Supplementary Material for
“2D lattice model of a lipid bilayer: Microscopic derivation and
thermodynamic exploration”
Davit Hakobyan and Andreas Heuer
Note 1
Since temperature has only a minor influence on the DLiPC behavior we restrict ourselves
to the discussion of DPPC. The dependence of the pairwise DPPC – DPPC interaction energy on
temperature is shown in Fig. S1a. As expected, the average interaction energy (red) for low order
parameters resembles the curve resulting from the high temperature simulations (i.e. 330 – 360 K,
disordered state) while on the right side it basically follows the low temperature data (290 – 320
K, ordered or gel state).
Thus, the minor temperature dependence for small and large order parameters,
respectively, is not relevant because the population of these states is basically restricted to high
and low temperatures, respectively.
It is interesting to note that in all lipid interactions the magnitude of the interaction energy
increases toward the very low order parameter values. To see which part of the total interaction
energy contributes to this effect the most the DPPC – DPPC interaction energy (red curve in Fig.
S1a) is decompose in tail – tail, head – tail and head – head interactions (Fig. S1b). As expected,
the order-dependent enthalpy is mostly formed by the tail-tail interaction. Interestingly, all three
components show an increase of interaction to the left of the plot for very disordered systems. It
may be suggested that in this limit the increase of the tail-tail interactions may be a results of
intercrossing of lipid tails (i.e. increasing number of contacts) as well as a decrease of the headhead distance. A precise answer to this question requires a more detailed analysis and is beyond
the scope of this paper.
FIG. S1. The DPPC – DPPC interaction enthalpies in pure DPPC MD systems. (a) The red curve
is the average over all temperature ranges. The black thick curve is the average enthalpic
interaction between DPPC lipids in liquid-disordered phase while the orange curve is the average
enthalpic interaction in gel (sub-gel) phases. (b) The decomposed energies of DPPC – DPPC
interaction in the pure system averaged over temperatures from 290 to 360 K.
Note 2
The difference between the initially guessed and the final estimation of the entropic
functions for DPPC lipid are shown in Fig. S2. This graph clearly shows that the initial guess for
the entropy is not too bad. However, the subsequent iteration steps are essential for a correct
estimation of the entropy function. As discussed in the main text, the difference between the
initial and final curve reflect the fact that the order parameters of the lipids are not identical for
all lipids but fluctuate around the average value and are spatially correlated.
FIG. S2. The initial and final entropic functions of DPPC lipid in the lattice model.
Note 3
The entropic (and enthalpic) polynomial for DPPC lipid was defined for the order
parameter range from -0.3 to 1. Below the value -0.3 the entropy was set to zero. Likewise, the
order parameter for DLiPC lipid was limited to the range from -0.3 to 0.8. The DPPC – DLiPC
interaction was defined in the range from -0.3 to 0.9.
Table I. Enthalpy polynomial coefficients. Shown in least-significant order first.
DPPC – DPPC
DLiPC – DLiPC
DPPC - DLiPC
-66.192
14.296
-17.170
12.547
-115.641
84.159
-68.883
24.251
-46.427
-15.850
-66.875
13.171
-27.968
21.086
-33.121
Table II. Coefficients of number of nearest neighbor function derived from respective pure
systems. The coefficients for DPPC and DLiPC correspond to sigmoid and polynomial functions,
respectively. The sigmoid function for DPPC has the following form: c1 / (1 + exp (-c2 * (x –
c3))) + c4, where the coefficients c1, c2, c3, and c4 correspond to the values shown in the table.
The polynomial coefficients for DLiPC are specified in least-significant order first.
DPPC
DLiPC
0.941
18.293
0.636
3.880
3.528
0.123
-0.091
Table III. The polynomial coefficients of the natural logarithm of the entropic Ω functions. Shown
in least-significant order first. The polynomial coefficients for DPPC and DLiPC are of order 17
and 15, respectively. These polynomial functions demonstrate the best fitting to rather complex
entropic curves.
DPPC
DLiPC
-5.7
14.9
-19.6
1.3
-394.4
951.1
10814.2
-42054.8
-88050.3
560608.8
-47538.1
-3285689.9
4550402.2
5461708.0
-20882403.7
23395800.4
-12094000.9
2459815.1
-5.5
19.9
-43.0
-9.2
-227.6
1465.1
3659.1
-43665.8
52365.1
336517.7
-1060533.1
200078.8
3716993.4
-7060679.2
5435987.0
-1585802.4
Table IV. The DPPC exchange time dependence on the order parameter
SCD range
Liner fitting coefficients
0.30 - 0.65
-6.236
23.178
-47.185
94.092
0.65 - 0.73