Entropic Forces in the Cell

Abstract

In physics we are accustomed to four fundamental forces governing every phenomenon in the Universe. However, when dealing with heterogeneous, multiphase systems, showing aggregation and self-organisation at length scales between nanometers and micrometers, other interactions seem to appear mysteriously, inducing strange effects such as osmosis, diffusion, depletion, hydrophobicity, settling, viscous drag, and so on. Certainly, also these effects must ultimately find their origins in the four fundamental forces. But in order to master them we need to introduce statistical thermodynamics concepts, conveniently embodied in the notion of “entropic” forces. The internal dynamics of a cell, a dense fluid crowded by hundreds of different proteins, molecules, charged ions, multiple lipid membranes, appears as an ideal laboratory to study such exotic physical phenomena.

Appendices

Appendix D: Membranes, Micelles and Liposomes

All the membranes that are found in the cell are constituted by phospholipids, for not less than 50 % in mass. Artificial membranes composed by only phospholipids are also stable, and can occur in a variety of shapes. The main characteristic of phospholipid molecules, besides the large number of variants, is that they are amphiphilic, being constituted by a polar head, therefore with hydrophilic character, and two non-polar, thereby hydrophobic, tails. Each tail is composed by two long aliphatic chains (each one a fatty acid), containing about 13–23 CH$_2$ groups, plus one terminal CH$_3$ (Fig. 5.12a, b). If all bonds between the carbon atoms in the chain are simple bonds (called trans), the fatty acid is said to be saturated; on the contrary, if one or more bonds are double (or cis, with the removal of one H from each group) the fatty acid is unsaturated. Each double bond implies formation of a “knee”, or bend in the chain, which affects the capacity of the phospholipid to assemble into a densely packed structure. Often in a phospholipid, one of the lipids is saturated and the other is unsaturated. Typical chain lengths of the lipids found in biological membranes are 16–18. While the most energetically favoured state is a straight, all-trans carbon chain, deviations of 120$^{\circ }$ (called gauche bonds) cost a bending energy of only about 0.8 $k_BT$, and so can be be thermally excited. These thermally excited kinks of a normal carbon chain are not to be confused with the permanent kinks provided by the double (cis) bonds.

Fig. 5.12

Atomistic model and formula of: a saturated, and b unsaturated palmitic acid. The saturated molecule has all single C–C bonds, and has a linear shape. The unsaturated molecule has (at least) one double C=C bond, which forms a kink in the carbon chain. c The structure of typical phospholipids: a polar head (choline or ethanolamine) linked via a glycerol to two fatty acids of variable length and saturation

As shown in Fig. 5.12c, the phospholipid molecule is built by attaching the terminal -OH oxygens from the two lipid chains (after liberating the H) to a central phosphate, RPO$_4$ (hence the prefix phospho), to which a side group R is attached. The nature of the latter is variable, the two most commonly found in biological membranes being ethanolamine, CH$_2$CH$_2$N$^+$H$_3$, and choline, CH$_2$CH$_2$N$^+$(CH$_3$)$_3$. The head groups, with the negatively charged oxygen and the positively charged nitrogen, have a dipole which interacts with the dipoles of water, thereby making the phospholipid head strongly polar. Also, the choline is significantly bigger than the ethanolamine, as the H attached to the nitrogen is replaced by the much larger methyl group, CH$_3$.

When phospholipids are mixed with water, their hydrophobic tails try to minimise the contact with water molecules, leaving only the hydrophilic heads exposed. Above some threshold concentration, $c_{mc}\simeq 10^{-10}$ M, phospholipids start to assemble spontaneously into a patch, which further gets curved into a globular aggregate , or micelle, with the hydrophobic tails grouped together so as to exclude contact with water (Fig. 5.13, left). Upon increasing the concentration, many spherical micelles can come together and pack into a dense lattice. Further concentration increase leads to elongated micelles, which eventually fusion together into a double layer (Fig. 5.13, right). This structure is the beginning of the cell membrane, a sandwiched structure with all the hydrophobic tails facing each other, while the hydrophilic heads are in contact with water on both sides. This is an outstanding result of the hydrophobic attraction effect, as described in Sect. 5.4, despite the fact that assembling the molecules together reduces their entropy.

Fig. 5.13

Self-assembly of phospholipids. Left Above some critical micellar concentration, phospholipids minimise the interaction energy with the solvent (water) by clustering into spherical micelles. The non-polar tails point toward the interior of the sphere, to avoid contact with water, while the polar heads make up the spherical surface. Right If the concentration is further increased, the spheroidal micelles grow and the layer of phospholipids splits, forming a double-layer membrane. Within the two layers the tails are facing each other, providing a water-exclusion zone; heads form two outer surfaces, with which water can make contact on both sides of the membrane. The membrane bends and folds into a closed shape, thus forming a closed vesicle that resembles the structure of a cell

Amphiphilic species can be obtained by attaching a polar headgroup to one, two, even three lipid tails. All such molecules have the capability of self-assembling, when the respective critical concentration is attained. The value of $c_{mc}$ can be inferred, at least qualitatively, from considerations about the free energy of assembly of one individual molecule to an existing cluster: the single molecule gains binding enthalpy in attaching to the cluster, but loses entropy of its free motion in the solvent. By schematising a long-tail molecule as a cylinder of radius r and length $nl_C$, with n the number of CH$_2$ groups and $l_C\simeq 1.25$ Å the length of a C-C carbon bond (it is 1.54 Å projected along the vertical direction), its lateral surface coming into contact with the other molecules in the cluster can be estimated as $S=2\pi r n l_C$; the gain in binding enthalpy is $E_b=\varSigma A$, for a generic lipid-lipid interfacial tension $\varSigma$ (see p. 338, for the membrane surface tension). The entropy gain can be estimated by considering the single amphiphile in the solvent as an “ideal gas” molecule. Then, from Eq. (2.11) in Chap. 2, by setting $N=1$, $c=N/V$, and $E=\tfrac{3}{2}k_BT$, the free molecule entropy is:

$$\begin{aligned} S_f = k_B \left\{ \ln \left[ \frac{1}{c} \left( \frac{2\pi mk_BT}{h^2} \right) ^{3/2} \right] + \frac{5}{2} \right\} \end{aligned}$$

(5.76)

The critical concentration $c_{mc}$ corresponds to the equilibrium, $G=E_b-TS_f = 0$, from which we obtain the condition:

$$\begin{aligned} \frac{E_b}{k_BT} = \frac{5}{2} + \ln \left[ \frac{1}{c_{cm}} \left( \frac{2\pi mk_BT}{h^2} \right) ^{3/2} \right] \end{aligned}$$

(5.77)

or else:

$$\begin{aligned} c_{cm} = \left( \frac{2\pi mk_BT}{h^2} \right) ^{3/2} \exp \left( \frac{5}{2} - \frac{E_b}{k_BT} \right) \end{aligned}$$

(5.78)

This shows that the critical concentration decreases rapidly with the binding enthalpy from the surface tension. This latter depends on the lateral contact surface of the amphiphile in the cluster, whose radius r to a first approximation can be taken to vary as $r=0.2\,\,\text {nm}, \sqrt{2}r, \sqrt{3}r$, for 1, 2, or 3 lipid tails in the molecule, respectively. With $\varSigma \simeq 30$ mJ/m$^2$ (see below), the binding enthalpy of amphiphilic molecules with $n=15$ is, respectively, $E_b=17, 24, 30 \,\, k_BT$, for 1, 2, or 3 tails. As a result, the critical concentration of 2-chain phospholipids, from Eq. (5.78) above, is about $10^{-3}$ smaller than for single-chain amphiphiles (and about 500 times larger than for 3-chain ones). It takes a much higher molar concentration for single-chain amphiphiles to assemble into a double-layer, compared to 2-chain phospholipids. This is the main reason why cell membranes should be composed by 2-chain, and not by single-chain amphiphilic molecules. The latter prefer to form spherical micelles, unless the molar concentration becomes very high.

The bilayer is a highly stable configuration, in which the phospholipid molecules maintain a considerable in-plane mobility thus giving the membrane a fluid-like consistency. However, also in this configuration the free edges of the bilayer expose some of the fatty acid tails to the contact with water molecules. This is the driving force for folding the flat bilayer into a curved structure, which will eventually close to form a vesicle, superficially similar to a closed cell wall. In this configuration, the water confined inside the double layer is in contact with the hydrophilic heads, as well as the water outside, while the hydrophobic tails are completely screened by any contact with water.

The thickness of a bilayer is usually about 5 nm (50 Å). A number of factors can affect the membrane free energy, by changing either $E_b$ or $S_f$. Notably, the total membrane surface can change by cutting away or adding portions of surface. Since no free patches can be found in solution because of the too high energy cost of the free borders, the patch will suddenly fold into a spherical shape (see Chap. 8). Such double-layer lipid spheroids, called vesicles or liposomes, can be found at several stages during the cell life, for example in the processes of exocytosis and endocytosis, by which a protein or a neurotransmitter is expelled from, or incorporated in the cell membrane. Also micelles can be observed in cells, when a hydrophobic species must be transported within the cell plasma: the single-lipid layer of the micelle provides a hydrophobic inner cavity, which can host and transport hydrophobic species in water. The membrane tension $\varSigma$ is affected by the mechanics of the underlying cytoskeleton, with the actin cortical layer exerting more or less pressure on the cell membrane during various stages of cell life (see Chap. 6). Also, transmembrane and cytoskeletal proteins can affect the contractility or elasticity of the membrane itself, and of the cortical layer immediately in contact with it. Figure 5.14 summarises some such processes.

Fig. 5.14

Interaction between the tension and area of the cell membrane, and various cellular processes. When the term $\varSigma A$ in the free energy is too high, the events on the right side occur; when it is too low, the events on the left take place. a In exocytosis, a patch of the membrane under low tension is detached and forms a closed liposome, or vesicle (blue), the remaining membrane (black) increases its tension. b The actin cortical layer (red) can increase its pressure on the membrane by fast polymerisation, leading to elongation of the actin filaments. c The expulsion of myosin (purple) from the actin layer decreases the contractility, leading to an increased actin tension on the membrane

In a real cell, the membrane is composed not only by lipids of various types, but contains in notable proportion cholesterol molecules, as well as a variety of proteins, which perform numerous specialised functions at the interface of the cell with the external world. Since the bilayer is in a fluid state, the diffusion of the proteins is sufficiently rapid. At low temperature, a phase of pure lipids undergoes a structural transition to a gel phase, in which the tails are more ordered, (i.e., have fewer gauche bonds) as are the headgroups. Presumably the heads possess long-range orientational order, but no long range positional order (this is called a hexatic phase, and is usually indicated as S$_o$). The increased order of the tails permits the lipids to pack more efficiently, with the consequence that the diffusion constant of the proteins decreases. At higher temperatures, this phase starts forming surface undulations, or “ripples”; the bilayer in this phase, indicated as P$_{\beta }$, is still very ordered despite the wavy appearance. Because the proteins cannot do their job in a timely fashion, these gel-like phases are biologically useless. Notably, lipids with two fully saturated tails of length about 16 are in this useless state already at body temperature. Replacement of one of the saturated chains by an unsaturated one causes a permanent kink in that chain and makes the system difficult to pack. With a large enough fraction of such “defects”, the transition temperature is lowered to well below 300 K, likely providing a reason for the widespread presence of unsaturated fatty acids in the cell membrane of all living organisms.

At yet increasing temperatures, the bilayer goes into a state with the tails very disordered. This is a “tail-melting”, and the corresponding phase, indicated as L$_d$, retains the properties of a fluid, with increased in-plane disorder.

Cholesterol, like any sterols, is a lipid with a very small polar head (just the OH hydroxyl termination), next to a planar structure of 3 hexagonal + 1 pentagonal aromatic rings, and terminated by short a hydrophobic tail, (CH$_2$)$_4$CH$_3$. Adding cholesterol to a lipid bilayer in the fluid phase decreases the membrane permeability to water, since cholesterol tends to occupy part of the free volume within the long lipid chains, thus decreasing their flexibility. A partial phase diagram of a synthetic membrane of DPMC phospholipids with increasing molar concentrations of cholesterol is shown in Fig. 5.15. The pure lipid phases are found by looking along the vertical line at zero concentration. Adding cholesterol to the gel phase disrupts the local order, increasing the in-plane diffusivity and reducing the membrane elastic modulus. A liquid-ordered phase is formed, on the right side of the phase diagram. This would be the “normal” phase also for a biological membrane, however considering that in a cell membrane islands of different lipids can exist, with locally variable concentrations of cholesterol. The average cholesterol concentration in the cell membrane can be about 40–50 % on a molar basis (about 15–20 % when expressed in weight fraction, since cholesterol is a smaller molecule compared to typical phospholipids). In the simple phase diagram in the figure, liquid-disordered and liquid-ordered phases, as well as liquid-ordered and solid-ordered phases are seen to coexist in the homogeneous lipid bilayer, at the normal temperatures $T\sim 36{-}38\,^{\circ }$C.

Fig. 5.15

Partial phase diagram of DPMC phospholipids with cholesterol, in excess water solvent (experimental data taken from [9]). The curved lines indicate coexistence points (temperature, concentration) between the various phases. Within each area of the diagram one phase is formed or, between two coexistence lines, a mixture of two phases (A+B) appears. The S$_o$ phase is not visible, since it appears at lower temperatures

The elastic properties of the membrane chiefly derive from the competition between attractive (Van der Waals + electrostatic) interactions between the side chains, and their entropy. Since each phospholipid occupies about 0.4 nm$^2$, and the interaction energy is $\sim$ $3 k_BT$, the energy required to stretch a patch of membrane is of the order of 7.5 $k_BT$/nm$^2$, or 30 mJ/m$^2$. The elastic moduli of membranes will be better defined in Chap. 8. Typical cell membranes have a low shear modulus, 4–$10 \times 10^{-3}$ N/m; a high elastic modulus, due to the small stretching allowed in lipid bilayers, $10^3$ N/m$^2$; a variable viscosity, which depends on membrane composition, 0.36–$2.1\times 10^{-3}$ Pa-s for red blood cells; and a bending stiffness $\kappa _b$, strongly influenced by the presence of membrane proteins and cytoskeleton elements, of the order of $10^{-19}$ N-m, or $\sim$ $100$ pN-nm.

Problems

5.1

Stationary flux

Show that the time-independent solution of the diffusion equation (5.42) corresponds to a constant flux across the membrane.

5.2

Artificial blood

In your laboratory, someone is trying to make artificial blood. Therefore, they start preparing spherical vesicles from a phospholipid suspension, with average size R. A concentration of about 30 % vol. of haemoglobin is introduced in the vesicles. When such artificial “red blood cells” are placed in pure water, the membrane is ripped open, and the protein diffuses in the water. After some test, you discover that if the vesicles containing haemoglobin are placed in a 1mM solution of NaCl, they do not explode and remain quite spherical. Explain the result. Moreover, if 1mM is good, do you think that 2 mM should be better?

5.3

A cell spewing glucose

Take a spherical cell of radius $R=10$ $\upmu$m, whose membrane has a permeability for glucose of $P_M=20\,\upmu$m/s. Calculate the time variation of the glucose concentration inside the cell, after it is immersed in a large tank of pure water at time $t=0$.

5.4

A breathing bacterium

Consider a bacterium as a sphere of radius $R_0$. Our bacterium lives in a pond, from where it takes the oxygen to breathe at a concentration $c_0$. Take that as soon as the oxygen molecules pass the outer bacterial membrane they are instantaneously turned into CO$_2$, and compute the oxygen concentration profile around the bacterium.

5.5

Haute cuisine

You are preparing a strawberry pie in the kitchen. So, you cut your berries in half and sprinkle them with powdered sugar. After just a few minutes, your fruits are softened and float in juice.What happened? Where the water comes from?

5.6

Separation by sedimentation

The sedimentation coefficient of a species in solution, $s=v/a$ , is the ratio between its sedimentation velocity and the acceleration applied (causing the sedimentation); it is measured in a special unit, the Svedberg, 1 S = 10$^{-13}$ seconds. Consider a centrifuge turning at $10^3$ rpm (rounds per minute). At time $t=0$ a beaker containing a solution of two mixed proteins A and B is placed at a position $r_0=5$ cm away from the central axis of the centrifuge. The two proteins have sedimentation coefficients of 10 and 30 S, respectively. At what time the protein A is found at $r=10$ cm? What will be the position of the protein B at that time?

5.7

Membrane permeability

Consider a spherical cell of radius $R=10\,\upmu$m, with some initial concentration [$c_{in}$] of a species, immersed in pure water. By knowing the permeability of the membrane to glycerol (10$^{-8}$ m/s) and glucose (10$^{-12}$ m/s), estimate the time necessary for all the molecules of each type to void completely the cell.

5.8

Blood flow in the arteries

Compute the increase in cardiac pressure necessary to transport the same amount of blood, from a single artery of radius R, into two branched arteries of equal radius R / 2.

5.9

The osmose on Mars

You and your friend who lives on the planet Mars are repeating the osmotic pressure experiment of Van t’Hoff. You both build the water container, a glass cylinder sealed by the same type of semi-permeable membrane, and do the experience of adding variable concentrations of glucose inside the cylinder. However, after a Skype call to Mars, you discover that the level variations of the water in the cylinder are very different between the Earth and Mars. Can you explain to your martian friend why?

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