are significantly modified by the thin-layer effect for a range of assumptions about molecular sizes .', 'Active forces are sufficient to drive proximity .', 'The thin-layer effect has the consequence of introducing two timescales ( milliseconds and microseconds ) in response to the two length scales inherent in the system .', 'We find that membrane hydraulic permeability overcomes the thin-layer effects , but only for values larger than previous physiological estimates .'], ['Receptor-ligand contact for the TCR occurs when the membranes are separated by Δ z 0 ⋆ ≈ 13 nm .', 'For the remainder of the manuscript , we refer to “membrane proximity” or “receptor-ligand contact” , defined as membrane configurations with separation Δ z 0 ≤ Δ z 0 ⋆ .', 'Other parts of the membranes are separated by a distance Δz∞ , where estimates range from 22 nm to 150 nm [7 , 31–37] for ectodomains of signaling molecules like CD45 , non-specific binding pairs like LFA-ICAM and cadherins , and the glycocalyx .', 'At the same time , cells themselves are rcell ∼ 2 μm for the smallest T cells [3] .', 'To explore the consequences of the thin layer geometry , plus the incompressibility of fluid , we are required to simulate a 3D system with a resolution of receptor-ligand size Δ z 0 ⋆ in a domain larger than the cell , which has radius rcell .', '( The analogous system in 2D would be insufficient since the opportunity for evacuating from the interface is fundamentally dependent on dimensionality of the boundary . )', 'The two cell surfaces are represented by elastic disks , as shown in Fig 1B , subject to bending resistance and approximate inextensibility .', 'These disks are held by boundary tension σ0 in their plane , and separated by approximately inextensible nonspecific molecules of size Δz∞ , which we refer to as the far-field separation .', 'These non-specific adhesions are absent from a region of radius rfree around the center of the disk , which we identify as the site of the receptor .', 'We assume both intracellular and extracellular fluids are Newtonian with viscosity of water , η = 10−3 Pas .', 'At the small length scales in our simulation , of ∼nm , the viscosity of the cytosol can be one or two orders of magnitude larger [38] , and at large length scales in our simulation , the viscosity is even larger .', 'The variability of viscosity , and its dependence on length scale of observation , is an active area of research and is attributed to the heterogeneous content of the cytoplasm and ordering of water [38] .', '( We do not a priori include here the concept of “effective viscosity” to describe the phenomenon of slower timescales due to fluid confinement [22] , since we anticipate these emerge from the dynamics naturally . )', 'Thus , all times we report are underestimates , and dynamics at more realistic viscosity and cell separation are expected to be slower .', 'Due to the linear nature of the fluid dynamics equations we use , all times scale linearly with viscosity .', 'To simulate this model , we use an implementation of the Stochastic Immersed Boundary framework .', 'We largely overcome the numerical challenges mentioned above , allowing us to simulate with parameters within the order-of-magnitude of experimentally estimated values , shown in Table 1 .', 'This framework , discussed in more detail in Methods and in the Supplemental Material , numerically approximates the fluid in an Eulerian representation , discretized in a rectangular Cartesian grid , while approximating the structure ( in this case , the one or two membranes ) in a Lagrangian representation , discretized as a triangulated mesh .', 'As a control , we simulate a single membrane with thermal undulations , being held in place by adhesion molecules attached to fixed points in the fluid , as if it were attached to a “ghost” membrane .', 'This simulation could be identified , for example , with a situation in which a cell is adhered to a highly permeable surface like a sparse network of extracellular matrix [43 , 44] that provides minimal hydrodynamic confinement .', 'A snapshot top view is shown in Fig 2B .', 'We find that the position of the receptor fluctuates as a Gaussian with standard deviation σ = 3 . 12nm ( 95% confidence interval [3 . 10 , 3 . 15]nm ) and an autocorrelation well-described by a single exponential decay with timescale τ = 1 . 05 × 10−6 s ( 95% confidence interval [1 . 02 , 1 . 08] × 10−6 s ) .', 'Dynamics ( both deterministic and stochastic ) of a single membrane can be decomposed into modes , each with a timescale that , in some geometries , can be solved for explicitly [23 , 25] .', 'The timescale of the nth mode associated with tension scales as ηrfree/σ0 n ∼ 10−6 s/n , and that the nth mode associated with bending scales as ηrfree/Bn3 ∼ 10−8 s/n3 , although both with significant prefactors .', 'Thus , our computational results are broadly consistent with the dominant mode being the first mode associated with tension .', 'We next simulate the interface with two membranes , as shown in Fig 3D .', 'The membranes are held at Δz∞ = 60nm outside the free radius .', 'We run simulations for 1 s .', 'We observe a stationary probability with mean separation 〈Δz〉 = 70 . 0nm .', 'This blistering by 10nm is due to an entropic repulsive pressure arising from thermal fluctuations [23 , 45 , 46] and is not observed in simulations where thermal fluctuations are removed ( shown below ) .', 'We observe a relatively small change in the amplitude of fluctuation compared to the single-membrane case , from 3 . 2nm to 4 . 6nm ( 95% confidence interval [4 . 5 , 4 . 6]nm ) .', 'The autocorrelation of Δz0 does not fit a single exponential , but rather fits a two-timescale decay ( black curve , Eq 21 ) with a fast timescale τfast = 5 . 3 × 10−7 s ( 95% confidence interval [5 . 0 − 5 . 4] × 10−7 s ) comparable to the single-membrane autocorrelation above , but also a slow timescale τslow = 8 . 2 × 10−5 s ( 95% confidence interval [8 . 1 , 8 . 3] × 10−5 s ) .', 'The fraction of the autocorrelation function described by the slow process is c2 = 0 . 48 ( 95% confidence interval [0 . 43 , 0 . 53] ) .', 'The double exponential equation we use to fit the autocorrelation is not a perfect fit , reflecting the inherent complexity of this process and the need for such computational modeling .', 'This finding is in agreement with previous experimental work [11 , 23] showing that spatial amplitudes are not changed significantly , but fluctuation timescales are significantly altered by confinement .', 'We again compare the timescale with previous estimates for a similar case that has been previously studied: a membrane near a wall [23] .', 'The timescale of the nth mode associated with tension scales as η r free 4 / σ 0 ( Δ z ∞ ) 3 n 4 ∼ 10 - 4 s / n 4 ( see Eq . 2 . 18 in [23] ) .', 'Again , our computational results are broadly consistent with the dominant mode being the first mode associated with tension .', 'Since the rate of receptor triggering is determined by the timescale of proximity ( i . e . , sufficient for close contact between receptor and ligand ) , we next want to use the fluid dynamics simulations to estimate the mean first-passage time ( MFPT ) to proximity .', 'Since these simulations include the target ligand only implicitly , we can infer the mean time to proximity for several values of Δ z 0 ⋆ .', 'For the simulations with Δz∞ = 60nm , we ran fluid dynamics simulations for 1 second .', 'For Δ z 0 ⋆ = 13 nm , proximity of Δ z 0 < Δ z 0 ⋆ was not observed , suggesting it is a rare event in the sense that it occurs on a timescale much larger than the fluctuation timescale .', 'Computational expense prohibits us from simulating significantly longer times .', 'To overcome this computational challenge of observing such rare close contacts , in this section , we develop an approximation based on Ornstein-Ulhenbeck ( OU ) processes [47] , and then use the Weighted Ensemble [48 , 49] computational method to find the mean first time to a particular state of the system , here defined as the first time for the membranes to be within a distance of Δ z 0 ⋆ of each other .', 'Full details are in Methods and S1 Appendix .', 'For the interface , we find that membranes will displace by 20 nm ( i . e . , the separation distance deviated from its mean of 70 nm down to 50 nm ) in approximately 10−2 s .', 'The time until a displacement larger than this grows super-exponentially: for a displacement of 25 nm ( i . e . , down to separation Δ z 0 ⋆ = 45 nm ) , it takes ∼1 s .', 'For the single membrane case , an analytical approximation exists for the single-component OU [50] , solid black line in Fig 4 , allowing us to confirm our computational method ( further validation is provided in S1 Appendix ) .', 'In Fig 4 , the interface case apparently has a larger ( i . e . , slower ) MFPT for the single membrane .', 'However , we note that these numbers are not directly comparable .', 'The single-component OU describes the position of a single membrane , which has standard deviation σ1 = 3 . 1 nm , while the two-component OU describes the distance between two membranes , which has a standard deviation σ2 = 4 . 6 nm .', 'A more direct comparison would be a hypothetical simulation in which two “single” membranes were held at a distance of 70 nm , but did not interact via fluid therefore would fluctuate independently .', 'In such a case , the separation between these membranes would be σ indep = 2 σ 1 2 ≈ 4 .', '4 nm , approximately the same as the interface standard deviation .', 'Cells , including the T cell , continuously extend active processes driven by F-actin like filopodia and microvilli [42 , 51] that facilitate receptor binding [37 , 52] .', 'To explore the effect of hydrodynamics on active processes at an interface , we simulate a force F at the receptor site , spread over a disk of radius a = 10 nm 1B .', 'In Fig 5 , we find that a force of F = 20 pN is sufficient to drive proximity from a far-field separation of Δz∞ = 50 nm for both single membranes and interfaces .', 'We perform deterministic simulations with thermal forces omitted ( black curves in Fig 5 ) .', 'The dynamics are quantitatively similar , and the simulations are much less computationally taxing .', 'For this reason , for the remainder of this section we perform simulations without thermal fluctuations .', 'Note in Fig 5 the stochastic and deterministic simulations approach equilibrium on approximately the same slow timescale , but the equilibrium separation is larger when thermal forces are included due to the entropic repulsion discussed above .', 'The shape of the protrusion is shown in Fig 5C and 5D .', 'Membrane profiles are reminiscent of micrographs of microvilli in T cells ( see , e . g . , [37] Fig . 3G ) : The edges are rounded due to membrane bending resistance , and closest proximity is at the tip , with cell separation distance gradually tapering off .', 'To isolate the influence of the thin-layer effect , we perform identical simulations with and without a second membrane , for various active forces , in Fig 6A .', 'For a single membrane , the distance approaches a new equilibrium rapidly , ∼10−5 s .', 'For an interface , Fig 6A demonstrates a rapid initial movement , ∼10−5 s , followed by a slower approach to the same equilibrium separation .', 'In Fig 6B we explore this further by plotting the position of both membranes for F = 10 pN: We find that there is an initial rapid movement of the top membrane , i . e , the driven membrane ( blue curve in B ) ∼10−5 s , however this is accompanied by a rapid depression of the bottom membrane , i . e . , the passive ( red curve ) .', 'Then , on a slower timescale ∼10−3 s , the passive membrane returns .', 'We attribute the rapid depression to the incompressibility of the extracellular fluid , and the slow timescale to the thin-layer timescale identified above , as the excess fluid must drain from the interface .', 'The plasma membrane is under tension , maintained by hydrostatic pressure and regulation of exocytosis , endocytosis and membrane ruffles [40 , 53] and is in the range of 3 − 300 pN/ μm [40 , 41] and is spatially nonuniform [54] .', 'We apply membrane tension in our simulation as a boundary surface tension with magnitude σ0 .', 'We find that higher surface tension necessitates more force for the equivalent equilibrium displacement , as shown in Fig 7A .', 'This demonstrates that the system is above the critical length scale below which surface tension is insignificant compared to membrane bending [17 , 55] .', 'Note that these these data show the equilibrium position in response to a constant force , therefore there is no effect of fluid dynamics , and thus no thin-layer effect .', 'The results we report are sensitive to the properties of the large surface molecule , such as the size of the region rfree around the receptor that is free of these molecules .', 'To put this parameter in a form more readily comparable with molecular surface densities , we define the parameter ρ = 1 / r free 2 , which has units of nm−2 .', 'We previously estimated that receptor-ligand contact occurs in depletion zones with rfree ∼ 100 nm [6] .', 'In Fig 7B , we explore the equilibrium separation as a function of force for various surface densities , rfree = 80 nm ( ρ = 1 . 6 × 10−4 nm−2 ) , rfree = 150 nm ( ρ = 4 . 4 × 10−5 nm−2 , rfree = 300 nm ( ρ = 1 . 1 × 10−5 nm−2 .', 'As expected , higher density of surface molecules reduces the equilibrium displacement .', 'The results we report are also sensitive to the molecular size Δz∞ of the large surface molecule .', 'The large range of estimates for Δz∞ arises from the uncertainty about which molecules dominate the process of keeping the membranes apart .', 'Molecules like CD45 may sterically maintain membrane separation by as little as 22nm .', 'Non-specific adhesion molecules like LFA-ICAM and cadherins are estimated to span a range from 28nm [31] to 43nm [32 , 33] .', 'Estimates for the thickness of the glycocalyx range from 40 − 50 nm [34 , 35] to 150 nm [36 , 37] .', 'So , we explore receptor proximity driven by active forces , varying the far-field separation Δz∞ from 30nm to 120nm in Fig', '8 . To clarity , as with previous simulations , the majority of the membranes ( everything outside the radius rfree ) are held apart at a fixed distance Δz∞ .', 'From this fixed distance , an active protrusion is extended from one membrane towards the other .', 'We examine the time dynamics of the protrusion .', 'As the starting distance is increased , the time before proximity Δ z 0 < Δ z 0 ⋆ increases , shown in Fig 7B .', 'However , we find that above a critical far-field separation Δz∞ ≈ 80 nm , membrane deformation speed increases , even though force is kept constant at F = 10 pN .', 'The effect is modest but sufficient so that , over a large range of far-field separations ∼60 − 130 nm , the time to proximity does not increase for increasing separation , Fig 7C .', 'Heuristically , this plateau arises because of a significant thin-layer effect dominates motion .', 'Since this effect depends sensitively on the thickness of the thin layer , increasing the thickness reduces the effect , and the active protrusion can push more easily through the free fluid .', 'On the other hand , although speed increases , the distance to the target cell also increases .', 'Thus there is an optimal “attack distance” from which to extend a protrusion .', 'If a membrane were perfectly water permeable ψ = ∞ there would be no thin-layer effect .', 'Biological membranes are sufficiently permeable that , in fast motile cells , fluid velocity appears stationary in the lab frame of reference when myosin contractility is inhibited [19] .', 'Therefore , it is a priori reasonable to expect that there is a magnitude of permeability above which the slow-timescale behavior of the interface is removed , leaving only fast dynamics .', 'We repeat the active force simulations at F = 20 pN , and explore permeabilities at each order of magnitude , in Fig', '9 . We find that the first significant deviation from impermeability ( ψ = 0 ) occurs at ψ ∼ 102 nm/sPa ( red dotted ) .', 'By ψ = 104 nm/sPa , the top membrane time series is comparable to the single-membrane case ( Fig 6B ) , i . e . , very little thin-layer effect is observed .', 'The transition to fast-only dynamics occurs through a reduction in timescale , and only a weak reduction in amplitude ( green curve is compressed horizontally ) .', 'Previous theoretical studies [22 , 23] have estimated the parameter regimes in which permeation ( i . e . , flow across membranes ) will dominate parallel flows ( i . e . , flow parallel to the interface ) .', 'Specifically , [22] finds that the critical scale of permeability , above which permeation is dominant , is', 'ψ ∼ ( Δ z ∞ ) 3 η ( r free ) 2 ∼ 10 4 nm / s Pa ( 2 )', 'where we have changed their equation to our notation , and assumed that the dominant wavelength is ∼ ( 1/rfree ) .', 'This scaling is in agreement with our simulations .', 'Moreover , this is orders of magnitude larger than the permeability estimates', '10−2 nm/sPa [38] .', 'The largest indirect estimates from motile epithelial keratocytes gives ∼101 nm/sPa [19] .', 'So , taken together , our results suggest the thin-layer effect cannot be abrogated by physiological levels of permeability .']]
['Introduction', 'Results']
['In many biological settings , two or more cells come into physical contact to form a cell-cell interface .', 'In some cases , the cell-cell contact must be transient , forming on timescales of seconds .', 'One example is offered by the T cell , an immune cell which must attach to the surface of other cells in order to decipher information about disease .', 'The aspect ratio of these interfaces ( tens of nanometers thick and tens of micrometers in diameter ) puts them into the thin-layer limit , or “lubrication limit” , of fluid dynamics .', 'A key question is how the receptors and ligands on opposing cells come into contact .', 'What are the relative roles of thermal undulations of the plasma membrane and deterministic forces from active filopodia ?', 'We use a computational fluid dynamics algorithm capable of simulating 10-nanometer-scale fluid-structure interactions with thermal fluctuations up to seconds- and microns-scales .', 'We use this to simulate two opposing membranes , variously including thermal fluctuations , active forces , and membrane permeability .', 'In some regimes dominated by thermal fluctuations , proximity is a rare event , which we capture by computing mean first-passage times using a Weighted Ensemble rare-event computational method .', 'Our results demonstrate a parameter regime in which the time it takes for an active force to drive local contact actually increases if the cells are being held closer together ( e . g . , by nonspecific adhesion ) , a phenomenon we attribute to the thin-layer effect .', 'This leads to an optimal initial cell-cell separation for fastest receptor-ligand binding , which could have relevance for the role of cellular protrusions like microvilli .', 'We reproduce a previous experimental observation that fluctuation spatial scales are largely unaffected , but timescales are dramatically slowed , by the thin-layer effect .', 'We also find that membrane permeability would need to be above physiological levels to abrogate the thin-layer effect .']
['The elastohydrodynamics of water in and around cells is playing an increasingly recognized role in biology .', 'In this work , we investigate the flow of extracellular fluid in between cells during the formation of a cell-cell contact , to determine whether its necessary evacuation as the cells approach is a rate-limiting step before molecules on either cell can interact .', 'To overcome the computational challenges associated with simulating fluid in this mechanically soft , stochastic and high-aspect-ratio environment , we extend a computational framework where the cell plasma membranes are treated as immersed boundaries in the fluid , and combine this with computational methods for simulating stochastic rare events in which an ensemble of simulations are given weights according to their probability .', 'We find that the membranes fluctuate independently with a characteristic timescale of approximately microseconds , but that as the cells approach , a new , slower timescale of approximately milliseconds is introduced .', 'Thermal undulations nor typical amounts of membrane permeability can overcome the timescale , but active forces , e . g . , from the cytoskeleton , can .', 'Our results suggest an explanation for differences in molecular interactions in live cells compared to in vitro reconstitution experiments .']
['classical mechanics', 'fluid mechanics', 'permeability', 'viscosity', 'developmental biology', 'membrane receptor signaling', 'materials science', 'molecular development', 'materials physics', 'cellular structures and organelles', 'chemical properties', 'physical chemistry', 'adhesion molecules', 'chemistry', 'fluid dynamics', 'cell membranes', 'continuum mechanics', 'hydrodynamics', 'physics', 'signal transduction', 'membrane characteristics', 'cell biology', 'biology and life sciences', 'physical sciences', 'material properties', 'cell signaling']