## Abstract

Recent experimental studies provide evidence for the existence of a spatially non-uniform temperature field in living cells and in particular in their plasma membrane. These findings have led to the development of a new and exciting field: thermal biology at the single-cell level. Here, we examine theoretically a specific aspect of this field, i.e. how temperature gradients at the single-cell level affect the phase behaviour and geometry of heterogeneous membranes. We address this issue by using the Onsager reciprocal relations combined with a simple model for a binary lipid mixture. We demonstrate that even small temperature variations along the membrane may introduce intriguing phenomena, such as phase separation above the critical temperature and unusual shape response. These results also suggest that the shape of a membrane can be manipulated by dynamically controlling the temperature field in its vicinity. The effects of intramembranous temperature gradients have never been studied experimentally. Thus, the predictions of the current contribution are of a somewhat speculative nature. Experimental verification of these results could mark the beginning of a new line of research in the field of biological membranes. We report our findings with the hope of inspiring others to perform such experiments.

## 1. Introduction

Recent experimental techniques, such as nanogels, nanoparticles or thermo-sensitive dyes injected into cells [1–3], enable scientists today to measure intracellular temperature with high spatial resolution. For example, intracellular temperature mapping of COS7 cells, with a fluorescent polymeric thermometer [4], showed a significant elevated temperature in the vicinity of the nucleus and the mitochondria. These experimental studies provide evidence for the existence of an intracellular temperature field with spatial variations of a few degrees Kelvin. Taking into consideration the typical cellular length scale implies that extreme temperature gradients exist at the single-cell level.

The inhomogeneous temperature distribution is intrinsically related to fundamental cellular processes [4]. For example, during cell metabolism, nutrients are converted into ATPs through irreversible biochemical processes. The heat released by these processes, for example aerobic respiration in the mitochondria [5], influences the internal temperature field. In this view, each mitochondrion is conceived as a heat source [1,4]. In cancer cells, on the other hand, the highly exothermic anaerobic processes become dominant, known as the Warburg effect [6]. Therefore, it is expected that the intracellular temperature field in cancer cells differs significantly from healthy cells. Gene expression and cell division are additional examples for localized exothermic processes that affect the intracellular temperature field [1].

The aforementioned findings have led to the development of a new and exciting field: *thermal biology at the single-cell level* [1]. Here, we examine theoretically a specific aspect of this field. In particular, we study how temperature gradients at the single-cell level affect the biomembrane. This is motivated by intracellular and intramembranous temperature measurements [4,7], which indicate that the cell membrane is subjected to a non-uniform temperature field. Further, temperature gradients can be introduced artificially to the membrane by irradiating trapped metallic nanoparticles with a laser light [8–10] and dynamically controlling their location [11]. Although temperature gradients in cell membranes have been observed experimentally, the effects of these gradients on the phase behaviour and shape of multi-component lipid membranes have never been studied experimentally. Thus, the predictions of the current contribution are of a somewhat speculative nature. Experimental verification of these results will shed new light on the way biological membranes are understood. Further, as intramembranous temperature gradients may insinuate thermal anomalies at the single-cell level, understanding their influence on membrane phase behaviour and shape might be used to indirectly diagnose cellular malfunctions. We report our findings with the hope of inspiring others to perform such experiments.

Lipids in biomembranes can be arranged in coexisting phases that possess different characteristics (chemical, mechanical, etc.). For example, the molecular structure of the lipid molecules brings about a property of spontaneous curvature [12], which depends on local mole fraction. It is thus evident that biomembranes have a unique and captivating characteristic, namely their shape is affected by composition and, on the other hand, the composition is modulated by shape [13–15]. Theoretical works have studied the effect of temperature on phase behaviour of membranes assuming a uniform temperature environment. In reality, biomembranes are often subjected to temperature gradients and operate in thermal non-equilibrium, as discussed above.

Figure 1*a* schematically illustrates a lipid membrane subjected to mechanical loads as well as to internal and external heat sources, e.g. mitochondria, nucleus and irradiated particles. The non-uniform distribution and uneven magnitude of the heat sources induce a non-uniform intramembranous temperature field. These influence membrane composition and shape. In turn, the change in shape and composition may affect the temperature field in the membrane. Thus, shape, composition and temperature are all intimately coupled. For simplicity, we study here a one-dimensional model of a membrane subjected to a prescribed temperature field, as illustrated in figure 1*b*. The model reveals that even small spatial temperature variations along the membrane may result in intriguing phenomena associated with the existence of temperature gradients, such as phase separation above the critical temperature, non-intuitive dynamic response and the possibility of temporospatial manipulation of the membrane.

## 2. Theoretical model

We consider a membrane of length *L*, as illustrated in figure 1*b*. The membrane is subjected to an external pressure *p*(*x*,*t*) and a temperature field *T*(*x*,*t*). We assume that the membrane is composed of two types of lipids, which we shall refer to as type-I and type-II. We introduce *c*(*x*,*t*): *c* → [0, 1], which describes the mole fraction of type-I in each point. Mass conservation dictates (1\*L*) *c*(*x*,*t*)d*x* = *c*_{0}, with *c*_{0} being the average initial mole fraction. Denoting *w*(*x*,*t*) as vertical displacement and assuming small deflections, the generalized Helfrich free energy (per unit width) [16] takes the form
2.1where the comma denotes partial derivative. The bending energy involves the contribution of the mean curvature with respect to the mole fraction-dependent spontaneous curvature, denoted by , while the contribution of the Gauss curvature is eliminated by the one-dimensional nature of the model. The stretching energy is ignored because its contribution to the energy is approximately constant [16]. The mole fraction *c* is considered as an order parameter according to the Ginzburg–Landau theory for phase transitions [17], with *f* being the free energy of mixture. *k*_{c} determines the resistance for composition gradients and *k*_{b} is the bending modulus, both assumed constant. The membrane is not in thermal equilibrium. Yet, we assume that each infinitesimal element in the membrane is in local thermodynamic equilibrium with its immediate surrounding [18] and consider the balance of entropy. Using the first and second laws, and considering only heat and mass fluxes along the membrane, the intensive entropy source *σ* = *σ*(*x*, *t*) can be identified as [18]
2.2where *δF*/*δc* and *T* are the local chemical potential and temperature, respectively, and (1/*T*) _{,x} and (*δF*/*δc* 1/*T*) _{,x} are the components of the generalized thermodynamic force ** Ψ**. We shall adopt the conventional phenomenological methodology of the thermodynamic reciprocal relations, the Onsager relations [19,20]. Those postulate a linear relation between the thermodynamic force and the mass flux,

*J*

_{m}, and heat flux,

*J*2.3with

_{q}**D**being a positive-semidefinite matrix which satisfies (2.2) for any

*J*

_{m}and

*J*. Mass conservation at each point gives

_{q}*c*

_{,t}+ (

*J*

_{m})

_{,x}= 0. Note that the mass flux

*J*

_{m}is also affected by the thermal force (1/

*T*)

_{,x}, unlike conventional systems which involve only chemical potential

*δF*/

*δc*. This type of transport mechanism is known as thermophoresis or the Soret effect [18,21]. As the membrane is subjected to a prescribed temperature field, the temperature gradient is not affected by the mass flux. Thus, the temperature gradient is enforced on the membrane. According to (2.3), the mass flux

*J*

_{m}is 2.4where

*D*is a generalized diffusion coefficient and

_{mm}*D*is the thermal reciprocal coefficient. For simplicity, we assume that

_{mq}*D*and

_{mm}*D*are constant. We set the characteristic length, energy per unit area, temperature and time to be

_{mq}*L*,

*k*,

_{c}ρ*T*

_{0}and

*L*

^{2}

*T*

_{0}/

*D*

_{mm}k_{c}

*ρ*, respectively.

*ρ*is the lipids' density and

*T*

_{0}is the critical temperature in an isothermal system, i.e. the temperature at which

*f*(

*c*,

*T*) switches from single to double well form. These definitions give rise to a new non-dimensional quantity,

*λ*≡ (

*D*/

_{mq}*D*)(1/

_{mm}*k*

_{c}

*ρ*), which is of major importance. As we will see,

*λ*is responsible for a class of non-intuitive behaviours that deviate from predictions of current biomembrane theories. These effects are the focus of this study. Henceforth, all quantities appear in a non-dimensional form, and the composition equation reads 2.5with the four boundary conditions

*c*

_{,x}|

_{x}_{=0,1}= 0 and

*J*

_{m}

*|*

_{x}_{=0,1}= 0 [22]. The variation of the free energy with respect to the deflection dictates the following quasi-equilibrium deflection equation [23] 2.6with four boundary conditions

*w|*

_{x}_{=0,1}= 0 and (

*w*

_{,xx}−

*H*

_{0}(

*c*))

*|*

_{x}_{=0,1}= 0. We consider a particular case of a uniform load,

*p*(

*x*,

*t*) =

*p*

_{0}= const., and after some manipulation on (2.6), (2.5) takes the form 2.7where

*k*≡

*k*

_{b}/

*k*

_{c},

*α*≡ 1/

*ρL*

^{2}and ()′ ≡ ∂/∂

*c*. This equation is decoupled from the deflection equation, or in essence from the geometrical features of the membrane. Yet, it remains ‘slaved’ to the mechanical properties of the system, such as the spontaneous curvature and the pressure. Importantly, (2.6) is coupled to the composition, and once

*c*(

*x*,

*t*) is known the deflection is easily acquired.

## 3. Results

In order to demonstrate the phenomena exhibited by (2.6) and (2.7), we set the following: the spontaneous curvature is linear with the composition, *H*_{0}(*c*) = *K*_{c}(*c* − *c*_{0}) [23]; the free energy of mixture is assumed to behave according to the regular solution model [24] *f*(*c*, *T*) = *k*_{B}*T*_{0}*ρ*(*T*(*c* ln *c* + (1 − *c*)ln(1 − *c*)) + 2*c*(1 − *c*)), where *k*_{B} is the Boltzmann constant, and the interaction parameter is expressed in terms of *T*_{0}; the temperature field takes a Gaussian form, *T*(*x*,*t*) = *T*_{edge}/*T*_{0} + (Δ*T*/*T*_{0})exp(−(*x* − *x _{p}*(

*t*))

^{2}/2

*R*

^{2}); and finally we use the following typical values [13,14,16,25–27]:

*T*

_{0}= 297° K,

*ρ*= 10

^{16}m

^{−2},

*k*

_{b}= 10

^{−19}J,

*k*

_{c}= 2 × 10

^{−18}J,

*K*

_{c}= 2 × 10

^{4}m

^{−1},

*L*= 50 µm,

*R*= 1/15 L,

*c*

_{0}= 0.5,

*p*

_{0}= 0. The dynamic behaviour of the system typically involves intervals of fast and slow dynamics. Slow dynamics may be misinterpreted as steady state because very small changes take place during the typical time of the experiment. Thus, we define

*c*

_{s}≡

*c*(

*x*,

*t*=

*t*

_{s}) and

*w*

_{s}≡

*w*(

*x*,

*t*=

*t*

_{s}) as the solution after time

*t*

_{s}of rapid changes followed by slow dynamics, i.e. . In addition, steady-state solution is defined as

*c*and

_{∞}*w*, associated with ‖

_{∞}*J*

_{m}‖ = 0.

Figure 2 shows the membrane configurations at *t* = *t*_{s} and at steady state for the case where the *entire membrane* is subjected to a temperature *lower than the critical temperature*. Note that the temperature field is uniform in region A. Thus, in this region, equation (2.7) reduces to the conventional model of multi-component lipid membranes, which has been thoroughly studied numerically and analytically [23,28,29]. This is illustrated in figure 2*a*, where region A displays modulations in composition associated with slow dynamics similar to those observed in conventional spinodal decomposition. Region B displays a thermophoresis-related phase separation. Note that region B shows a response to a relatively small variation in temperature. This temperature incline is centred at the middle of the membrane and spreads through a distance of about 0.4 L. Clearly, a distinct broad region of a separated phase (rich with type-I) appears in the centre of region B, which corresponds to the effective region of the temperature gradient regulated by *R*. In figure 2*b*, it is notable that the deflection presents a tilted yet roughly straight line in region A. This is explained by the fact that *p*_{0} = 0: in region A, the undulation wave length of *c*_{s} is nearly equal, thus equation (2.6) dictates minute curvature change in that region. Still, the deflection increases towards region B owing to the broad region of separated phase. Figure 2*c*,*d* shows the steady-state composition and deflection. Here, the modulations in composition observed in region A of figure 2*a* have vanished and the central region of distinct higher concentration has broadened. The slow transition to the steady state (from figure 2*a*,*b* to *c*,*d*) is governed by the behaviour in region A where the temperature is uniform. As in conventional spinodal decomposition, the transition to the steady state is through coarsening rather than reduction in the amplitude of oscillations. The width of the transition zone between the two distinct phases is governed by *α*. A uniform composition dictates a constant curvature (2.6). Hence, the deflection shown in figure 2*d* exhibits a uniform curvature at the sides and a region of roughly constant curvature near the centre.

Interestingly, even when the membrane is *entirely above* the critical temperature, phase separation occurs (figure 3). Unlike the case where the entire membrane is below the critical temperature, the composition does not exhibit spatial modulations during the period of slow dynamics, and the deflection is monotonous from edge to centre (figure 3*a*,*b*). The steady-state configuration is shown in figure 3*c*,*d*. As expected, the system has a uniform composition in regions where the temperature gradient vanishes. Note the similarity between the shape at *t*_{s} and at steady state, indicating that during the stage of slow dynamics only minor changes to the membrane shape take place. We have demonstrated a phase separation above the critical temperature even with small Δ*T*. This is a result of the Soret effect. The importance of this effect with respect to biological membranes is the strong influence of composition on the membrane geometry. Thus, membrane shape may be manipulated by imposing temperature gradients, as illustrated in figure 3*b*,*d*. The reason behind the phase separation above the critical temperature *T*_{0} is understood by comparing the effective chemical potential, *μ*_{e}(*c*) ≡ (*f′* + *λ*)/*T*, at the middle and at the edge of the domain. At steady state, *μ*_{e} must be uniform over the domain (2.7). This is illustrated by the horizontal line in the insets of figure 4. The height of this line dictates two different values of *c* at the middle and at the edge. Mass constraint roughly states that the average of those mole fractions is equal to *c*_{0}. Further, the cubic dependence of *μ*_{e} with respect to *c* (to the leading order) gives rise to a significant difference between the values of *c* at the middle and at the edge. Importantly, the magnitude and sign of *λ* dominate this behaviour. Figure 4 shows the influence of *λ* on the mole fraction at the edge. Higher magnitudes of *λ* lead to more significant phase separation, while no separation is observed when *λ* is small. Also, the sign of *λ* determines whether the mole fraction at the mid-span is higher than that at the edge, or vice-versa. It is possible to show, see appendix A for further details, that . As the product Δ*T* · *λ* has a cube-root effect on the phase separation, Δ*T* can be conceived as an amplifying variable for the purpose of measuring *λ* experimentally. Note that this cube-root relation is valid as long as Δ*T* is small, and the other parameters merely influence the proportion magnitude. The antisymmetry of the graph shown in figure 4 and of the approximated analytical result (see appendix A) is broken when *p*≠0. This is seen from equation (2.7), where the sign of *λ* can be changed while the term related to the mechanical load, *p*, keeps the same sign. Membranes of living cells are subjected to mechanical loads, such as osmotic pressure and forces applied by the cytoskeleton that depend on composition [29]. This suggests that even a broader variety of phenomena may take place in membranes of living cells.

The existence of *λ* forces a spatial variation in composition that, roughly speaking, follows the temperature field. This is illustrated by the dome-like function exhibited by the composition in the central region of figure 3*c*. The features of this dome result from a competition between two phenomena. One is related to the magnitude of *λ* and makes the composition follow the spatial shape of the temperature field. The other is associated with the single-well structure of *f* which tends to flatten the composition (make it uniform), as seen far from the centre in figure 3*c*. Evidently, this competition also exists in the central region of figure 2*c* where composition effectively lies in only one of the energy wells of *f*, leading to a dome-shape similar to that of figure 3*c*. In principle, these insights suggest that one may dictate the steady-state composition, and hence shape, by prescribing an appropriate temperature field. This concept is demonstrated in figure 5, which shows the steady-state configuration associated with a temperature field having multiple peaks. These results imply that the biological membrane is a unique platform that can also be dynamically manipulated into desired mechanical responses by means of small thermal stimuli. Figure 6 demonstrates the membrane response to a time-dependent temperature field. Here, a temperature field similar to the one used in figures 2⇑–4 is applied with *x _{p}* moving at constant velocity

*V*. This can be thought of as simulating a focused laser beam moving along the membrane and irradiating trapped metallic nanoparticles [10,11]. Note that two different wave types are propagating to the right. The first is a composition wave

*c*(

*x*,

*t*), illustrated by the colour map, which drives a deflection wave

*w*(

*x*,

*t*). Note that this result does not account for the dynamic interaction with the surrounding fluid, which is beyond the scope of this work. This interaction may alter the deflection dynamics but is not expected to significantly affect the composition wave. Also, steady-state configurations, such as those presented earlier, are not likely to change.

## 4. Summary

The work presented here is motivated by new findings in the field of thermal biology at the single-cell level. We formulated a new model that accounts for the existence of temperature gradients in membranes with a binary lipid mixture. The intriguing results demonstrated by the model are all an aftermath of the postulated thermodynamic transport force (1/*T*)_{,x}. The significance of such behaviour in biological cells depends on the magnitude of *λ*, yet to be explored. Our model and results suggest a conceptual set-up for quantifying *λ* experimentally. The classic Soret effect results from collisions of particles with different size or mass. Thus, the magnitude of *λ* is expected to differ for different lipid mixtures, depending on various factors such as head size, tail size, level of saturation and more. In addition, mixtures of lipids and integral proteins, which significantly differ in size, will most likely produce even higher values of *λ*. Modelling binary mixtures of lipid–protein is very similar to the model presented in this paper. In a more general context, one may view the biological membrane as a boundary that its response can shed light on thermal processes taking place inside the cell. If so, an interesting question arises: is it possible to determine the in-cell heat source distribution by correlating it to the membrane geometry and spatial composition? Along the same lines of the Warburg effect [6], this might enable the diagnosis of cellular malfunctions by detecting thermal anomalies at the single-cell level.

## Funding statement

This work was supported by the Israel Science Foundation, grant no. ISF 1500/10.

## Appendix A

The purpose of this appendix is to describe the derivation of the analytic approximation for the steady-state mole fraction, *c _{∞}*(0), depicted in figure 4. We approximate

*c*(

_{∞}*x*) by the function

*c*

_{approx}(

*x*) shown in figure 7, which have a width similar to the prescribed temperature field.

For brevity, we omit in what follows the lower case notation ‘approx’. The mass conservation enforces
A1we write:
A2Employing the mass flux boundary conditions, *J*_{m}|_{x}_{=0,1} = 0, equation (2.7) in steady state reduces to
A3

Following the approximation above, *c*_{,xx}|_{x}_{=0.5} = *c*_{,xx}|_{x}_{=0} = 0. Therefore, in steady state, the following relation must be satisfied
A4

The Taylor expansion of *f*′, in the vicinity of *c* = *c*_{0} = 0.5, is
A5

By using *R* ≈ 1/15 L and substituting (A 5) into (A 4), solving (A 4) for *λ* gives
A6

By substituting the parameters used for the numerical result shown in figure 4, we deduce that the leading term in (A 6) is the cubic one. By disregarding the higher order terms, we conclude with (see figure 8) A7

- Received December 29, 2013.
- Accepted March 6, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.