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A model for the temperature dependence of membrane excitability
J. theor. Biol. (1978) 73, 609-614
A Model for the Temperature Dependence of Membrane Excitability P. K. GHOSH AND D. SENGUPTA Department of Physics, Visva-Bharati, Santiniketan 731235, West Bengal, India (Received 28 November 1977)
Based on a decorated Ising model, the possible existence of multiple phase transitions in biological membranes is; explored in this paper. Attempt is made to relate the stimulus-response behavior of membranes with structural order in different temperature regions.
1. htrodaction Nerve excitation without threshold has been observed in squid axon in a recent interesting work (Cole, Guttman & Bezanilla, 1970). It was shown that the maximum slope of the stimulus-response curve decreases from 10’ to 500 to 6 as the temperature was raised from 6*3”C to 20°C to 40°C (Guttman, 1972). It has been suggested that this sudden transition from the all or none law to a fractional or graded response might represent a phase transition in the membrane (Changeux, Thiery, Tung & Kittel, 1967; Tasaki, 1968; Adam, 1968). It is clear that in a biological system there will also be alower temperature bound to the domain of excitability. X-ray studies of the myelin in the frog sciatic nerve showed that maximum changes occur in the di&action pattern of fresh nerve between 0°C and -2°C and between 58°C and 61°C (Chapman, 1967). Evidence of thermal phase transition in Mycoplasma Laidlawii has been clearly established by differential scanning calorimetry (Steim, Tourtellotte, Reinert, McElhaney, & Rader, 1969) and X-ray diffraction studies (Engelman, 1970). This corresponds to a transition from hexagonal close-packed to liquid crystalline phase of the fatty acyl chains of lipid bilayers. It was shown that this reversible sharp transition takes place in the temperature range -2O-+ 5O”C, depending on the method of preparation. In the present work we hope to relate this phase transition to the all or none response in a membrane. 609 0022-5193/78/0821-0614$02.00/O 0 1978 Academic press Inc. (London) Ltd.
610
P.
K.
GHOSH
AND
D.
SENGUPTA
2. The Decorated Ising Model To discuss such co-operative phase transitions in biomembranes, the two-dimensional Ising model has been utilized by several authors (Changeux, Thiery, Tung & Kittel, 1967; Almeida, Bond & Ward, 1971). However, very little attempt has been made to correlate this phase transition to the temperature dependence of membrane response or to the structural details of the membrane. In the present model, the membrane is considered as a two-dimensional array with each site occupied by single polar molecule or a composite group having a net electric dipole moment. This electric dipole model has been justified by Wei (1971) on the basis of the fact that trimethylamine ions -N+ E (CH,), in membrane phospholipid has a permanent electric dipole moment. Liener (1966) showed that trimethylamine ions do occur at the surface of the membrane. The basic interaction in this case among the dipoles is clearly electrostatic. In order to make this model compatible with the Ising formalism (Newell & Montroll, 1953; Huang, 1963), we assume only nearest-neighbour (nn) interaction. In spite of the long range nature of electrostatic forces this model is expected to give good results for dipoles in an electrolytic solution (Hill, 1960). The simple two-dimensional Ising model with nn interaction always leads to a single order-disorder phase transition. Our earlier discussion suggests the possibility of two phase transitions: (a) from no-response to all-or-none response and (b) from the all or none to the graded response. The possibility of multiple phase transitions motivated us to explore the applicability of two-dimensional decorated Ising model in the present problem. Besides the direct dipole-dipole interaction, in the decorated Ising model there is an indirect interaction (super-exchange) between the dipoles through other atoms or molecules that do not have a net electric dipole moment. The existence of such non-polar molecules in the membrane is well known (Goodwin, 1974). If the direct and the super-exchange interactions are of opposite signs, then it is possible to obtain multiple phase transitions. 3. Mathematical
Formulation
Let J1 be the coupling constant for the direct interaction between nn electric dipoles and J2 the coupling constant for the super-exchange interaction via the intermediate atoms or molecules. The direct interaction is obviously repulsive (J1 < 0). For multiple phase transitions to occur, the super-exchange
TEMPERATURE
DEPENDENCE
OF
MEMBRANE
EXCITABILITY
611
interaction must be attractive and slightly stronger than the direct one (1.J21>lJJ). Both these interactions may be characterized by a unique effective interaction (Fradkin & Eggarter, 1976) with coupling constant J eR= J, + $kT In cash (2J,/kT). (la> where k is the Boltzmann constant and T is the absolute temperature. Instead of a unique super-exchange interaction there might be n intermediate atoms involved, each coupled to the sites A and B (Fig. 1) with coupling
FIG. 1. Direct and super-exchange (n intermediatepaths)interactionsbetweenelectric dipolesat sitesA and B.
constant J&.
The effective interaction is now modified to J,, = J, +&zkT In cash (2J,/nkT).
(lb) values for low and high temperatures are J, + IJzj > 0, for kT -%J, J eff (2) = { J1 <0, for kT % Jz. It is seen that the effective interaction changes sign at temperature T = IJ&k, and it is this change of sign which is responsible for multiple phase transitions. Such a decorated Ising model would have the same Hamiltonian as the regular one, with J replaced by Jes, and is of the following form The limiting
H=
-Jeff & CPjp
(3)
where (q) indicates nn and pi, Oj assume the values + 1. Exact solution by Onsager (Huang, 1963) shows that phase transitions occur whenever
IJeffl = 0.44
kT For large n there are three phase transitions and T,,, T,, around jJzI/nk. A calculation biological membranes is given in the next
* at temperatures TcI = 2.27 IJ1 I/k, using typical values of J1, Jz for section.
612
P. K. GHOSH
AND
D. SENGUPTA
4. Results We estimate J1, the direct coupling constant, starting from the values of the dipole moment (25 to 1000 Debye) as given by Wei (1969). The expression for J1 is as follows +2
J,
-- - -3 $1
(in esu),
(5)
where Ip^I is the electric dipole moment associated with each lattice site, jPl is the distance between nn dipoles and Eisthe dielectric constant of the membrane fluid. The above formula (5) holds for dipoles parallel to each other and oriented normal to the plane of the membrane. Now, substituting @I = 50 D (1 D = 10-l’ esu), l?l =5x10-*cm,ands=SO,weget J,/k z - 3600 (K). The exact order of magnitude of Jz cannot be estimated since the microscopic mechanism is not yet clear. Using Jl/k = - 34741K and J2/k = 4650 K we graphically solve equation (4) for the critical temperatures in Fig. 2. The values obtained are T,, z 7880 K, T,, = 317 K and T, = 255 K. It is to be noted that the calculation has been done for it = 12, indicating twelve intermediate superexchange paths. In the real membrane, there is a lattice of dipoles. In equation (5) we have ignored the presence of all other atoms and dipoles, besides the dipoles A and B. The effect of the rest of lattice can be incorporated either by an exact calculation (Vaks, Larkin &
T (OK)
FIG.
2. The critical temperatures for a decorated Ising model with n = 12.
TEMPERATURE
DEPENDENCE
OF
MEMBRANE
EXCITABILITY
613
Ovchinnikov, 1966) or by a mean field theory (Fradkin & Eggarter, 1976). The same qualitative result as that by the above authors is achieved by taking large value of n in our simplified version. 5. Conclusions Our results indicate that the dipoles are oriented either all up or all down (ferroelectric order) up to the critical temperature T,, = 255 K. In the temperature range Tea < T < T,,, the system is in disordered state. At T,, = 317 K, the system goes into antiferroelectric phase. There is a further phase transition at T,, Y 7880 K, which is of little importance to us, since at such high temperature the membrane is expected to be irreversibly damaged. At low temperatures (T < T,,) the system is “frozen” in ferroelectric order. To disturb this long-range order a considerable amount of energy has to be expended. Consequently at low temperatures there is no response to a given stimulus. At T,, the system goes into a disordered state. From the Ising Hamiltonian it is clear that the behavior of such a system can be represented by a “two level maser”. Interpretation of the four fundamental properties of nerve impulse, namely the all-or-none response, the strength-duration relation, refractoriness and refractory period and frequency modulation, has been exhaustively discussed by Wei (1971). However, the non-threshold region is not very clearly explained in Wei’s model. Experimental evidence already suggests that the transition into the graded response region is possibly due to a phase transition. This comes out naturally in the present model as a transition into antiferroelectric state at T,,. The correlation of this phase with the graded response is not clear. The confirmation of these phase transitions requires closer experimental study of the various membrane parameters near the expected critical points. In particular, we hope that the present paper would stimulate further experiments to relate the temperature dependence of membrane excitability to structural order. REFERENCES Zeit. fiir Naturforsch. 23b, 181. ALMEIDA, S. P., BOND, T. D. &WARD, T. C. (1971). Biophys. J. 11,995. CHANGEIJX, J. P., THIERY, J., TUNG, Y. & K~ITEL, C. (1967). Proc. nut. Acad. Sci. U.S.A. 57, 335. CHAPMAN, D. (1967) In Thermobiology, p. 131, A. H. Rose, ed. New York: Academic
ADAM, G. (1968).
Press. COLE, K. S., GUTI-MAN, R. & BEZANELA, F. (1970). ENGELMAN, D. M. (197O)L mol. Biol. 47,150. FRADKIN, E. H. 8r EGGAR~R, T. P. (1976) Phys.
Proc. nat Acad. Sci. U.S.A. 65,884. Rev. A14,495.
614
P. K. GHOSH AND D. SENGUPTA
GOODWIN, B. C. (1974).Fwaday Symp. Chem. Sot. 9,226. GUTTMAN, R. (1972) In Perspectives in Membrane Biophysics, pp. 159-160,D. P. Agin, ed. NewYork: Gordon & BreachSC.Publishers, HILL, T. L. (1960). An Introduction to Statistical l%ermodynamics, p. 265. New York: Addison-Wesley PublishingCo. Inc. HUANG, K. (1963). Stutistical Mechanics. NewYork: JohnWiley. LENER,T. E. (1966). Organic and Biological Chemistry, p. 165.New York: The Ronald Press. NEWELL, G. F. & MONTROLL, E. W. (1953).Rev. mod. Phys. 25,353. STEIM, J. M., TOURTELLOTTE, M. E., REINERT, J. C., MCELHANEY, R. N. & RADER, R. L. (1969). Proc. nat. Acad. Sci. U.S.A. 63, 104. TASAKI, I. (1968). Nerve Excitation. Illinois: C. C. Thomas. VAKS, V. G., LARKIN,A. I. & O~~HMNIK~~, Yu. N. (1966). Sou. Phys. JETP 22, 820. WEI,L. Y. (1969).Bull. math. Biophys. 31,39. WEI,L. Y. (1971).Bull. math. Biophys. 33, 187.
A Model for the Temperature Dependence of Membrane Excitability P. K. GHOSH AND D. SENGUPTA Department of Physics, Visva-Bharati, Santiniketan 731235, West Bengal, India (Received 28 November 1977)
Based on a decorated Ising model, the possible existence of multiple phase transitions in biological membranes is; explored in this paper. Attempt is made to relate the stimulus-response behavior of membranes with structural order in different temperature regions.
1. htrodaction Nerve excitation without threshold has been observed in squid axon in a recent interesting work (Cole, Guttman & Bezanilla, 1970). It was shown that the maximum slope of the stimulus-response curve decreases from 10’ to 500 to 6 as the temperature was raised from 6*3”C to 20°C to 40°C (Guttman, 1972). It has been suggested that this sudden transition from the all or none law to a fractional or graded response might represent a phase transition in the membrane (Changeux, Thiery, Tung & Kittel, 1967; Tasaki, 1968; Adam, 1968). It is clear that in a biological system there will also be alower temperature bound to the domain of excitability. X-ray studies of the myelin in the frog sciatic nerve showed that maximum changes occur in the di&action pattern of fresh nerve between 0°C and -2°C and between 58°C and 61°C (Chapman, 1967). Evidence of thermal phase transition in Mycoplasma Laidlawii has been clearly established by differential scanning calorimetry (Steim, Tourtellotte, Reinert, McElhaney, & Rader, 1969) and X-ray diffraction studies (Engelman, 1970). This corresponds to a transition from hexagonal close-packed to liquid crystalline phase of the fatty acyl chains of lipid bilayers. It was shown that this reversible sharp transition takes place in the temperature range -2O-+ 5O”C, depending on the method of preparation. In the present work we hope to relate this phase transition to the all or none response in a membrane. 609 0022-5193/78/0821-0614$02.00/O 0 1978 Academic press Inc. (London) Ltd.
610
P.
K.
GHOSH
AND
D.
SENGUPTA
2. The Decorated Ising Model To discuss such co-operative phase transitions in biomembranes, the two-dimensional Ising model has been utilized by several authors (Changeux, Thiery, Tung & Kittel, 1967; Almeida, Bond & Ward, 1971). However, very little attempt has been made to correlate this phase transition to the temperature dependence of membrane response or to the structural details of the membrane. In the present model, the membrane is considered as a two-dimensional array with each site occupied by single polar molecule or a composite group having a net electric dipole moment. This electric dipole model has been justified by Wei (1971) on the basis of the fact that trimethylamine ions -N+ E (CH,), in membrane phospholipid has a permanent electric dipole moment. Liener (1966) showed that trimethylamine ions do occur at the surface of the membrane. The basic interaction in this case among the dipoles is clearly electrostatic. In order to make this model compatible with the Ising formalism (Newell & Montroll, 1953; Huang, 1963), we assume only nearest-neighbour (nn) interaction. In spite of the long range nature of electrostatic forces this model is expected to give good results for dipoles in an electrolytic solution (Hill, 1960). The simple two-dimensional Ising model with nn interaction always leads to a single order-disorder phase transition. Our earlier discussion suggests the possibility of two phase transitions: (a) from no-response to all-or-none response and (b) from the all or none to the graded response. The possibility of multiple phase transitions motivated us to explore the applicability of two-dimensional decorated Ising model in the present problem. Besides the direct dipole-dipole interaction, in the decorated Ising model there is an indirect interaction (super-exchange) between the dipoles through other atoms or molecules that do not have a net electric dipole moment. The existence of such non-polar molecules in the membrane is well known (Goodwin, 1974). If the direct and the super-exchange interactions are of opposite signs, then it is possible to obtain multiple phase transitions. 3. Mathematical
Formulation
Let J1 be the coupling constant for the direct interaction between nn electric dipoles and J2 the coupling constant for the super-exchange interaction via the intermediate atoms or molecules. The direct interaction is obviously repulsive (J1 < 0). For multiple phase transitions to occur, the super-exchange
TEMPERATURE
DEPENDENCE
OF
MEMBRANE
EXCITABILITY
611
interaction must be attractive and slightly stronger than the direct one (1.J21>lJJ). Both these interactions may be characterized by a unique effective interaction (Fradkin & Eggarter, 1976) with coupling constant J eR= J, + $kT In cash (2J,/kT). (la> where k is the Boltzmann constant and T is the absolute temperature. Instead of a unique super-exchange interaction there might be n intermediate atoms involved, each coupled to the sites A and B (Fig. 1) with coupling
FIG. 1. Direct and super-exchange (n intermediatepaths)interactionsbetweenelectric dipolesat sitesA and B.
constant J&.
The effective interaction is now modified to J,, = J, +&zkT In cash (2J,/nkT).
(lb) values for low and high temperatures are J, + IJzj > 0, for kT -%J, J eff (2) = { J1 <0, for kT % Jz. It is seen that the effective interaction changes sign at temperature T = IJ&k, and it is this change of sign which is responsible for multiple phase transitions. Such a decorated Ising model would have the same Hamiltonian as the regular one, with J replaced by Jes, and is of the following form The limiting
H=
-Jeff & CPjp
(3)
where (q) indicates nn and pi, Oj assume the values + 1. Exact solution by Onsager (Huang, 1963) shows that phase transitions occur whenever
IJeffl = 0.44
kT For large n there are three phase transitions and T,,, T,, around jJzI/nk. A calculation biological membranes is given in the next
* at temperatures TcI = 2.27 IJ1 I/k, using typical values of J1, Jz for section.
612
P. K. GHOSH
AND
D. SENGUPTA
4. Results We estimate J1, the direct coupling constant, starting from the values of the dipole moment (25 to 1000 Debye) as given by Wei (1969). The expression for J1 is as follows +2
J,
-- - -3 $1
(in esu),
(5)
where Ip^I is the electric dipole moment associated with each lattice site, jPl is the distance between nn dipoles and Eisthe dielectric constant of the membrane fluid. The above formula (5) holds for dipoles parallel to each other and oriented normal to the plane of the membrane. Now, substituting @I = 50 D (1 D = 10-l’ esu), l?l =5x10-*cm,ands=SO,weget J,/k z - 3600 (K). The exact order of magnitude of Jz cannot be estimated since the microscopic mechanism is not yet clear. Using Jl/k = - 34741K and J2/k = 4650 K we graphically solve equation (4) for the critical temperatures in Fig. 2. The values obtained are T,, z 7880 K, T,, = 317 K and T, = 255 K. It is to be noted that the calculation has been done for it = 12, indicating twelve intermediate superexchange paths. In the real membrane, there is a lattice of dipoles. In equation (5) we have ignored the presence of all other atoms and dipoles, besides the dipoles A and B. The effect of the rest of lattice can be incorporated either by an exact calculation (Vaks, Larkin &
T (OK)
FIG.
2. The critical temperatures for a decorated Ising model with n = 12.
TEMPERATURE
DEPENDENCE
OF
MEMBRANE
EXCITABILITY
613
Ovchinnikov, 1966) or by a mean field theory (Fradkin & Eggarter, 1976). The same qualitative result as that by the above authors is achieved by taking large value of n in our simplified version. 5. Conclusions Our results indicate that the dipoles are oriented either all up or all down (ferroelectric order) up to the critical temperature T,, = 255 K. In the temperature range Tea < T < T,,, the system is in disordered state. At T,, = 317 K, the system goes into antiferroelectric phase. There is a further phase transition at T,, Y 7880 K, which is of little importance to us, since at such high temperature the membrane is expected to be irreversibly damaged. At low temperatures (T < T,,) the system is “frozen” in ferroelectric order. To disturb this long-range order a considerable amount of energy has to be expended. Consequently at low temperatures there is no response to a given stimulus. At T,, the system goes into a disordered state. From the Ising Hamiltonian it is clear that the behavior of such a system can be represented by a “two level maser”. Interpretation of the four fundamental properties of nerve impulse, namely the all-or-none response, the strength-duration relation, refractoriness and refractory period and frequency modulation, has been exhaustively discussed by Wei (1971). However, the non-threshold region is not very clearly explained in Wei’s model. Experimental evidence already suggests that the transition into the graded response region is possibly due to a phase transition. This comes out naturally in the present model as a transition into antiferroelectric state at T,,. The correlation of this phase with the graded response is not clear. The confirmation of these phase transitions requires closer experimental study of the various membrane parameters near the expected critical points. In particular, we hope that the present paper would stimulate further experiments to relate the temperature dependence of membrane excitability to structural order. REFERENCES Zeit. fiir Naturforsch. 23b, 181. ALMEIDA, S. P., BOND, T. D. &WARD, T. C. (1971). Biophys. J. 11,995. CHANGEIJX, J. P., THIERY, J., TUNG, Y. & K~ITEL, C. (1967). Proc. nut. Acad. Sci. U.S.A. 57, 335. CHAPMAN, D. (1967) In Thermobiology, p. 131, A. H. Rose, ed. New York: Academic
ADAM, G. (1968).
Press. COLE, K. S., GUTI-MAN, R. & BEZANELA, F. (1970). ENGELMAN, D. M. (197O)L mol. Biol. 47,150. FRADKIN, E. H. 8r EGGAR~R, T. P. (1976) Phys.
Proc. nat Acad. Sci. U.S.A. 65,884. Rev. A14,495.
614
P. K. GHOSH AND D. SENGUPTA
GOODWIN, B. C. (1974).Fwaday Symp. Chem. Sot. 9,226. GUTTMAN, R. (1972) In Perspectives in Membrane Biophysics, pp. 159-160,D. P. Agin, ed. NewYork: Gordon & BreachSC.Publishers, HILL, T. L. (1960). An Introduction to Statistical l%ermodynamics, p. 265. New York: Addison-Wesley PublishingCo. Inc. HUANG, K. (1963). Stutistical Mechanics. NewYork: JohnWiley. LENER,T. E. (1966). Organic and Biological Chemistry, p. 165.New York: The Ronald Press. NEWELL, G. F. & MONTROLL, E. W. (1953).Rev. mod. Phys. 25,353. STEIM, J. M., TOURTELLOTTE, M. E., REINERT, J. C., MCELHANEY, R. N. & RADER, R. L. (1969). Proc. nat. Acad. Sci. U.S.A. 63, 104. TASAKI, I. (1968). Nerve Excitation. Illinois: C. C. Thomas. VAKS, V. G., LARKIN,A. I. & O~~HMNIK~~, Yu. N. (1966). Sou. Phys. JETP 22, 820. WEI,L. Y. (1969).Bull. math. Biophys. 31,39. WEI,L. Y. (1971).Bull. math. Biophys. 33, 187.