摘要:

1IntroductionElectron paramagnetic resonance (EPR) spectroscopy is particularly suitable for studying reorientational dynamics in lyotropic liquid crystalline phases. The spectral lineshape is quite sensitive to the overall rotational dynamics because the time scale is in the slow‐motion or near slow‐motion regime.1Dynamical and structural information about the lipid molecules has been extracted from EPR lineshape analysis of a variety of lyotropic phases, including lamellar,2micellar3,4and hexagonal5phases. In lamellar phases the theoretical emphasis has been on the internal and restricted reorientational lipid motion. However, for curved lipid/water interfaces, as in the micellar and the hexagonal phases, an additional lipid motion is also active, namely the surface or lateral diffusion. In ref. 5 lateral diffusion around hexagonal cylinders clearly modifies the EPR lineshape. By assuming a fixed lipid diffusion coefficient it was possible to estimate the cylinder radius. All these lineshape analysis diffusion models were assumed. The synthesis of molecular reorientation description and the slow‐motion lineshape theory were developed in the eigenfunction formalism in which the stochastic Liouville equation (SLE) in the Fokker–Planck form was solved.6,7Present address: Institute of Electronics NANB, Logojsky Tract 22, 220090, Minsk, Belarus.Department of Mathematics, Umeå University, 901 87 Umeå, Sweden.In this work we turn to another approach, the direct method based on the Langevin form of SLE, summarized inTable 1. The direct method of calculating EPR slow‐motion lineshapes is suitable when molecular dynamics are available from classical dynamics simulation techniques. The lineshape theory may be synthesized with Brownian dynamics (BD) or molecular dynamics (MD) simulation techniques. In this work we use BD simulation techniques in order to include lipid lateral diffusion along curved lipid/water interfaces in the lineshape model. The second lipid motion taken into account is the restricted intramolecular reorientational diffusion of the lipid chain.Theoretical levels of the SLE in the Langevin formOur main objective here is to develop an EPR lineshape approach which takes into account the influence of curved lipid/water interfaces. We present explicit calculations of both local lipid reorientational diffusion and lateral diffusion by means of Brownian dynamic (BD) simulation techniques. The synthesis of slow‐motion ESR lineshape theory and BD simulations is illustrated for two types of curved surfaces, the rippled surface, given in three‐dimensional Cartesian coordinates (x,y,z) byz=asin(bx), and what we like to call the “Baltic Sea” surface given byz=a[sin(bx)+sin(by)], as shown inFig. 1, whereaandbare parameters describing the amplitude and period of the surface.Two model surfaces are displayed: the “Baltic Sea” surfacez=a[sin(bx)+sin(by)] and the “Rippled” surfacez=asin(bx).1.1Theory of slow‐motion EPR lineshapesThere are two theoretical approaches in analyzing slow‐motion EPR lineshapes. One approach is based on the stochastic Liouville equation (SLE) in the Fokker–Planck (FP) form. This approach was developed by Kubo6in the early 1960s and later further developed for EPR by Freed and coworkers.7The second approach, which is used in this paper, is based on the Liouville–von Neumann equation in the semiclassical approximation. It is called the Langevin form of SLE and is astochastic differential equation. The FP‐SLE equation describes a quantum mechanical subsystemSimbedded in a “large” classical latticeLwhere the classical degrees of freedom are described by a Fokker–Planck equation. The system density operatorσ(t,Λ) then satisfies1where the Liouville operatorL0Sdescribes the isolated spin subsystemSandΓΛis the diffusion operator. The interaction between the spin subsystemSand the environment, the latticeL, is given byLSL(Λ), which depends explicitly on classical random variablesΛ, but not explicitly on time. HereΛis used as a short notation for all stochastic variables present in the spin–lattice interactionLSL(Λ). The time dependence ofΛis given by a probability density,P(Λ,t) satisfying a Fokker–Planck (diffusion) equation involving the operatorΓΛ, thus representing Markov processes which are relevant with respect to the spin–lattice couplingLSL(Λ). In EPR lineshape analysis, (1) is most conveniently solved in the frequency space using a direct product basis constructed from eigenfunctions of the Fokker–Planck operator and eigenoperators of the time independent electron spin LiouvillianL0S. The problem of solving the system of coupled differential equations is thus transformed into a problem of inverting a huge Liouville matrix at a large number of frequencies, which is most often solved by an iterative procedure due to Lanzcos.8The second approach, the Langevin SLE developed for slow‐motion EPR lineshapes has recently been discussed.9–11In refs. 9 and 11 the Liouville–von Neumann equation is transformed into a system of coupled stochastic differential equations with explicit time dependence in the spin–lattice coupling. Consequently, for a nitroxide spin label with electron spinS=1/2 coupled with a nuclear spinI=1, a complete set of spin operators is introduced and the stochastic Liouville equation may be reduced to a system of nine coupled stochastic differential equations112where the spin–lattice coupling is explicitly time dependent. However, in this formulation no explicit description is given of the stochastic process present in the spin–lattice coupling. Thus, in order to proceed, an explicit computational model must be designed for the stochastic process present inLSL(t). Such a model, equivalent to the one used in (1), is obtained by transforming the Fokker–Planck equation ∂P(Λ,t)/∂t=ΓΛP(Λ,t) into the corresponding Langevin equation. From direct simulation of the Langevin equation the fluctuation ofΛ(t) is obtained, and consequently the fluctuation ofLSLis obtained in terms of trajectories. It is also possible to construct suitable trajectories ofΛ(t) and consequentlyLSL(t) from sufficiently long molecular dynamics (MD) simulations. In the Langevin‐SLE approach (seeTable 1) we thus have to simulate a system of coupled stochastic differential equations. Comparing the two approaches: using the FP‐SLE one obtains the EPR lineshapes directly in the frequency domain, whereas in the Langevin‐SLE the time dependent spin density matrix is obtained,i.e. the electron spin time correlation functions. EPR lineshapes are obtained by applying a Fourier–Laplace transform. This is a more informative approach. It is also a more flexible approach with respect to the stochastic processes of the spin–lattice coupling and the more direct relation to computer simulation methods.Here we apply the Langevin form of the SLE to calculate slow‐motion EPR lineshapes of curved lamellar systems. The spin bearing species are allowed to diffuse along a curved surface. We consider two statistically independent dynamic processes of the lipid molecules in the bilayer. First, the local reorientational diffusion of the lipid acyl chain is obtained by BD simulation, using the cone model. Secondly, translational diffusion of the spin probe along the curved surface is obtained by performing BD simulations of force‐free diffusion along curved surfaces. These two dynamic processes are modulatingLSL(t) of the Langevin‐SLE, which is solved following the approach described in detail in ref. 11. However, here we are faced with an extra dynamic process, namely the lateral diffusion of lipid molecules along curved interfaces. One aim of this paper is to develop this method of including several independent dynamic processes in the Langevin‐SLE. Secondly, we are interested in the methodology of describing translational diffusion on curved bilayers and investigating in what sense EPR slow‐motion lineshapes are influenced by the curvature of bilayer surfaces.

ISSN:1463-9076     

出版商:RSC    

年代:2000

数据来源: RSC