Saurav G.
Varma
,
Argha
Mitra
and
Sumantra
Sarkar
*

Center for Condensed Matter Theory, Department of Physics, Indian Institute of Science, Bengaluru, Karnataka 560012, India. E-mail: sauravvarma@iisc.ac.in; argha.mitra02@gmail.com; sumantra@iisc.ac.in

Received
20th June 2024
, Accepted 20th August 2024

First published on 22nd August 2024

Molecular transport maintains cellular structures and functions. For example, lipid and protein diffusion sculpts the dynamic shapes and structures on the cell membrane that perform essential cellular functions, such as cell signaling. Temperature variations in thermal equilibrium rapidly change molecular transport properties. The coefficient of lipid self-diffusion increases exponentially with temperature in thermal equilibrium, for example. Hence, maintaining cellular homeostasis through molecular transport is hard in thermal equilibrium in the noisy cellular environment, where temperatures can fluctuate widely due to local heat generation. In this paper, using both molecular and lattice-based modeling of membrane transport, we show that the presence of active transport originating from the cell's cytoskeleton can make the self-diffusion of the molecules on the membrane robust to temperature fluctuations. The resultant temperature-independence of self-diffusion keeps the precision of cellular signaling invariant over a broad range of ambient temperatures, allowing cells to make robust decisions. We have also found that the Kawasaki algorithm, the widely used model of lipid transport on lattices, predicts incorrect temperature dependence of lipid self-diffusion in equilibrium. We propose a new algorithm that correctly captures the equilibrium properties of lipid self-diffusion and reproduces experimental observations.

Cellular transport primarily happens through two mechanisms: diffusion, which obeys the detailed balance principle, and active transport, which breaks detailed balance.^{8} The constraint of detailed balance is a defining feature of an equilibrium system. It states that the probability flux from a state i to a state j exactly equals the flux from state j to state i. Mathematically,

p_{i}π_{ij} = p_{j}π_{ji}, | (1) |

As a concrete example, let's consider the diffusion coefficient's temperature (T)-dependence. Diffusion of a free colloid obeys the Stokes–Einstein relationship, which states that D(T) = k_{B}T/γ, where k_{B} is the Boltzmann constant and γ is the friction coefficient. Also, the diffusion of molecules adsorbed on periodic surfaces is determined by escape from the periodic potential generated by the surface molecules, which results in an Arrhenius-like form for the diffusion coefficient:

(2) |

Recently, it was observed that, in live cells, Glycosylphosphatidylinositol anchored proteins (GPI-AP), a model cell surface protein and a known interaction partner of the ACS,^{13} shows T-independent diffusion.^{18} This observation contradicts prior works, which show, using very similar techniques (fluorescence correlation spectroscopy: FCS), that diffusion is T-dependent even in live cells.^{17,19–21} Therefore, it remains unclear whether the transport of molecules in live cells is T-independent or not. An important observation is that the T-independence of diffusion is dependent on the scale of observation.^{18} When the membrane is observed with a small FCS spot (3 × 10^{4} nm^{2}), diffusion is T-independent. However, as the spot size increases (6 × 10^{4} nm^{2}), diffusion becomes T-dependent.^{18} Therefore, we hypothesize that processes that are important at the nanometer scale, such as the interaction of the membrane with the ACS, are the key drivers of T-independent transport of molecules. This paper tests this hypothesis using a coarse-grained molecular dynamic (CG-MD) model and a lattice model of model membranes.

Now, we briefly describe the results and the conclusions of the paper. The paper is broadly divided into two parts. In the first part, we ran equilibrium CG-MD simulations to study lipid self-diffusion in model membranes (pure DPPC, pure DAPC, and 5:3:2 mixture of DPPC:DAPC:CHOL), which showed that diffusion is an activated process irrespective of the observation scale. In the second part of the paper, we tested the effect of underlying nonequilibrium driving on the activated diffusion process using a lattice model of lipid self-diffusion. Specifically, we studied the effect of ACS on the lipid self-diffusion using a lattice model of lipids. We find that coupling to the ACS makes lipid diffusion T-independent. Surprisingly, lipids that do not couple to the ACS can also show near T-independence because of their interactions with the ACS-coupled lipids. We comment on the consequences of the T-independence and its biological relevance to end the paper. We have also found that the Kawasaki algorithm, used extensively to simulate lipid diffusion, does not accurately capture diffusion's thermodynamics. We propose an alternate algorithm, called the barrier-hop dynamics, which accurately captures the equilibrium kinetics of activated diffusion.

The rest of the paper is organized as follows. In Section II, we describe the simulation and analysis methods in detail. In Section III, we describe the results. Finally, we conclude this paper by discussing our results in Section IV. Some additional figures and a movie are shown in the ESI.†

1. System.
For the equilibrium simulation of PM models using CG-MD, we have used three symmetric model membranes with box size 30 nm × 30 nm. The models were built using the INSANE martini tool^{22} and simulated in GROMACS using MARTINI 3.0 forcefield to achieve faster dynamics.^{23,24} Two membranes were built using pure lipids: (1) the saturated DPPC (di-palmitoyl phosphatidylcholine, C_{16:0}) and (2) the polyunsaturated DAPC (di-arachidonoyl phosphatidylcholine, C_{20:4}). A third mixed composition membrane (DPPC:DAPC:CHOL, 5:3:2) was also simulated to understand lipid transport in more realistic membranes.^{25} For example a snapshot of mixed membrane with box dimension is shown in Fig. 1a. All systems were solvated with water, and 0.15 M NaCl was added to attain the physiological ionic condition.^{22} The system topologies were created in the GROMACS format. Five different temperatures (298, 302, 306, 310, and 314 K) were used to check the T dependency of the PM models. All the production simulation (Table 2) was conducted for 5 μs, except two pure DPPC systems that were run for 30 μs using GROMACS-2022.2^{26–28} with 20 fs timestep where each 100 ps snapshots were stored for analysis.

2. CG-MD simulation protocol to measure diffusivity.

a. Energy minimization and equilibration. The system's energy was minimized using the steepest descent and the conjugate gradients methods in the next step. We ran at least 5000 steps of energy minimization to ensure proper geometry and to minimize steric clashes. Following energy minimization, all systems were subjected to a six-step equilibration protocol (Table 1), where the stiffness of the positional restraint on the lipid head groups (PO_{4} bead of DPPC and DAPC) was in gradually decreased to avoid collapse of the system. Five different temperatures (298, 302, 306, 310, and 314 K) were controlled by velocity-rescale thermostats^{29} with a coupling constant of 0.1 ps, which was used. For the equilibration, we used Berendsen barostat^{30} with the compressibility of 4.5 × 10^{−5} bar^{−1} semi-isotropic scaling.^{31,32}

b. Production runs. For these simulations velocity rescale temperature bath was applied.^{29} The pressure was controlled with the Parrinello–Rahman barostat^{33} with a 12 ps coupling constant, and the compressibility of 3 × 10^{−4} bar^{−1}.^{34} However, due to the longer spatial scale during production, the chance of undulation increases, which was significantly reduced by using a mild positional restraint (2 kJ mol^{−1} nm^{−1}) on the PO_{4} beads in the Z-direction.^{35,36} All the production simulation was conducted for 5 μs, except two pure DPPC systems that were run for 30 μs using GROMACS-2022.2^{26–28} with 20 fs timestep where each 100 ps snapshots were stored for analysis. The cumulative simulation time was 1 ms.

a. Energy minimization and equilibration. The system's energy was minimized using the steepest descent and the conjugate gradients methods in the next step. We ran at least 5000 steps of energy minimization to ensure proper geometry and to minimize steric clashes. Following energy minimization, all systems were subjected to a six-step equilibration protocol (Table 1), where the stiffness of the positional restraint on the lipid head groups (PO

Step | Restraint (kJ mol^{−1} s^{−1}) |
Timestep (fs) | Time (ns) |
---|---|---|---|

1 | 200 | 2 | 1 |

2 | 100 | 5 | 1 |

3 | 50 | 10 | 1 |

4 | 20 | 15 | 0.75 |

5 | 10 | 20 | 1 |

6 | 2 | 20 | 50 |

Total | 54.75 |

b. Production runs. For these simulations velocity rescale temperature bath was applied.

PM | Replicas | Time (μs/run) | Time (μs/temp) | Total time (μs) |
---|---|---|---|---|

Pure DPPC | 2 | 30 | 60 | 300 |

6 | 5 | 30 | 150 | |

Pure DAPC | 10 | 5 | 50 | 250 |

Mixed | 12 | 5 | 60 | 300 |

Total | 1000 |

3. Lipid flow simulation.
Another set of simulations was conducted to study the nanometer scale correlated flows of lipids. The T range was kept the same (298–314 K), and only the pure DPPC model was simulated. The starting structures were taken from the long simulations (after 3 μs) to their respective temperatures. All the parameters were kept the same, but the run time was reduced to 1 ns with 20 fs timesteps. These short-time simulations were run for 20 replicas at each temperature, and each 20 fs coordinates were stored.

4. Analysis.
The analysis was conducted with in-house developed Python 3.11 programs using the MDAnalysis package^{37} and GROMACS-2022.2 tools^{28} and visualization was performed with the molecular graphics viewer VMD.^{38}

1. Bulk diffusivity measurement.
The MSD for lipids at different temperatures was calculated over the last 2 μs of each 5 μs runs, whereas the last 5 μs were used for the 30 μs trajectories. The analyzed part of the trajectories was further divided into 1 μs blocks^{35,39} to calculate MSD. The MSD is defined as:

where 〈X^{2}〉 is the ensemble average of time averaged MSD (TAMSD), τ is the lag-time, N is the number of tracer particles in the system (it is also the number of trajectories that constitute the ensemble). is the time-averaged MSD for particle i for a lag-time τ, defined as:

where D_{0} is a T-independent constant that depends on the lipid properties, k_{B} is the Boltzmann constant, and E_{A} is the activation energy. E_{A} was determined by calculating the slope of lnD(T) vs. 1/T curve for different PM models. Per experimental convention, 1000/T was used instead of 1/T for the plots.^{17} The errors were estimated from the standard deviation among the replicas.

(3) |

(4) |

The gmx msd tool was used to calculate the MSD of the PO_{4} (PC lipids) and ROH (for CHOL) beads of the lipids (only the headgroup of lipids were considered for the MSD calculation). The results were plotted on a log–log scale to determine the power-law dependence of diffusion. The diffusion or transport coefficient was calculated by multiplying the slope with the coarse-grained (CG) conversion factor of 4, as suggested in.^{31,40} Subsequently, the transport coefficients at different temperatures were plotted to derive the activation energy of specific lipid molecules using the Arrhenius equation:

(5) |

(6) |

2. Local diffusivity measurement.
To understand the short-time spatial behavior of lipids, we analyzed one ns simulation trajectories. The system was divided into grid boxes measuring 3 nm × 3 nm, each containing approximately 10–15 PO_{4} beads of DPPC lipids, which were tracked to measure the local MSD (Fig. S3, ESI†). The lag time τ was varied between 0 and 1 ns with a resolution of 20 fs. The MSD for the PO_{4} groups within each grid box was computed, which showed subdiffusive transport with the anomalous exponent of 0.6. Therefore, the transport coefficient was not the diffusion constant. However, we could still determine the transport coefficient, D(T), from the slope of the MSD curve (check the Results section for details). Following the D(T) calculation, we analyzed its distribution across the grids for the abovementioned temperatures. This method was repeated for the remaining replicas, and we compared the spatial average of D(T) at different T.

On a fixed time interval τ, the displacement vectors for the ith and the jth lipids, Δr_{i}(τ) and Δr_{i}(τ) were calculated. The correlation function is then given by:

(7) |

(8) |

(9) |

1. Interaction between lipids.
Lipids of the same type (passive–passive or inert–inert) interact with each other through attractive Lennard-Jones (LJ) potential with a cutoff radius of 2.5σ, where we took one lattice unit as 2^{1/6}σ for excluded volume interaction between unlike lipids. Hence, we have at most two lattice units (=2 × 2^{1/6}σ < 2.5σ) attractive interaction between like lipids as shown in Fig. 4c. The interaction strength J is chosen so that J/k_{B}T = 1/1.43 at T = 293 K. Lipids of different species interact with excluded volume interaction only, such that J = 0 for unlike lipids. Hence, the total interaction energy, U, of a given pair of lipids varies between −6 to 0k_{B}T, which is in the same ballpark as the activation energies calculated from the molecular simulation. The neighbor configurations and the interaction energy distribution (−U) for different possible neighbor configurations are shown in Fig. 4d.

2. Equilibration using Kawasaki algorithm.
To equilibrate the lipids on the lattice, we used Kawasaki exchange moves,^{55} which exchange two neighboring lipids with a probability p determined by the following formula:

where ΔE is the difference in the interaction energy before and after the swap. Kawasaki moves guarantee the system reaches the correct equilibrium state if run sufficiently long. However, due to kinetic effects, local Kawasaki moves often lead to an arrested state. Hence, we used global Kawasaki moves for equilibrating a random initial lipid configuration. In the global algorithm, two lipids were chosen randomly from the entire system and swapped using the same criteria, which avoided the kinetic traps and rapidly converged to the global equilibrium state at a given temperature.

(10) |

3. Diffusive (passive) moves.
The equilibrated state served as the initial condition for the simulations to study the transport of the lipids.

The algorithm, which we call barrier-hop dynamics or BHD, is as follows:

1. Pick a random lipid and a neighboring lipid

2. Calculate the total interaction energy of the lipid and the neighboring lipid, U.

3. Measure the barrier height B = −U.

4. Swap the lipids with probability p = e^{−B/kBT}.

Using the BHD algorithm, we model the equilibrium self-diffusion of lipids on the lattice. The lipids interact with each other through Lennard-Jones potential, from which we calculate the total interaction energy, U, of a pair of lipids (Fig. 1b). U depends on the local lipid configurations and there are 2^{18} ≈ 2.6 × 10^{5} of them. From a random sampling of these configurations, we obtain the distribution of barrier heights, which vary between 0 and 6k_{B}T at 310 K (Fig. 1c).

Barrier heights chosen this way are unrealistic as they do not capture the various entropic effects that determine the free energy of the transition state. Obtaining the “correct” barrier heights would require detailed molecular simulation followed by systematic coarse-graining of the barrier heights, which is beyond the scope of this manuscript.^{53,54} Instead, our model provides a simple alternative that can be used to understand the effect of active fluctuations on lipid transport, which is the main focus of this paper. Indeed, similar models have been used previously with similar intentions.^{55–60} Another concern is that if U is positive, such as in the presence of long-range repulsive forces, B will become negative. There, we need to identify a better definition for B. Here, we do not consider long-range repulsive forces, and B is always non-negative.

Once a passive lipid is inside the circular region defined by the aster, it can be advected by active moves if it also binds to the ACS. Because passive lipids stochastically bind to the ACS, we assume that the passive lipids have some binding probability, p_{b}, which controls how often they bind to the ACS. A passive lipid bound to the ACS is advected to its core.^{51} The detailed algorithm and simulation parameters are given below.

1. Aster remodeling.
Simultaneously with lipid dynamics, asters follow life-death processes with remodeling rate (τ_{a}) such that the area fraction of asters (A_{frac}) remains constant. The number of aster discs at any time is the same and defined by . Once asters are initialized, a lifetime (t_{life}) sampled from an exponential distribution with mean lifetime (τ_{a})^{61} is designated to each aster from the time of birth (t_{birth}). No overlap between asters was ensured while carrying out the aster birth-death process. When t − t_{birth} = t_{life} aster disappears and appears with some other independent location and lifetime.

2. Algorithm for active moves.
The detailed balance-breaking or active moves are implemented on the lattice through Kinetic Monte Carlo exchange moves, slightly modified from ref. 51. The algorithm chosen to perform the exchange move is as follows:

1. A lipid is chosen randomly from L × L sites.

2. Next, a neighbor is chosen for the exchange.

• If the chosen lipid is passive and is bound to the ACS (with binding probability p_{b}), then the neighbor lipid for exchange is selected to lie in the aster core direction with probability p > 0.25. Other neighbors are chosen with probability (1 − p)/3.

• Otherwise, any of the four neighboring lipids are chosen with equal probability.

3. p_{b} is defined from the two-state model of the (un)bound states of passive lipid with ACS. Hence, where ΔE is the energy difference between the bound and the unbound state.

4. Let _{i}, _{f} and _{a} be the position vectors of the chosen passive lipid, neighbor site for exchange, and the aster core, respectively. To perform the directional move on the lattice, we have defined three cases from _{a} − _{i} = Δ = Δx + Δyŷ, where Δ is the position of aster core w.r.t. the chosen passive lipid.

(a) |Δx| > |Δy|: the position of exchange site is given by, _{f} = (x_{i} + sgn(Δx), y_{i}).

(b) |Δx| < |Δy|: similarly, _{f} = (x_{i}, y_{i} + sgn(Δy)).

(c) |Δx| = |Δy|: the direct diagonal move is not possible. Therefore, (x_{i} + sgn(Δx), y_{i}) or (x_{i}, y_{i} + sgn(Δy)) is chosen randomly.

5. Also, passive lipids near the outer aster edge feel interaction with ACS, and radial move is performed with probability (1 − p_{X})p_{b} where X is the species lying on the aster core before the swap. p_{X} = 0 if X is an inert lipid, else it is p_{b}.

6. If a passive lipid is found on the aster core, it cannot move on its own, and no exchange move was performed.

7. For passive lipids away from any aster, BHD move is performed.

Simulation parameters | Values |
---|---|

Lattice size (L × L) | 100 × 100 |

Radius of aster (R_{a}) |
8 |

Areal density of asters (A_{frac}) |
0.2 |

Mean lifetime of aster remodeling (τ_{a}) |
10 |

Equilibration steps | 500 steps |

Time to reach steady state | 1000 steps |

Production run | 18500 steps |

No. of replicas for each (T) | 40 |

Probability to move towards aster core (passive lipids) (p) | 0.8 |

Temperatures (T) | 281–321 K (spacing = 3 K) |

Binding probability of ACS to PM (p_{b}) |
0.0–1.0 (spacing = 0.2) |

Interaction strength of LJ (J/k_{B}T at T = 293 K) |
1/1.43 (like lipids) & 0 (unlike lipids) |

LJ interaction cutoff radius | 2.5σ |

1 lattice unit | 2^{1/6}σ |

Fig. 2 Activated diffusion in CG-MD model. (a) and (b) MSD (〈X^{2}〉) and diffusivity (semilog scale) of pure DPPC membrane at different T (298–314 K). (c) The activation energy (E_{A}) of lipids for different PM models k_{B}T at 310 K. All the results from long time scale simulations. Error bars are the standard deviation of 8 replicates. In (b) logD(T) is plotted vs. 1000/T, as per experimental convention.^{17} |

We found that individual lipids showed correlated motion with their neighbors and next-nearest neighbors when the lag time was short (ps–ns) (Fig. 3a), which has also been shown before.^{68} In contrast, in the long time scales where we examined ensemble-averaged MSD curves (Fig. 2a), we noticed diffusive behavior implying random movement of lipids. Therefore, the bulk, long-time behavior of the lipid transport, as seen in the MSD curves, is strikingly different than their local short-time transport. Hence, we analyzed the local nanoscale transport through means different from ensemble-averaged MSD.

First, we measured the local diffusivities of the lipids using their MSD averaged over grids of size 3 nm × 3 nm (Fig. S3, ESI†). From these distributions, we computed the distribution of the transport coefficient, D(T), at different temperatures and measured their average value. Both showed weak but clear Arrhenius-like T-dependence (Fig. 3b and c). Therefore, in equilibrium, even at the nanoscale, we observed clear signatures of activated diffusion, which consolidated our hypothesis that active driving is needed to make activated diffusion T-independent.

Because MSD-based measurements of transport coefficients are defined in the t → ∞ limit, it is unclear whether MSD-based measurement of D(T) is accurate or not. A more accurate alternative is to measure the T-dependence through the correlation functions (Section II C) of the lipid transport. For this purpose, we measured the displacement–displacement correlation function,^{41–48} which showed two prominent peaks at around 0.51 and 0.83 nm distance (Fig. 3d). The peaks showed weak but systematic variation with temperature: both decreased in height with increasing temperature (Fig. 3d-i and d-ii). However, the variation is so small that it will be impossible to detect in present experiments.

Our observations suggest that equilibrium correlated flows cannot render the transport T-independent. Hence, we conjectured that nonequilibrium flows of lipids must be the origin of T-independence in the FCS experiment. At the smallest lengthscales probed by the FCS experiments,^{5,69} this nonequilibrium flow must affect tens of thousands of lipids simultaneously, indicating that the cortical ACS is a possible driver of such flows.^{13,52,70} Unfortunately, it is tough to investigate the coupled dynamics of the ACS with the membrane using CG-MD because of the prohibitive computational cost. To the best of our knowledge, even with the best supercomputers, the interaction between membrane and only a single actin filament has been studied.^{71} Hence, we resort to a mesoscopic lattice model^{51} of the membrane to investigate the effect of the ACS on the membrane–lipid transport (Section II D).

In the membrane, a lipid interacts with its neighboring lipids. When two adjacent lipids swap places, there are significant rearrangements of the local lipid configurations. Such transition states usually have higher free energy than the configurations before and after the swap. From a modeling perspective, it implies that every lipid swap requires overcoming a kinetic barrier with activation energies correlated with the diffusing lipid's total interaction energy with its neighbors. In Kawasaki algorithm, we ignore this activation energy. The resultant transport process is unphysical: the coefficient of self-diffusion shows nonmonotonic variations with T (Fig. 4a). For temperatures higher than 312 K, diffusivity decreases with 1/T, and below 312 K, diffusivity increases with 1/T, violating experimental and computational observations,^{17,73,74} where diffusivity decreases monotonously with 1/T.

Fig. 4 (a) Kawasaki dynamics predicts unphysical variation of D(T) with T. (b) BHD gives us the desired activated diffusion given by Arrhenius kinetics^{17}E_{A} = 5.81k_{B}T at 310 K. |

Kawasaki dynamics gives such inaccurate results because it incorrectly increases the swap rate of like lipids in lipid domains, where swapping does not cost any energy. However, because like lipids attract each other, the energetic cost of kinetic rearrangements is high, resulting in a high activation barrier. Therefore, kinetically, we expect like lipids to slow down in a lipid domain, as seen in experiments.^{5} However, Kawasaki algorithm is agnostic of this cost, leading to inaccurate T-dependence at low temperatures (high 1/T), where lipid domains form easily. At higher temperatures (low 1/T), lipids of various species are well-mixed, and the kinetic barriers are close to zero. Hence, in such a situation, the Kawasaki algorithm predicts the correct temperature dependence.

B = −U | (11) |

The self-diffusion coefficient obtained from this model follows Arrhenius kinetics with an activation barrier of E_{A} = 5.81k_{B}T at 310 K (Fig. 4b), which is within a factor of three of the mixed-membrane E_{A} and comparable to single component membrane E_{A} obtained from CG-MD simulations (Fig. 2c). Therefore, in the lattice model, we have used the BHD algorithm to model all equilibrium diffusive moves.

The appearance and disappearance of the asters introduce active fluctuations in the lipid movement, which changes their transport properties. To illustrate this change, we studied an extreme system where the passive lipids, once inside an aster patch, remained always bound to the aster and moved only through active moves. We found that the MSD of the passive lipids showed negligible T-dependence (Fig. 5a), implying the T-independence of D(T) (Fig. 5a-inset). Therefore, our hypothesis that active fluctuations lead to T-independent diffusivity seems correct.

To check whether such weak variation can be captured in experiments, we computed the Q_{10} metric.^{18,77} To have significant T-dependence Q_{10} must be greater than 1.2.^{18} However, we found that Q_{10} for the passive lipids was lower than this threshold. Hence, its T-dependence will not be detected in experiments.

The inert lipids, which never couple to the ACS, also show altered transport properties, with weak superdiffusive behavior at small τ and weak subdiffusive behavior at large τ as shown in Fig. 5b. Therefore, to investigate the T-dependence of the transport, we cannot use diffusivity and must use a general transport coefficient. In general, we can write:

〈X^{2}〉 = 4D(T)τ^{α}, | (12) |

To test our hypothesis further, we systematically reduced the binding probability, p_{b} of the passive lipids with the ACS. We found that for low p_{b}, the equilibrium T-dependence was recovered, with the exponential variation of D(T) with temperature for both passive and inert lipids (Fig. 6). At high p_{b}, D(T) became T-independent for passive lipids and weakly T-dependent for inert lipids. These observations provided the final confirmation that active, nonequilibrium, fluctuations are responsible for T-independent transport in our system.

It is worthwhile to make some comments about membrane transport of lipids in light of these observations. First, lipids may move superdiffusively at short times even when directly not coupled to active driving forces. Hence, when developing reaction–diffusion-type models for membrane transport, we need to be aware of this fact. Second, as noted earlier,^{51} lipid segregation is enhanced in the presence of ACS-driven active fluctuations, which may help cells maintain membrane homeostasis in the presence of thermal shock or when thermal fluctuations are not sufficient to create segregated membrane structures, such as lipid rafts.

We have made two striking observations. First, at the nanometer scale, the flow of the lipids is strongly correlated yet shows clear signs of activated diffusion. On the other hand, the macroscopic activated diffusion of lipids shows no such correlated movements. Where the microscopic correlation is lost is unclear. It is puzzling that even though the lipid movements are strongly correlated at the nanoscale, the standard metric of transport, such as the MSD or the diffusion coefficient, does not show it. A theoretical study would require understanding the many-body transport of lipid clusters that move together at the nanometer scale. The collective motion of the lipid clusters may be random, which is why, perhaps, the MSD or the diffusion coefficient does not capture it. However, we do not have any proof for this hypothesis. We are also unsure about the origin of the weak T-dependence in lipid self-diffusion at nanosecond timescales, where such correlated motion is observed. At that scale, it is difficult to calculate the transport coefficients unambiguously, as they are defined in the τ → ∞ limit. However, using theoretical approaches to model single-particle tracking data, it may be possible to resolve this puzzle.^{83–85}

The second striking observation is that nonequilibrium flow from the ACS makes the transport coefficient, D(T) weakly T-dependent. Even though we have demonstrated this T-independence for a particular nonequilibrium driving, namely the presence of the ACS, it is easy to see that our model is not fine-tuned to any specific biological system. Instead, the key ingredient is the stochastic switching between active and passive moves. As shown in Fig. 6, reducing the propensity of the active moves to the passive moves increases T-dependence. Therefore, the T-independence of D(T) originates from the breaking of detailed balance and is a general observation that is not surprising by itself. Indeed, it is well-known that the fluctuation–dissipation theorem, which relates a transport coefficient to the ambient temperature, is generally violated far from thermal equilibrium.^{8,86,87} What is remarkable is that the T-independence of D(T) provides a novel mechanism to ensure the robustness of cell signaling.

Specifically, let's consider the precision of molecular sensing on the cell membrane in the slow diffusion (fast reaction) limit, where the precision of sensing concentration, c, of a ligand^{88} varies as

(13) |

(14) |

(15) |

≈1.26 | (16) |

T-Independence in our model arises generically when lipids are transported through correlated flows. Hence, we have reasons to believe that this observation is general. However, it is important to note that our model does not explore all possible variations that can be imagined. For example, we have made the simplifying assumption that all asters are of the same size. In contrast, existing data suggest that asters have a nontrivial size distribution.^{89} Our data shows that changing the asters' size without changing the area fraction does not change the results qualitatively because it does not change the ratio of the active to passive moves (Fig. S4, ESI†). In contrast, it is unclear whether the correlated spatial creation and destruction of the asters can change the results qualitatively. Because it does not change the active-to-passive moves ratio, we have reasons to believe that it should not change the results qualitatively.

Finally, we have used a new algorithm called barrier-hop dynamics (BHD) to simulate diffusion. We have shown that Kawasaki dynamics, the default choice to model lattice diffusion, does not reproduce the qualitative features of equilibrium diffusion, but BHD does. We have demonstrated that Kawasaki dynamics gives unphysical transport properties. Although BHD reproduces the general features of the equilibrium D(T), the choice of barriers is unphysical. That being said, the purpose of introducing BHD was to reproduce the general features of lipid diffusion in thermal equilibrium, which we succeeded in. However, an appropriate choice of diffusion barriers would have allowed us to make quantitative comparisons with the CG-MD model calculations.

Irrespective of these drawbacks, our calculations provide valuable insights about the transport of lipids on the plasma membrane. It is surprising that even lipids not coupled to the ACS can have D(T) that is almost T-independent. As we have shown, a consequence of such temperature-independent transport processes is the robustness of cell sensing. Because of their ubiquity, transport affects not just sensing but all cellular processes. Therefore, it is not hard to imagine that their temperature independence will impact the stability of all cellular processes.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp02470b |

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