流体膜の典型例は生体膜における脂質二重膜である。膜におけるタンパク質やラフト様の液体領域の拡散係数は, 生体化学反応の重要な因子として, 実験的理論的に研究されてきた。物体が流体中を小さな速さで並進運動するときに流体から受ける力の大きさは速さに比例し, その比例係数を抵抗係数という。抵抗係数と拡散係数は関連がつく。
本研究では, 両側を三次元流体に囲まれた平面流体膜にある円形液体領域の抵抗係数を計算した。剛体円板や, 内外の膜粘性が等しい円形液体領域の抵抗係数は, 他の研究者により, 既に計算されている。内外の膜粘性がわずかに異なる円形液体領域の抵抗係数は, 報告者自身が過去に計算したが, そこで指摘されたように, 内外の膜粘性が異なる場合, 応力の連続のために, 膜における二次元流速場は領域境界で滑らかでない。ところが, まわりの三次元流体の流速場は滑らかであって, この流速場と二次元流速場は粘着境界条件で結びついている。報告者は既にこのような解を構成する方法を開発していたが, 本研究では, 抵抗係数を内外の膜粘性差に関して展開し, その係数の漸化式を導出し, それに含まれる無限積分を数値的に評価して, 級数の和を求めることで, 実際に抵抗係数を計算した。内部の膜粘性が無限の極限では, 級数の和が剛体円板に対する結果に一致することも確認された。まわりの三次元流体が, 壁によって囲われている場合についても, 抵抗係数を計算し, 内部の膜粘性が無限の極限では, 従来の剛体円板の結果に一致することを確認している。
三次元流体中のブラウン運動の計測の分解能は, 近年飛躍的に向上している。流体膜において同様の分解能で, 円形液体領域の拡散係数が計測できれば, 本研究の結果を使って, 円形液体領域や流体膜の物質定数を正確に決定できるようになると期待される。
A typical example of the fluid membrane is a lipid-bilayer membrane contained in the biomembrane. The self-diffusion of a membrane protein and that of a raft-like region have been studied experimentally and theoretically because they are considered to be important in the biochemical reaction. The force which the ambient fluid exerts on an object in its stationary translational motion has the magnitude proportional to the speed if the motion is sufficiently slow. The constant of the proportion is referred to as drag coefficient, which can be related with the diffusion coefficient.
In this study, the drag coefficient of a circular liquid domain is calculated on the assumption that it lies in a flat fluid membrane surrounded by three-dimensional fluids on both sides. The coefficient of a rigid disk is well known, while that of a circular liquid domain is also well known when the membrane viscosity inside the domain equals the one outside the domain. I previously calculated the drag coefficient by assuming that the ratio of the membrane viscosity of the domain interior to that of the domain exterior is slightly different from unity. When the ratio is not unity, the continuity of the stress makes the velocity gradient discontinuous across the domain perimeter in the membrane. On the other hand, the velocity gradient is continuous in the ambient fluids, whose velocity field should agree with that of the membrane as the spatial point approaches the membrane. I previously devised a calculation procedure leading to fields satisfying the conditions. The procedure is extended in the present study ; the recurrence relations for the coefficients of the series expansion of the drag coefficient with respect to the dimensionless difference between the membrane viscosities are derived, the numerical integrations involved in the coefficients are performed, and the drag coefficient of a circular liquid region with a distinct membrane viscosity is calculated. As the ratio increases to infinity, the sum of the series is numerically shown to approach the drag coefficient of the disk calculated previously. The approach is also found when the three dimensional fluid on each side is confined by a flat wall.
Recently, the resolutions in measuring the Brownian motion in the three-dimensional fluid are much improved. The results of the present study, together with future measurement of the self-diffusion of a circular liquid domain with high resolutions, should help in determining the material constants involved in a fluid membrane having a circular liquid domain.