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Theory of biomembrane phase transitionsJ. F. Nagle

Citation: The Journal of Chemical Physics 58 , 252 (1973); doi: 10.1063/1.1678914 View online: http://dx.doi.org/10.1063/1.1678914 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/58/1?ver=pdfcov Published by the AIP Publishing

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T H E O R Y O F B I O M E M B R A N E P H A S E T R A N S I T I O N S 253

interactions between chains will prevent most singlebond trans gauche changes until relatively high temperatures when a cooperative melting. of the hydrocarbon chains can take place. Such a cooperative

transition is observed calorimetrically.3.4.6.12.14 Measurements 6 on mycoplasma laidlawii showed the transition width to be about 20C centered at about 35Cfor a growth temperature of 37C. I t has been emphasized by many authors that this transition takes placeat temperatures of prime importance to biologicalsystems. Apparently membranes need a certain amountof chain fluidity to be viable.) Recent measurements 22

on synthetic pure lecithins show a much narrowertransition as shown in Fig. 2 with a transition enthalpyl .H,,-,9.5 kcal/mole for dipalmitoyl-L-a-Iecithin (DPL).This figure also shows an anomalous specific heat at34C which will be discussed in Sec. V. The sharpnessof the main specific heat anomaly at 41C in Fig. 2indicates the possibility of a first or higher order phase

transition for pure, uniform lecithins and, at least, isindicative of a strongly cooperative process.

The nature of the transition is most clearly revealedby x-ray diffraction experiments.6.7.8.12Below the transition the hydrocarbon chains produce a sharp diffraction ring which corresponds to a nearest neighbor chaindistance of 4.8 A Above the transition this ring isquite diffuse as in liquid hydrocarbons and correspondsto distances of roughly 5.3 A

From these two sets of experiments it seems reasonable to propose an order-disorder model for which wewrite an effective Hamiltonian

H = H a t t Hste r ic+ H r t

where att is the sum of the attractive interactionsholding the bilayer together and Hswric prevents anytwo molecules from being closer together than theirvan der Waals or hard core radii. The model is calledan order-disorder model because, as in polymer theory 29H rot allows each carbon-carbon bond only the transconformation with energy 0 or the two gauche conformations with energy E=O.5 kcal/mole. The partitionfunction is then a sum over discrete states and kineticenergy plays no role in the model: motion over irans-gauche rotational barriers is assumed in order to achieveequilibrium but it has negligible effect on the partit ionfunction and thermodynamic properties.

The order-disorder assumption in our model is reasonable if and only if there is not much motion, i.e.,if the correlation times o are long. Evidence supportingthis assumption has been obtained by NMR and spinlabeling experiments above the transition. In particular, experiments 5. 6 on multilamellar bilayers indicatethat the correlation times are of the order of 10 - 6 secfor methylene groups nearest the phosphate head groupsand for methylenes near the terminal methyl of the

hydrocarbon chains the correlation times are of the

z~ 6 0 r - - - - - . - - - - - . - - - - - - - - - - . - - - -:3oCf)

11..50o

E2lng(cp1, c/>2 , (A1)where g CP1,cI>2 2=detM(CP1, c/>2)which is obtained following Kastelijn's method.

40

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T H E O R Y O F B I O M E M B R A N E P H A S E T R A N S I T I O N S 263

For model A with unit cell shown in Fig. 13

0 x[1-exp( -i-Pl J 1 + y exp i4>2) 0

x[exp 4>t) 1 J 0 0 y-exp i4>2)M 4)l,4>2)=

l y exp -i4>2 0 0 x[1-exp( -i4>t) J A2)

0 - y + e x p - ~ )x[exp 4>1 1 J 0and

g 4)l, 4>2)= 2r(1-cos4>l) e x p ~ )+2y_y2 exp ~ ) . A3)Using Eq. 3.2) we have

A4)

and

AS)

The -P2 integrals in A4) and AS) can be performed by changing variables to w= ei4>2 and using the calculus ofresidues. The different analytic behavior which appears in the different regions in the x-y plane in Fig. 7 is dueto the poles of the denominators in A4) and AS) crossing from inside to outside of the unit circle. For example,for y< lone pole is always inside the unit circle and the second pole is inside the unit circle iff 2r(1-cos-Pl) 2in A4) and AS) then turns out to be simply trigonometric rather than the generally expected elliptictype due to cancellations between the denominator and the numerator. The results for Po and Pu are given in 3.6).

Similarly for model B one has for the unit cell in Fig. 14

A6)

and

Using 3.2) we have

A7)

and

P u = t i ~ ) 2r 1 r2o d4>2 . A8)2 11 J J y2+ r 2x cos4>l+ 1) e- 2

The -P2 integrals in A7) and A8) are easily performed using w= e i< >2and the calculus of residues. The -PI integralin A8) is then trivial and the one in A7) can be found in standard tables. Again the results are found in thetext in 3.7).

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