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A theoretical study of biological membrane response to temperature gradients at the single-cell level rsif.royalsocietypublishing.org
Lior Atia and Sefi Givli Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
Research Cite this article: Atia L, Givli S. 2014 A theoretical study of biological membrane response to temperature gradients at the single-cell level. J. R. Soc. Interface 11: 20131207. http://dx.doi.org/10.1098/rsif.2013.1207
Received: 29 December 2013 Accepted: 6 March 2014
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.
Subject Areas: biophysics, biomathematics, biocomplexity Keywords: lipid membrane, phase separation, thermophoresis, spontaneous curvature
Author for correspondence: Sefi Givli e-mail: [email protected]
1. Introduction Recent experimental techniques, such as nanogels, nanoparticles or thermosensitive 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,
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Figure 1. A lipid bilayer subjected to mechanical and thermal loads. (a) A schematic of a closed vesicle subjected to mechanical loads and heat sources of various types leading to a non-uniform temperature field. The different colours represent spatial variations in lipid composition. (b) A simplified one-dimensional model for a membrane comprised from two types of lipids and subjected to a prescribed temperature field T(x,t) (represented by the bottom thick line with varying intensity) and a distributed pressure p(x,t). (Online version in colour.) 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 1a 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 1b. 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 1b. 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 Ð point. Mass conservation dictates (1\L) L c(x,t)dx ¼ c0, with c0 being the average initial mole fraction. Denoting w(x,t)
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geometric and composition undulations
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where dF/dc and T are the local chemical potential and temperature, respectively, and (1/T ) ,x and (dF/dc 1/T ) ,x are the components of the generalized thermodynamic force C. 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, Jm, and heat flux, Jq J ¼ DC
(2:3)
with D being a positive-semidefinite matrix which satisfies (2.2) for any Jm and Jq. Mass conservation at each point gives c,t þ (Jm),x ¼ 0. Note that the mass flux Jm is also affected by the thermal force (1/T ),x, unlike conventional systems which involve only chemical potential dF/dc. 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 Jm is 1 dF 1 þDmq , (2:4) Jm ¼ Dmm T dc ,x T ,x where Dmm is a generalized diffusion coefficient and Dmq is the thermal reciprocal coefficient. For simplicity, we assume that Dmm and Dmq are constant. We set the characteristic length, energy per unit area, temperature and time to be L, kcr, T0 and L 2T0/Dmmkcr, respectively. r is the lipids’ density and T0 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, l ; (Dmq/Dmm)(1/kcr), which is of major importance. As we will see, l 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
3 (2:5)
with the four boundary conditions c,xjx¼0,1 ¼ 0 and Jmjx¼0,1 ¼ 0 [22]. The variation of the free energy with respect to the deflection dictates the following quasi-equilibrium deflection equation [23] k(w,xx H0 (c)),xx p ¼ 0,
(2:6)
with four boundary conditions wjx¼0,1 ¼ 0 and (w,xx 2 H0(c))jx¼0,1 ¼ 0. We consider a particular case of a uniform load, p(x,t) ¼ p0 ¼ const., and after some manipulation on (2.6), (2.5) takes the form 0 f ac,xx p0 x(x 1)H 0 0 þ l c,t þ ¼ 0, (2:7) T ,xx where k ; kb/kc, a ; 1/rL 2 and ()0 ; @ /@ 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, H0(c) ¼ Kc(c 2 c0) [23]; the free energy of mixture is assumed to behave according to the regular solution model [24] f(c, T ) ¼ kBT0r(T(c ln c þ (1 2 c)ln(1 2 c)) þ 2c(1 2 c)), where kB is the Boltzmann constant, and the interaction parameter is expressed in terms of T0; the temperature field takes a Gaussian form, T(x,t) ¼ Tedge/T0 þ (DT/ T0)exp(2(x 2 xp(t))2/ 2R 2); and finally we use the following typical values [13,14,16,25–27]: T0 ¼ 2978 K, r ¼ 1016 m22, kb ¼ 10219 J, kc ¼ 2 10218 J, Kc ¼ 2 104 m21, L ¼ 50 mm, R ¼ 1/15 L, c0 ¼ 0.5, p0 ¼ 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 cs ; c(x, t ¼ ts) and ws ; w(x, t ¼ ts) as the solution after time ts of rapid changes followed by slow dynamics, i.e. jjJm (x, t ¼ ts )jj jjJm (x, t ¼ 0)jj. In addition, steady-state solution is defined as c1 and w1, associated with kJmk ¼ 0. Figure 2 shows the membrane configurations at t ¼ ts 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 2a, 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
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where 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 H0 , 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. kc determines the resistance for composition gradients and kb 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 s ¼ s(x, t) can be identified as [18] dF 1 1 s ¼ Jm þ Jq 0 (2:2) T ,x dc T ,x
composition equation reads 1 dF 1 c,t þ þl ¼ 0, T dc ,xx T ,xx
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as vertical displacement and assuming small deflections, the generalized Helfrich free energy (per unit width) [16] takes the form ð 1 1 F¼ kb (w,xx H0 (c))2 pw þ f(c, T) þ kc (c,x )2 dx, 2 L 2 (2:1)
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Figure 2. The solution for four different DT (equally spaced between 1 4 58 K). The entire membrane is subjected to a temperature below the critical temperature with Tedge ¼ 2878 K, xp(t) ¼ 0.5 and l ¼ 0.2. The different colours correspond to different DT. (a) The composition cs (solid lines) associated with the temperature field (dashed lines). (b) The deflection ws. (c) Steady-state composition c1. (d) Steady-state deflection w1. (Online version in colour.)
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 2b, it is notable that the deflection presents a tilted yet roughly straight line in region A. This is explained by the fact that p0 ¼ 0: in region A, the undulation wave length of cs 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 2c,d shows the steady-state composition and deflection. Here, the modulations in composition observed in region A of figure 2a have vanished and the central region of distinct higher concentration has broadened. The slow transition to the steady state (from figure 2a,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 a. A uniform composition dictates a constant curvature (2.6). Hence, the deflection shown in figure 2d 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 3a,b). The steady-state configuration is shown in figure 3c,d. As expected, the system has a uniform composition in regions where the temperature gradient vanishes. Note the similarity between the shape at ts 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 DT. 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 3b,d. The reason behind the phase separation above the critical temperature T0 is understood by comparing the effective chemical potential, me(c) ; (f 0 þ l )/T, at the middle and at the edge of the domain. At steady state, me 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 c0. Further, the cubic dependence of me 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
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Figure 3. The solution for four different DT (equally spaced between 25 4 58 K). The entire membrane is subjected to a temperature above the critical temperature with Tedge ¼ 3078 K, xp(t) ¼ 0.5 and l ¼ 0.2. The different colours correspond to different DT. (a) The composition cs (solid lines) associated with the temperature field (dashed lines). (b) The deflection ws. (c) Steady-state composition c1. (d) Steady-state deflection w1. (Online version in colour.) 0.70
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Figure 4. Tedge/T0 ¼ 1.033, DT ¼ 18 K and xp(t) ¼ 0.5. The minimum composition value in steady state depends on the value of l. Insets: me(c) at x ¼ 0 and at x ¼ 0.5. At steady state, me must be uniform over the domain (horizontal line). (a) Positive values of l. (b) Negative values of l. (Online version in colour.) edge. Importantly, the magnitude and sign of l dominate this behaviour. Figure 4 shows the influence of l on the mole fraction at the edge. Higher magnitudes of l lead to more significant phase separation, while no separation is observed when l is small. Also, the sign of l determines whether the
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Figure 5. Steady-state configuration associated with the temperature field (dashed line) T(x,t) ¼ Tedge /T0 þ 1=T0 (DT (1) exp((x xp(1) )2 /2R2 ) þ DT (2) exp((x xp(2) )2 /2R2 )), with xp(1) ¼ 0:3, xp(2) ¼ 0:7, Tedge/T0 ¼ 1.033, l ¼ 0.2. (a) DT (1) ¼ DT (2) ¼ 58 K, (b) DT (1) ¼ 2DT (2) ¼ 58 K. The solid black line represents the membrane deflection and the varying colours illustrate local composition according to the colour bar on the right. (Online version in colour.)
purpose of measuring l experimentally. Note that this cuberoot relation is valid as long as DT 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 l 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 l 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 3c. The features of this dome result from a competition between two phenomena. One is related to the magnitude of l 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 3c. Evidently, this competition also exists in the central region of figure 2c where composition effectively lies in only one of the energy
wells of f, leading to a dome-shape similar to that of figure 3c. 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 steadystate 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 xp 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.
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Figure 6. Membrane response to a time-dependent temperature field, xp(t) ¼ Vt, V ¼ 4 1023, Tedge/T0 ¼ 1.033, DT¼ 18 K, l ¼ 0.2. The thick black lines and asterisks represent the membrane deflection and heat source centre location at different times correspondingly. The varying colours represent local composition according to the colour bar. (Online version in colour.)
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 l, yet to be explored. Our model and results suggest a conceptual set-up for quantifying l experimentally. The classic Soret effect results from collisions of particles with different size or mass. Thus, the magnitude of l 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 l. 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, c1(0), depicted in figure 4. We approximate c1(x) by the function capprox(x) shown in figure 7, which have a width similar to the prescribed temperature field.
capprox (x)
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Figure 7. The function capprox(x), having a width of 6R.
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Figure 8. A comparison between the numerical result shown in figure 4 (solid line) and the analytical approximation (dashed line) (A 7). (Online version in colour.) For brevity, we omit in what follows the lower case notation ‘approx’. The mass conservation enforces ð1 c(x, t)dx ¼ c0 , (A 1) 0
we write: ð1
c dx ¼ 6Rcmax þ (1 6R)cmin ¼ c0
0
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(A 2)
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Employing the mass flux boundary conditions, Jmjx¼0,1 ¼ 0, equation (2.7) in steady state reduces to (A 3)
Following the approximation above, c,xxjx¼0.5 ¼ c,xxjx¼0 ¼ 0. Therefore, in steady state, the following relation must be satisfied 0 0 f þ l f þ l ¼ : (A 4) T T x¼0 x¼0:5
1 4:109 0:5) þ 6(c min C kB T0 B DT B C: l(cmin ) ¼ A kc @ 70 1 3 þ 1 (cmin 0:5) 3 DT
(A 6)
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) ffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 cmin (l) ¼ 0:5 0:22 DT l:
(A 7)
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10. Ma H, Bendix PM, Oddershede LB. 2012 Large-scale orientation dependent heating from a single irradiated gold nanorod. Nano Lett. 12, 3954 –3960. (doi:10.1021/nl3010918) 11. Urban A, Fedoruk M, Horton M, Radler J, O.dler J, Stefani F, Feld-mann J. 2009 Controlled nanometric phase transitions of phospholipid membranes by plasmonic heating of single gold nanoparticles. Nano Lett. 9, 2903 –2908. (doi:10. 1021/nl901201h) 12. Sprong H, Van Der Sluijs P, Van Meer G. 2001 How proteins move lipids and lipids move proteins. Nat. Rev. Mol. Cell Biol. 2, 504–513. (doi:10.1038/ 35080071) 13. Ursell TS, Klug WS, Phillips R. 2009 Morphology and interaction between lipid domains. Proc. Natl Acad. Sci. USA 106, 13 301 –13 306. (doi:10.1073/pnas. 0903825106) 14. Baumgart T, Das S, Webb W, Jenkins J. 2005 Membrane elasticity in giant vesicles with fluid phase coexistence. Biophys. J. 89, 1067–1080. (doi:10.1529/biophysj.104.049692) 15. Shlomovitz R, Schick M. 2013 Model of a raft in both leaves of an asymmetric lipid bilayer. Biophys. J. 105, 1406 –1413. (doi:10.1016/ j.bpj.2013.06.053) 16. Helfrich W. 1973 Elastic properties of lipid bilayers: theory and possible experiments. J. Biosci. C 28, 693 –703. 17. Falk F. 1982 Landau theory and martensitic phasetransitions. J. Phys. 43(NC-4), 3– 15. 18. Groot SRd, Mazur P. 1962 Non-equilibrium thermodynamics. Amsterdam, The Netherlands: North-Holland Publishing Co. 19. Onsager L. 1931 Reciprocal relations in irreversible processes II. Phys. Rev. 38, 2265 –2279. (doi:10. 1103/PhysRev.38.2265)
20. Onsager L. 1931 Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405 –426. (doi:10.1103/ PhysRev.37.405) 21. Kumaki J, Hashimoto T, Granick S. 1996 Temperature gradients induce phase separation in a miscible polymer solution. Phys. Rev. Lett. 77, 1990– 1993. (doi:10.1103/PhysRevLett.77.1990) 22. Alt HW, Pawlow I. 1992 A mathematical model of dynamics of non-isothermal phase separation. Phys. D Nonlinear Phenom. 59, 389–416. (doi:10.1016/ 0167-2789(92)90078-2) 23. Atia L, Givli S. 2012 Biological membranes from the perspective of smart materials: a theoretical study. Int. J. Solids Struct. 49, 2617 –2624. (doi:10.1016/j. ijsolstr.2012.05.030) 24. Hildebrand JH, Scott RL. 1962 Regular solutions. Prentice-Hall international series in chemistry. Englewood Cliffs, NJ: Prentice-Hall. 25. Seifert U. 1997 Configurations of fluid membranes and vesicles. Adv. Phys. 46, 13 –137. (doi:10.1080/ 00018739700101488) 26. Givli S, Giang H, Bhattacharya K. 2012 Stability of multicomponent biological membranes. SIAM J. Appl. Math. 72, 489– 511. (doi:10.1137/ 110831301) 27. Ma L, Klug WS. 2008 Viscous regularization and r-adaptive remeshing for finite element analysis of lipid membrane mechanics. J. Comput. Phys. 227, 5816– 5835. (doi:10.1016/j.jcp.2008.02.019) 28. Shlomovitz R, Gov NS. 2009 Membrane-mediated interactions drive the condensation and coalescence of FtsZ rings. Phys. Biol. 6, 046017. (doi:10.1088/ 1478-3975/6/4/046017) 29. Veksler A, Gov NS. 2007 Phase transitions of the coupled membrane –cytoskeleton modify cellular shape. Biophys. J. 93, 3798 –3810. (doi:10.1529/ biophysj.107.113282)
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The Taylor expansion of f 0 , in the vicinity of c ¼ c0 ¼ 0.5, is k B T0 16 (A 5) 4(T 1)(c 0:5) þ T(c 0:5)3 : f0 3 kc
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A theoretical study of biological membrane response to temperature gradients at the single-cell level rsif.royalsocietypublishing.org
Lior Atia and Sefi Givli Faculty of Mechanical Engineering, Technion, Haifa 32000, Israel
Research Cite this article: Atia L, Givli S. 2014 A theoretical study of biological membrane response to temperature gradients at the single-cell level. J. R. Soc. Interface 11: 20131207. http://dx.doi.org/10.1098/rsif.2013.1207
Received: 29 December 2013 Accepted: 6 March 2014
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.
Subject Areas: biophysics, biomathematics, biocomplexity Keywords: lipid membrane, phase separation, thermophoresis, spontaneous curvature
Author for correspondence: Sefi Givli e-mail: [email protected]
1. Introduction Recent experimental techniques, such as nanogels, nanoparticles or thermosensitive 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,
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
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Figure 1. A lipid bilayer subjected to mechanical and thermal loads. (a) A schematic of a closed vesicle subjected to mechanical loads and heat sources of various types leading to a non-uniform temperature field. The different colours represent spatial variations in lipid composition. (b) A simplified one-dimensional model for a membrane comprised from two types of lipids and subjected to a prescribed temperature field T(x,t) (represented by the bottom thick line with varying intensity) and a distributed pressure p(x,t). (Online version in colour.) 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 1a 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 1b. 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 1b. 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 Ð point. Mass conservation dictates (1\L) L c(x,t)dx ¼ c0, with c0 being the average initial mole fraction. Denoting w(x,t)
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where dF/dc and T are the local chemical potential and temperature, respectively, and (1/T ) ,x and (dF/dc 1/T ) ,x are the components of the generalized thermodynamic force C. 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, Jm, and heat flux, Jq J ¼ DC
(2:3)
with D being a positive-semidefinite matrix which satisfies (2.2) for any Jm and Jq. Mass conservation at each point gives c,t þ (Jm),x ¼ 0. Note that the mass flux Jm is also affected by the thermal force (1/T ),x, unlike conventional systems which involve only chemical potential dF/dc. 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 Jm is 1 dF 1 þDmq , (2:4) Jm ¼ Dmm T dc ,x T ,x where Dmm is a generalized diffusion coefficient and Dmq is the thermal reciprocal coefficient. For simplicity, we assume that Dmm and Dmq are constant. We set the characteristic length, energy per unit area, temperature and time to be L, kcr, T0 and L 2T0/Dmmkcr, respectively. r is the lipids’ density and T0 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, l ; (Dmq/Dmm)(1/kcr), which is of major importance. As we will see, l 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
3 (2:5)
with the four boundary conditions c,xjx¼0,1 ¼ 0 and Jmjx¼0,1 ¼ 0 [22]. The variation of the free energy with respect to the deflection dictates the following quasi-equilibrium deflection equation [23] k(w,xx H0 (c)),xx p ¼ 0,
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with four boundary conditions wjx¼0,1 ¼ 0 and (w,xx 2 H0(c))jx¼0,1 ¼ 0. We consider a particular case of a uniform load, p(x,t) ¼ p0 ¼ const., and after some manipulation on (2.6), (2.5) takes the form 0 f ac,xx p0 x(x 1)H 0 0 þ l c,t þ ¼ 0, (2:7) T ,xx where k ; kb/kc, a ; 1/rL 2 and ()0 ; @ /@ 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, H0(c) ¼ Kc(c 2 c0) [23]; the free energy of mixture is assumed to behave according to the regular solution model [24] f(c, T ) ¼ kBT0r(T(c ln c þ (1 2 c)ln(1 2 c)) þ 2c(1 2 c)), where kB is the Boltzmann constant, and the interaction parameter is expressed in terms of T0; the temperature field takes a Gaussian form, T(x,t) ¼ Tedge/T0 þ (DT/ T0)exp(2(x 2 xp(t))2/ 2R 2); and finally we use the following typical values [13,14,16,25–27]: T0 ¼ 2978 K, r ¼ 1016 m22, kb ¼ 10219 J, kc ¼ 2 10218 J, Kc ¼ 2 104 m21, L ¼ 50 mm, R ¼ 1/15 L, c0 ¼ 0.5, p0 ¼ 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 cs ; c(x, t ¼ ts) and ws ; w(x, t ¼ ts) as the solution after time ts of rapid changes followed by slow dynamics, i.e. jjJm (x, t ¼ ts )jj jjJm (x, t ¼ 0)jj. In addition, steady-state solution is defined as c1 and w1, associated with kJmk ¼ 0. Figure 2 shows the membrane configurations at t ¼ ts 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 2a, 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
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where 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 H0 , 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. kc determines the resistance for composition gradients and kb 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 s ¼ s(x, t) can be identified as [18] dF 1 1 s ¼ Jm þ Jq 0 (2:2) T ,x dc T ,x
composition equation reads 1 dF 1 c,t þ þl ¼ 0, T dc ,xx T ,xx
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as vertical displacement and assuming small deflections, the generalized Helfrich free energy (per unit width) [16] takes the form ð 1 1 F¼ kb (w,xx H0 (c))2 pw þ f(c, T) þ kc (c,x )2 dx, 2 L 2 (2:1)
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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 2b, it is notable that the deflection presents a tilted yet roughly straight line in region A. This is explained by the fact that p0 ¼ 0: in region A, the undulation wave length of cs 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 2c,d shows the steady-state composition and deflection. Here, the modulations in composition observed in region A of figure 2a have vanished and the central region of distinct higher concentration has broadened. The slow transition to the steady state (from figure 2a,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 a. A uniform composition dictates a constant curvature (2.6). Hence, the deflection shown in figure 2d 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 3a,b). The steady-state configuration is shown in figure 3c,d. As expected, the system has a uniform composition in regions where the temperature gradient vanishes. Note the similarity between the shape at ts 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 DT. 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 3b,d. The reason behind the phase separation above the critical temperature T0 is understood by comparing the effective chemical potential, me(c) ; (f 0 þ l )/T, at the middle and at the edge of the domain. At steady state, me 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 c0. Further, the cubic dependence of me 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
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Figure 5. Steady-state configuration associated with the temperature field (dashed line) T(x,t) ¼ Tedge /T0 þ 1=T0 (DT (1) exp((x xp(1) )2 /2R2 ) þ DT (2) exp((x xp(2) )2 /2R2 )), with xp(1) ¼ 0:3, xp(2) ¼ 0:7, Tedge/T0 ¼ 1.033, l ¼ 0.2. (a) DT (1) ¼ DT (2) ¼ 58 K, (b) DT (1) ¼ 2DT (2) ¼ 58 K. The solid black line represents the membrane deflection and the varying colours illustrate local composition according to the colour bar on the right. (Online version in colour.)
purpose of measuring l experimentally. Note that this cuberoot relation is valid as long as DT 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 l 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 l 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 3c. The features of this dome result from a competition between two phenomena. One is related to the magnitude of l 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 3c. Evidently, this competition also exists in the central region of figure 2c where composition effectively lies in only one of the energy
wells of f, leading to a dome-shape similar to that of figure 3c. 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 steadystate 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 xp 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.
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Figure 6. Membrane response to a time-dependent temperature field, xp(t) ¼ Vt, V ¼ 4 1023, Tedge/T0 ¼ 1.033, DT¼ 18 K, l ¼ 0.2. The thick black lines and asterisks represent the membrane deflection and heat source centre location at different times correspondingly. The varying colours represent local composition according to the colour bar. (Online version in colour.)
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 l, yet to be explored. Our model and results suggest a conceptual set-up for quantifying l experimentally. The classic Soret effect results from collisions of particles with different size or mass. Thus, the magnitude of l 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 l. 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, c1(0), depicted in figure 4. We approximate c1(x) by the function capprox(x) shown in figure 7, which have a width similar to the prescribed temperature field.
capprox (x)
cmax
cmin
6R
x
Figure 7. The function capprox(x), having a width of 6R.
0.60 0.55
c• (0)
4. Summary
0.50
0.45 0.40 0.35 –0.20 –0.15 –0.10 –0.05
0 l
0.05 0.10 0.15 0.20
Figure 8. A comparison between the numerical result shown in figure 4 (solid line) and the analytical approximation (dashed line) (A 7). (Online version in colour.) For brevity, we omit in what follows the lower case notation ‘approx’. The mass conservation enforces ð1 c(x, t)dx ¼ c0 , (A 1) 0
we write: ð1
c dx ¼ 6Rcmax þ (1 6R)cmin ¼ c0
0
cmax
c0 (1 6R)cmin ¼ 6R
(A 2)
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Employing the mass flux boundary conditions, Jmjx¼0,1 ¼ 0, equation (2.7) in steady state reduces to (A 3)
Following the approximation above, c,xxjx¼0.5 ¼ c,xxjx¼0 ¼ 0. Therefore, in steady state, the following relation must be satisfied 0 0 f þ l f þ l ¼ : (A 4) T T x¼0 x¼0:5
1 4:109 0:5) þ 6(c min C kB T0 B DT B C: l(cmin ) ¼ A kc @ 70 1 3 þ 1 (cmin 0:5) 3 DT
(A 6)
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) ffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 cmin (l) ¼ 0:5 0:22 DT l:
(A 7)
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The Taylor expansion of f 0 , in the vicinity of c ¼ c0 ¼ 0.5, is k B T0 16 (A 5) 4(T 1)(c 0:5) þ T(c 0:5)3 : f0 3 kc
0
8
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f 0 þ l c,xx ¼ const. T
By using R 1/15 L and substituting (A 5) into (A 4), solving (A 4) for l gives
