Estimate of entropy generation rate can spatiotemporally resolve the active nature of cell flickering

We use the short-time inference scheme (Manikandan, Gupta and Krishnamurthy, Phys. Rev. Lett. 124, 120603, 2020), obtained within the framework of stochastic thermodynamics, to infer a lower-bound to entropy generation rate from flickering data generated by Interference Reflection Microscopy of HeLA cells. We can clearly distinguish active cell membranes from their ATP depleted selves and even spatio-temporally resolve activity down to the scale of about one $\mu$m. Our estimate of activity is model--independent.


I. INTRODUCTION
At the dawn of biophysical research [1], it was already realized that the fundamental property of living cells is that they are not in thermal equilibrium even when they are statistically stationary -they consume energy and generate entropy, see e.g., Ref. [2] for a recent review.Nevertheless, tools of equilibrium physics are frequently used to interpret results of experiments on living cells, for example, equilibrium models are used to infer bending rigidity from flickering data -vibrating fluctuations of a cell membrane, see e.g., Ref. [3] for this and several other examples.Although the possible influence of active transport on flickering was first pointed at least seventy years ago [4] investigation into the essential non-equilibrium feature of flickering, by comparing flickering data from a healthy cell with its ATP (adenosine triphosphate)-depleted self, has peaked in the last two decades [5][6][7], see also Ref [8] and references therein.For RBCs (Red Blood Cells), the first unequivocal demonstration of their non-equilibrium nature was shown via the violation of the fluctuation-dissipation theorem [9].However, no attempt has been made so far to spatiotemporally resolve, from flickering data, the fundamental physical quantity that characterizes nonequilibrium, v.i.z., the rate of entropy production, σ.
On the one hand, recent spectacular progress in visualizing and tracking biological processes [10][11][12][13][14], with unprecedented accuracy and control raises hopes that measuring entropy production rate for cellular processes is indeed possible.On the other hand, there are several difficulties: (a) They are small in magnitude, of the order of few k B s per second, and lie below the threshold of detection for the existing calorimetric techniques [15].(b) Thermal fluctuations cannot be ignored hence any measurement results in noisy readings [16].(c) Typically, we have access to only a few degrees of freedom, e.g., flickering gives us access to the fluctuations of the cell membrane down to a certain length and time scale while fluctuations at smaller length and time scales and the motion of the cytoskeleton that drives flickering, are not accessible.(d) In far-from-equilibrium regimes, there are very few general principles.
The theoretical understanding of the far-fromequilibrium behavior of microscopic systems have undergone a revolution over the last two decades [17,18] giving rise to the subfield of statistical physics called Stochastic thermodynamics, which can in-principle extract entropy production rate from long-enough stochastic trajectories of all the degrees of freedoms of the system [18,19].With limited data -limited in both time and number of accessible degrees of freedom -the best we can do is to set bounds [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34].
In this paper, we use a recent addition to this list of techniques, the short-time inference scheme [29,35,36], together with flickering data of HeLa cells to estimate spatio-temporally resolved entropy production rate.As the entropy production rate is the fundamental characteristic of the active nature of the fluctuations, in the rest of this paper we shall use the word activity and entropy production rate interchangeably.
The short-time inference scheme is built on the thermodynamic uncertainty relation [22] which states that a lower bound to the entropy production rate can be obtained via fluctuations of any arbitrary current J in phase space, as, where ⟨•⟩, and Var (•) denotes mean and variance of a random variable calculated from a statistically stationary time-series of phase space trajectories over a time-interval t and k B is the Boltzmann constant.Refs.[29,35,36] extended this result to show that for a large class of nonequilibrium systems with an overdamped Langevin dy- namics, the inequality, (1), saturates in the short time limit, i.e., Here J ∆t is calculated over time ∆t and the mean and the variance is calculated over an ensemble of such time intervals.This scheme has certain key advantages over its competitors: one, it is model independent -the precise knowledge of the underlying overdamped Langevin equation is not required; two, it is not limited to stationary time series.This scheme has been successfully used to estimate the entropy production rate from a moderate number of realizations of stochastic trajectories of time-dependent systems [31].The scheme was also tested experimentally using the stationary trajectories of a colloidal particle in an stochastically-shaken optical trap [38].Here we apply Eq. ( 2) to analyze flickering data from living cell membranes.

II. METHODS
In Fig. (1) we pictorially summarize our method.We first obtain a movie of flickering data for the basal membrane, a part of the membrane attached to the substrate, of adherent HeLa cells using Interference Reflection Microscopy (IRM).HeLa cells (CCL-2, ATCC) were grown in Dulbecco's Modified Essential Medium (DMEM, Gibco, Life Technologies, USA) supplemented with 10% fetal bovine serum (FBS, Gibco) and 1% Anti-Anti (Gibco) at 37 • C under 5% CO 2 .Experiments were always performed after 16-18 hours of cell seeding.IRM imaging [37,39] was conducted using a motorized inverted microscope (Nikon, Japan) equipped with adjustable field and aperture diaphragms, a 60× water immersion objective (NA 1.22), 1.5× external magnifica-tion, an onstage 37 • C incubator (Tokai Hit, Japan), an s-CMOS camera (ORCA Flash 4.0, Hamamatsu, Japan), a 100 W mercury arc lamp, an interference filter (546±12 nm), and a 50-50 beam splitter [37].Pixel intensities were converted to relative heights (distance of the membrane from the coverslip) using a MATLAB (MathWorks, USA) code after obtaining the intensity-to-height conversion factor determined by a beads-based calibration method as reported previously [37].Every experiment was preceded by an independent calibration.
A typical image of a cell is shown in Fig. (1B).In addition to the cell the background is also visible.For the analysis, we divide this cell into several square patches of equal size (∼ 2µm 2 ).One such patch is shown as a blue square in Fig. (1B).Each patch is made of M ×M pixels.Thus the time evolution of each patch is completely described by M 2 stochastic variables I = (I 1 , I 2 , . . ., I M 2 ) -the light intensity at each of the pixels.We record movies with a time-step ∆t = 50ms between two consecutive snapshots.A typical movie consists of N f = 2048 snapshots -see Fig. (1C).In order to apply the inference scheme in (2), in principle, we can calculate currents from two consecutive snapshots and average over the number of snapshots to calculate the quantity inside the square brackets in (2).In that case, we have to perform numerical optimization over M 2 space which is a formidable problem.Following Ref. [28] we use Principal Component Analysis (PCA) to reduce the dimension of the problem in the following manner.Construct the covariance matrix C, with elements C ij ≡ ⟨I i I j ⟩ − ⟨I i ⟩ ⟨I j ⟩.The average is taken over the number of frames, N f .This is averaging over a time interval, in which the time series is assumed to be stationary.The method applies equally well for ensemble averaging [31].The projections of the data along the eigenvectors of this covariance matrix, in decreasing magnitude of eigenvalue, are the principal components.Fig. (1D) shows a typical example of the dynamics in the first two components, I 1 (t) and I 2 (t).In general, the state of the system can be projected to any N ≤ M 2 dimensional vector I = (I 1 , . . .I α , . . .I N ).Next, we define a scalar current: where d(I) is any arbitrary N dimensional function.Then we need to find the optimal function d * (I) which maximize the bound in (2).We use two different algorithms: (A) We use a linear function d α (I) = c αβ I β , where the matrix c αβ is a matrix of constant coefficients and perform the optimisation by the particle-swarm algorithm.(B) We use a non-linear function for d(I) represented by a feed-forward neural network d(I|θ), where θ are the parameters of the network.For a fixed choice of the parameters, the mean and the variance of the current are computed by averaging over the index i.The analysis can then be extended to cells under different physiological conditions and at different times, to obtain a spatio-temporally resolved entropy production map as in Fig. (1E).Both the algorithms give similar results; with (A) yielding a slighly lower estimate than (B).See Appendix A for details.

III. RESULTS
In Fig. (2A), we show how our method performs on the experimental data as the cells are ATP-depleted by incubating them in glucose-free ATP-depleting medium.We do this by adding 10 mM sodium azide and 10 mM 2deoxy D-glucose to the cells [40] in M1 Imaging medium (150 mM NaCl; Sigma-Aldrich), 1 mM MgCl2 (Merck, Kenilworth, NJ), and 20 mM HEPES (Sigma-Aldrich), and incubating them for 60 minutes for ATP depletion.In recent work by some of the authors, it was found that ATP-driven activities increase temporal fluctuations and flatten out spatial undulations -see Figure 3 in [37] and discussions therein for the detailed analysis and discussion on the effect of ATP depletion on membrane fluctuations.
In the leftmost row of Fig.
(2) we show the image of the cell -the cell boundaries are marked in yellow.The next column shows a pseudocolor plot of log 10 (σ) of the live cell -marked by control.The next three columns show how the entropy production rate changes after 20 minutes, 40 minutes and 1 hr of incubation.In each of these cases, we use a movie of N f = 2048 snapshots where consecutive snapshots are separated by ∆t = 50ms.Since the total duration of the movie is quite short compared to the time-scales over which ATP-depletion operates, each of these movies are considered statistically stationary.We have performed these experiments for a total 31 HeLa cells, with two independent repeats.Images of two typical cells are shown in Fig. (2A) and Fig. (2B).The images of several other representative cases are in the supplemental material.Our results clearly demonstrate that the entropy production rate computed using (2) spatiotemporally resolves the active nature of cell membrane fluctuations.Notably, we see many patches of high entropy production in the active cell membrane which is well contrasted with the background, and patches of low entropy production in the ATP-depleted membrane, less contrasted with the background.We also see an overall decay of σ in time.
To calculate reliable statistical properties of the entropy production rate -for the control and the ATPdepleted ones -we limit ourselves to patches that are inside the cell and away from the nucleus -an example is shown in Fig. (2C).The relative heights of these patches lie within the range ∼ 0 − 100 nm -termed as first-branch regions (FBRs).Henceforth we call these patches FBR-patches.We use rank-order methods to calculate the cumulative distribution function (CDF) of σ from all the FBR-patches of all the cells, see Fig. (2D).The rank order method does not suffer from the binning errors that plague histogram based methods to calculate the probability distribution function.We calculate the PDF of σ by calculating the derivatives of the CDF.The complement of the CDF, defined as 1 -CDF has an exponential tail to the right.This implies that the PDF of σ, also has an exponential tail.In Fig. (2E) we show how the mean value of the σ decreases under ATP depletion.A straightforward computation of the average σ over all FBR patches for the control cell yields a value of approximately 4×10 −3 k B s −1 across a patch area of 2µm 2 .The exponential tail of the PDF implies that σ shows large fluctuations.In particular, the average of the top 10% of patches is nearly five times higher than the mean, and the average of the top 50% of patches is nearly double -See Appendix A 3.
Which processes on the membrane are the major contributors to the measured activity?Of the many ATPdependent processes that may actively impact membrane fluctuations, forces originating at the underlying actomyosin cortex are important [37,41].The effect of actin polymerizing forces [42] as well as forces by myosin motor protein contracting the actin network are expected to be diminished if the cortex is weakened or removed.It has been shown [43] that the drug Cytochalasin D (Cyto D) suppresses actin polymerization and weakens the cortical network.This is achieved by treating the cells with 5 µM cytochalasin D (Cyto D, Sigma Aldrich) for 1 hour in serum-free media [37].It's effect on membrane fluctuations was further found to be highly spatially inhomogeneous -see Figure 4 in Ref. [37] and discussions therein for the detailed analysis and discussion on the effect of Cyto-D treatment on membrane fluctuations.The network is subsequently contracted and fragmented by myosin, the cortex ruptures, clears from most of the membrane area and accumulates at multiple foci.Thus we expect that a possible effect of cyto-D is to decrease the entropy production rate.In Fig. (3) we show the effect of cyto-D on the entropy production rate.We indeed find a decrease in activity.The decrease of the mean value of the measured activity is smaller than the decrease measured for ATP depletion.
Note that the probability distribition function of entropy generation rate has an exponential tail.We have found a significant number of patches with estimates of σ at least 5 times higher than the mean value (see Fig. 5 of supplemental material).As our measurements find set a lower bound to the actual entropy production rate, it is likely that in reality the PDF falls off even slower than an exponential.This underscores the heterogeneity nature of activity of the membrane.

IV. DISCUSSION
Our attempt to tease out the essential non-equilibrium feature from flickering by using the entropy production rate as a measure of activity has a clear advantage over earlier methods, such as the ones dependent on the breakdown of the fluctuation-dissipation relation e.g.Refs.[9,[44][45][46].First, the entropy production rate is the crucial measure of non-equilibrium -if a system A has a higher entropy production rate than a system B then A is further away from equilibrium than B. Second, we are able to spatiotemporally resolve the entropy production rate thereby able to identify regions that have higher entropy production rate.This has not been achieved before.Third, our method gives a model-independent estimate of the lower-bound of entropy production whereas Refs.[9,47] must use a model for active fluctuations.Finally, the measurements in Refs.[9,[44][45][46][47] are typically invasive in nature -they involve tracking the response of microscopic beads attached to systems and external perturbations [11] or attaching fluorescent proteins or filaments to the relevant degrees of freedom [12,13].In contrast, our analysis is less invasive and is applicable to movies made by exploiting the naturally existing intracellular contrast [14,28].
Several comments are now in order.First, on the role on noise in our data.Clearly, if we apply our algorithm to patches outside the cell we would obtain a small but nonzero number.We define a signal-to-noise ratio to be the activity obtained for patches inside the cell over patches outside the cell.We calculate this for patches of different sized and find that the signal-to-noise ratio is maximum for patches of size 2µm 2 (Fig 3A in supplementary material).All the results we show in this paper uses this size.Furthermore, for this patch size the activity grows linearly with area of the membrane.See Appendix A 1.
Second, in addition to the necessary spatio-temporal coarse-graining, which is part of any experimental measurement, we have performed a dimensional reduction by using principal component analysis and retained the first few components.How many principal components must we keep?We use a method proposed in Ref. [28].We randomly shuffle our data along the time axis and then again calculatate the eigenvalues of the correlation matrix.We find that the eigenvalues of this shuffled data are very close to one another.These set the noise floor of our measurements of principal components.Now going back to the eigenvalues calculated from the original data we find that only the first few, at most 10, eigenvalues are significantly larger than the noise floor (see Appendix A 2). Next, let us consider the algorithm where we assumed the optimal function to be a linear function of the currents.We limit our analysis to the first two principal components.Increasing the number of components to 10 do not have any significant effect on the estimated value of σ.For the second algorithm, where we used a neural network representation, the estimate of σ changes as the number of components are increased from 2 to 10, during the training stage.But yielded minimal change in the testing stage.Nonetheless, we retained the first 10 components for all the results shown in this paper.The estimates of σ obtained by the two algorithms are comparable; the neural network-based estimates are slightly higher.Furthermore, these also have better signal-tonoise ratio between the cell and the background.
Third, we compare our findings with the already known estimates of entropy production rates in microscopic biological systems.To the best of our knowledge there are no results for HeLA cells.Measurements that tracked a single filament of microtubules report an entropy production rate of 5k B min −1 [32] while a single actin fiber contracted by myosin motors report an entropy production rate, per unit length, to be ∼ 1 k B /(µms) [48].At the level of cells, e.g., in Ref. [33], the entropy production rate was quantified from experimental recordings of spontaneous hairbundle oscillations in mechanosensory hair cells from the ear of the bullfrog.Interestingly, by applying (1), without any optimization over the current, the study found entropy productions of the order of 10 3 k B s −1 .Another recent work [47] that analyzes flickering data from circumference of RBCs in a model dependent fashion finds a entropy production rate of ≈ 10 6 k B s −1 .This estimate depends on the details of the model used for cellflickering.However, it is argued to be compatible with microcalorimetric measurements of heat generation from packed RBCs in bulk [49,50].In contrast, we find an average entropy production rate of ∼ 4 × 10 −3 k B s −1 over a 2µm 2 patch of the membrane.The distribution of the entropy production rate over patches feature a fat tail to the right, and there are a significant number of patches with activities up to five times higher than the mean value (see Appendix A 3). Multiplying just the mean with the area of the basal membrane (∼ 10 3 µm 2 ) we find an estimate close to a few k B s per second.Although, this is at least an order of magnitude higher than the numbers reported in Ref. [32], who used measurements of calcium flicker trajectories from single cells reported in Ref. [51], these values are several orders of magnitude lower than Refs.[33,47].There may be several reasons for the low estimates we obtain, but the obvious reasons appear to be high noise levels in the experimental data and lower bounds of the estimate in Eq. ( 2) resulting from the necessary spatiotemporal coarse-graining and dimen- sional reduction.Furthermore, we reiterate that the goal of our paper is not to calculate the total entropy production by all non-equilibrium processes in a cell -our focus is solely on the cell membrane.We do expect values significantly smaller than the total rate of consumption of ATPs in a cell because the membrane fluctuations are just one non-equilibrium process among many that goes on in a living cell.
Fourth, can we estimate the time-scales of the active processes that play the most significant role in entropy generation?Unlike many of the cases listed above, the flickering data does not contain any clear "active spikes".This may be because multiple active processes with a range of time scales, (0.1-2s [9,37]) act on the membrane.Furthermore, an estimate of time-scales is necessarily model-dependent.We have analyzed such a model to obtain a closed form estimate for the entropy production rate, and argue why it cannot be reliably applied to experimental data.See Appendix B for details.
In summary, there are a couple of areas that could benefit from further studies.Firstly, the entropy generation rate we obtained may be too small to be relevant for the metabolic processes of the whole cell.This could be attributed to the high noise level in our data; at most, only the first ten, sometimes even just the first two, principal components contain useful information.Thus, we anticipate that with improved resolution, our approach may unveil a significantly larger entropy generation rate.Secondly, the effect of Cyto-D on the entropy generation rate is quite small.Despite the significant impact of the acto-myosin cytoskeleton, which is notably affected by Cyto-D, the most active fluctuations of the membrane remain largely unchanged.Consequently, although we expected Cyto-D to substantially alter the entropy production rate, this is not the case.At present, we do not fully comprehend this aspect of our results.
To end on a positive note, besides capturing the celllevel activity, we report for the first time, a spatiotemporally resolved activity.As shown in Fig. 2, cells have high activity regions which disappear on ATP depletion.This offers explicit evidence of heterogeneity of activity in the cell membrane pointing to a lack of rapid establishment of an statistically homogeneous (although non-equilibrium) state.This technique can be combined with fluorescence imaging of a variety of "active" proteins or structures to measure their localized action.Thus, comparing maps of lateral organization of molecules of interest with activity maps could potentially identify those involved in actively regulating processes like cell migration, endocytosis, and mechano-sensing.

∆tVar (J ∆t )
. (A1) Here, J ∆t is calculated over time ∆t, and the mean and the variance are calculated over an ensemble of such time intervals.Explicitly, we are searching over the phase space of the first few PCA components (see subsection A 2), and currents constructed using a linear basis.We defined an N -dimensional vector d(I) as a linear combination of the components of I, i.e., where the matrix c αβ consists of constant coefficients.At time t = i∆t, we define a scalar current: (A3) While using the particle swarm algorithm, we fixed N = 2. Increasing the number of components was did not have a significant effect on the estimates of entropy production rate, primarily due to the presence of experimental noise.The details of the algorithm can be found in Ref. [38].
Linear representations of the function d(I) such as the above severely limits the phase space of currents we are working with.Ideally, we should consider models which are more generic in the sense of having a non-linear basis, and straightforwardly scalable to arbitrary number of principle components.The most natural choice is to use a Neural Network.Hence we have performed and independent estimation of the entropy production rate by using a neural network to represent the vector d(I).The network architecture we used consists of two fully connected layers with a hyperbolic tangent (Tanh) activation function applied after each layer.The input dimension (x dim) is set to N , and the number of nodes in the hidden layer (nb nodes) is set to 256.For any input, this network returns an N dimensional, non-linear representation of the vector d := d(I|θ), where θ are the parameters of the nerual network.The presence of the activation function tanh makes this representation arbitrarily non-linear, and bounded.
In Algorithm 1, we provide a schematic summary of the inference scheme with the neural network model.The hyperparameters of the algorithm, such as the number of nodes in the hidden layer, the number of Principle components used and the number of steps used for training are chosen by trying to maximize the signal-to-noise ratio between the estimate of entropy production rate between the cell and the background.Crucially, we have also used a data-splitting scheme where the first half of the data is used for training the model and the second half of the data is used to estimate entropy production rate using the trained model.See [31] for details.This approach helps overcome issues of overfitting to the data, which can lead to very high estimates of dissipation.Overfitting was also found to happen when the training was done for a large number of steps.Hence, we limited the number of steps in training to 100.With the data-splitting scheme, we get estimates of dissipation that is consistent with the estimates obtained using the Particle Swarm algorithm.See Fig. 4 for the comparison of results between the two algorithms.Results for more cells using both algorithms are shown in the supplemental material.

Dependence of the entropy estimate on the patch size
In Fig. 5, we summarise our analysis of how entropy production estimates depend on the cell area.We identified an optimal patch size of 20×20 pixels (approximately 2; µ; m 2 ), which yielded the highest signal-to-noise ratio when comparing activity within the FBR region to background values.For this optimal patch size, we further found that entropy production computed over the entire cell membrane increases linearly with the membrane's area (number of patches).

Principle Component Analysis
Here we demonstrate that the principle components other than the first few do not significantly contribute to the dynamics.We consider the data corresponding to the first cell in Figure 2 of the maintext.In Fig. 6, we plot the first 20 eigenvalues corresponding to PCA components for the normalized experimental data (Blue) and the normalized data randomly shuffled along the time axis (Purple).The error bars represent the standard deviation over different patches.The PCA eigenvalues for the shuffled data show the noise level in the system [28], which is slightly above zero.We observe that the eigenvalues rapidly decrease in magnitude and reach the noise levels within the first few principal components.This indicates that the principle components other than the first few closely approach the noise floor and do not significantly contribute to the dynamics.

Percentile Averages
Due to significant inhomogeneity across the cell surface, relying solely on the mean value of the entropy production rate over patches may be inadequate.Hence here we look at percentile averages, where we plot the average σ over the top x% of patches of the control cells.As shown, the average of the top 10% of patches is nearly five times higher than the mean, and the average of the top 50% of patches is nearly double.

Appendix B: Entropy production rate in a model of active membrane
We calculate the entropy production rate for an elementary model of active membrane [52].Usually, flickering data is interpreted with a model [3,53,54] where the membrane is described by a height field h(x, y) above the x-y plane in Monge gauge described by the Hamiltonian in equilibrium at temperature T .Here γ is the surface tension and κ is the bending rigidity of the membrane.
As the fluctuations of h are small, the model uses a small deformation approximation of curvature.We shall call this the Gaussian model.The Hamiltonian is diagonal in Fourier space.In the same spirit as Refs.[8,9,52] we write an active version of this model by writing an equation of motion for each Fourier mode ĥ(q, t) coupled to an Ornstein-Uhlenbeck noise [55] λ(q, t) as, ∂ ĥ(q, t) ∂t + 1 τ (q) ĥ(q, t) where τ (q) = 4ηq κq 4 + γq 2 , (B2c) ⟨ζ(q, t)ζ(−q, s)⟩ = 2D(q)δ(t − s), (B2d) The membrane is assumed to undergo a stochastically driven overdamped motion.The viscosity of the fluid around the membrane is η which is related to the amplitude of the stochastic driving ζ by fluctuation-dissipation theorem (B2f).In addition, there is active noise, ζ A which is assumed to be white-in-space and an Ornstein-Uhlenbeck noise (B2b) in time with a correlation time τ A and an amplitude D A .This active noise models both active processes on the membrane [52], e.g., opening of ion channels, endo and exo cytosis etc, and driving by the cytoskelton [9].In (B2a), we use the parameter b (of dimension s −1 ) to determine the strength of the coupling of the membrane to the active fluctuations.Note that there are other models of time-correlated noises used to model active fluctuations in living systems.In Refs.[8,9,52], the active noise is modelled using a random telegraph process [56], which leads to the same exponential correlation in time for the noise as the Ornstein-Uhlenbeck While the neural network-based algorithm, without a data-splitting scheme, produced the highest estimate, discerning overfitting effects is challenging.Conversely, estimates acquired using neural networks following the data-splitting scheme were lower but comparable in order of magnitude and slightly higher in values to those obtained from the particle swarm algorithm.Thus, we have presented results only from the neural network with data splitting and the particle swarm algorithm.process.In Refs.[57][58][59] the Ornstein-Uhlenbeck model itself is used.
For simplicity, from here onwards, we drop the q dependence of ζ A , τ A , and D A .As we have ignored nonlinearities in the model for the membrane every ĥ(q) is independent of every other ĥ(q).Consequently, the total entropy production rate is a sum over all q, σ = σ q .(B3) The individual terms in the sum corresponding to the entropy production rate for Eqs.B2a and B2b can be calculated analytically [60,61], by mapping to a wellstudied model in stochastic thermodynamics [38,62,63].
We obtain, In (B4b) the subscript "A" denotes the variance being calculated for an active membrane (b ̸ = 0) whereas the subscript "Eq" denotes the variance being calculated for membrane in thermal equilibrium, i.e., b = 0.They can be computed from the knowledge of the stationary distribution of ĥ and λ as [62,64] S(q) A = ĥ * (q) ĥ(q) = Dτ (q) + b 2 D A τ (q) 2 τ 2 We observe that the average of the top 10% of patches is nearly five times higher than the mean, and the average of the top 50% of patches is nearly double.Data is from one set of experiments using 14 cells.Two independent repeats show a similar trend.
We remark that the entropy production rate remains finite in the white noise limit of the Ornstein-Uhlenbeck process.
Notice that, though the model provides an exact analytical expression (B4a) and an accessible form (B4b) for the entropy production rate, applying it on the experimental data is rather challenging.This is because, we do not have access to either the active time-scale τ A nor the equilibrium fluctuations of the membrane S(q) Eq .Furthermore, in reality, the mechanical properties of the membrane such as σ and κ could also change as a function of ATP depletion, and the model ignores any such possibility [65,66].Nevertheless, assuming the model is representative of the actual membrane dynamics observed in the experiments, the following qualitative features can be deduced.
1.The entropy production rate vanishes in the D A → 0 limit.
2. The entropy production rate must be positive.Hence, S(q) A ≥ S(q) Eq (B6) Assuming that the ATP depletion takes the cell membranes close to the equilibrium state, we can further argue that S(q) A > S(q) ATPD where the subscript stands for the ATP depleted cell membrane.
3. The summation in (B3) should converge.Hence, S(q) A → S(q) Eq for large q. (B7) As a corollary, we also obtain S(q) A → S(q) ATPD for large q.
Indeed, we find that fluctuations of the cell membranes we studied experimentally exhibit the qualitative features predicted by this model.We summarize our findings in Fig. (8) for the two cell samples in Fig. 2 of the main

2 CameraFIG. 1 .
FIG. 1.We use Interference reflection microscopy (A) to obtain flickering data from the basal membrane of HeLa cells (B) with a patch marked by a sky blue square.Zoomed in picture of the patch in (C) typically contains M 2 = 400 pixels.The 400 time series are dimensionally reduced to a few by Principal Component Analysis (PCA) (D) The stochastic trajectory of the first two principal components for a typical case.We obtain the entropy production rate, σ by numerical optimization of this reduced problem.Using the first six principal components instead of the first two has a negligible effect on our results.Color map of σ in (E) clearly shows the active cells compared to its background.The sketch in (A) is adapted from Ref. [37].

FIG. 2 .FIG. 3 .
FIG. 2. Activity of cell membrane.(A) and (B) From left to right: Typical IRM image of two live HeLa cells -the boundary of the cells is marked in yellow.Color map of ln(σ) before the cells are treated with ATP-depleting agents.Color map of log 10 (σ) after incubating the cells in ATP-depleting agents for 20min., 40min.and 60min.For clarity, the color map is limited to a fixed range.(C) Representative IRM image of a cell with FBRs (size: 20 × 20 pixels or 2µm 2 ) marked in yellow (D) Plot of (1 -CDF) of σ, for the FBR-patches, for control and three different durations of ATP-depletion.(E) Average of σ over FBR patches under ATP-depletion.

FIG. 4 .
FIG.4.The figure illustrates comparisons of entropy production rate estimates obtained using different algorithms: Particle swarm algorithm which takes in as input the first two principle components of IRM data, and neural network based algorithm which takes as input the first 10 principle components.While the neural network-based algorithm, without a data-splitting scheme, produced the highest estimate, discerning overfitting effects is challenging.Conversely, estimates acquired using neural networks following the data-splitting scheme were lower but comparable in order of magnitude and slightly higher in values to those obtained from the particle swarm algorithm.Thus, we have presented results only from the neural network with data splitting and the particle swarm algorithm.

FIG. 5 .
FIG. 5. (A)The signal-to-noise-ratio computed by comparing the ratio of activity inferred in FBR region with the corresponding value for the background.We find that the patch size 20 × 20 pixels or ∼ 2 µ m 2 has the best signal to noise ratio.(B) The dependence of the entropy production rate on the area of the cell-membrane for the optimal patch size ( 20 × 20 pixels ).We observe that the estimate increases approximately linearly with the area.

AτFIG. 6 . 2 FIG. 7 .
FIG.6.Plots of the first 20 eigenvalues corresponding to PCA components, for the normalized experimental data (blue) and the normalized data randomly shuffled along the time axis (purple), for a representative cell flickering data.The errorbars correspond to one standard deviation over different patches.The PCA eigenvalues for data shuffled along the time axis shows the noise level in the system[28], which lie slightly above zero.We see that the magnitude of the eigenvalues rapidly drop and reach the noise levels within the first few principle components.