MM Study

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J. Phys. Chem. B 2008, 112, 2445-2455

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Monitoring Biological Membrane-Potential Changes: A CI QM/MM Study Catalin F. Rusu,†,‡ Harald Lanig,† Olaf G. Othersen,† Carola Kryschi,*,‡,§ and Timothy Clark*,†,§ Computer-Chemie-Centrum der Friedrich-Alexander-UniVersita¨t Erlangen-Nu¨rnberg, Na¨gelsbachstrasse 25, 91052 Erlangen, Germany, Institut fu¨r Physikalische und Theoretische Chemie, Friedrich-Alexander-UniVersita¨t Erlangen-Nu¨rnberg, Egerlandstrasse 3, 91058 Erlangen, Germany, and Interdisciplinary Center for Molecular Materials, Friedrich-Alexander-UniVersita¨t Erlangen-Nu¨rnberg, Na¨gelsbachstrasse 25, 91052 Erlangen, Germany ReceiVed: July 10, 2007; In Final Form: NoVember 6, 2007

In recent decades, new less-invasive, nonlinear optical methods have been proposed and optimized for monitoring fast physiological processes in biological cells. One of these methods allows the action potential (AP) in cardiomyocytes or neurons to be monitored by means of second-harmonic generation (SHG). We now present the first, to our knowledge, simulations of the dependency of the intensity of the second harmonic (ISHG) on variations of the transmembrane potential (TMP) in a cardiomyocyte during an action potential (AP). For this, an amphiphilic potential-sensitive styryl dye molecule with nonlinear optical properties was embedded in a dipalmitoylphosphatidylcholine (DPPC) bilayer, replacing one of the phospholipid molecules. External electrical fields with different strengths were applied across the membrane to simulate the AP of a heart-muscle cell. We used a combined classical/quantum mechanical approach to model the structure and the spectroscopic properties of the embedded chromophore. Two 10 ns molecular dynamics (MD) simulations provided input geometries for semiempirical molecular orbital (QM/MM) single-point configuration interaction (CI) calculations, which were used to calculate the wavelengths and oscillator strengths of electronic transitions in the di-8-ANEPPS dye molecule. The results were then used in a sum-over-states treatment to calculate the second-order hyperpolarizability. The square of the hyperpolarizability scales with the intensity of the second harmonic, which is used to monitor the action potentials of cardiomyocytes experimentally. Thus, we computed changes in the intensity of the second harmonic (∆ISHG) as function of TMP changes. Our results agree well with experimental measurements.

Introduction Second-harmonic-generation (SHG) microscopy1-4 is a candidate technique aimed at elucidating membrane biophysics with voltage-sensitive dyes. It has been reported3-5 to be more sensitive than fluorescence-microscopy techniques based on electrochromism. A chronological overview of SHG as a promising investigation method is given by Millard et al.6 In order to follow fast physiological membrane processes, for example action potentials (APs) in cardiomyocytes or neurons, in biological tissues noninvasively, custom push-pull chromophores are required. These molecules are engineered to bind to and orient within a lipid bilayer in order to exhibit a direct electronic response to alterations in the membrane potential. The aminonaphthylethenylpyridinium (ANEP)1,4,6-12 dyes are currently among the most commonly used fast potentiometric probes1-4,6-11,13-16 and are capable of following submillisecond membrane-potential changes.1,4,7-11 The spectral characteristics of ANEP dyes are generally quite similar. However, structural variations among them makes some especially suitable for specialized applications. Di-4-ANEPPS4,6,7,9,13,14 and di-8-ANEPPS4,6,8,9,12,14-16 show the most consistent poten* To whom correspondence should be addressed. Telephone: +49 9131 852 7305 (C.K.); +49 9131 852 2948 (T.C.). Fax: +49 9131 852 8796 (C.K.); +49 9131 852 6565 (T.C.). E-mail: [email protected] (C.K.); [email protected] (T.C.). † Computer-Chemie-Centrum. ‡ Institut fu ¨ r Physikalische und Theoretische Chemie. § Interdisciplinary Center for Molecular Materials.

tiometric responses in different cell and tissue types.17 The latter, being more lipophilic, is less susceptible to internalization17 and therefore stabilizes better in the hydrophobic interior of phospholipids, permitting extended observation.17 Such chromophores undergo changes in their electronic structure and, consequently alteration of their spectroscopic properties, in response to changes in the surrounding electric field.7,9-11,13,16 Excitation-ratio measurements allow membrane potentials to be quantified via a potential-dependent shift in the excitation spectra.7-13,16 Furthermore, electrooptic hyperpolarizability modulation makes the second-harmonic signal displayed by these dyes sensitive to changes in the membrane potential.1,4,9,12 SHG in organic molecules is a second-order nonlinear process and is influenced by molecular structure and orientation. For individual molecules, the quantities of interest are the tensor elements, βijk, of the second-order molecular hyperpolarizability tensor β. The arrangement of individual molecules with respect to an external field, and in the case of crystals to each other, determines how the βijk terms interact and contribute to the macroscopic second-order nonlinearity. Oudar18 has identified the structural properties that have been shown to generate large β values in organic crystals and has classified them as π-conjugation, charge transfer, and noncentrosymmetric crystal structure. A noncentrosymmetric crystal structure allows the generation of optical phenomena arising from even-order susceptibilities χ(n). In a molecular framework, the lack of center-of-inversion

10.1021/jp075372+ CCC: $40.75 © 2008 American Chemical Society Published on Web 02/05/2008

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symmetry is essential for the nonzero even-order susceptibility contributions to the overall nonlinearity. Both the increase of conjugation in a highly polarizable π-electron system and an increment of the charge-transfer process, which requires the presence of push-pull substituents on an aromatic ring or other π-conjugated system, enhance β. It has been shown, for nitroanilines for example, that β can be separated into two parts:19,20

β ) βadd + βCT

(1)

where βadd is the additive term, which corresponds to the interaction between the substituents and the π-conjugated system, and βCT is the contribution due to the donor-acceptor charge transfer. According to the theory, βCT should make the strongest contribution to the total β. Using the two-level model,19 Morrell and Albrecht21 computed βCT for p-nitroaniline. Considering only the ground state and first excited charge-transfer state, they obtained 75% of the experimental hyperpolarizability value (βexp). Furthermore, for more extensively conjugated π-systems, the charge-transfer state is lowered in energy with respect to other excited configurations, and one would expect the two-level system to agree even better with experimental data. Oudar18 showed for aminonitrostilbene and dimethylaminonitrostilbene that βadd represents only a 10% contribution to the total β and can therefore be neglected. Thus, considering the two-level system, eq 1 can be written as

β = βCT )

3e2p2 F(ω)f∆µg,e 2m

(2)

with

F(ω) )

W [W - (2pω) ][W2 - (pω)2] 2

2

(3)

The term F(ω) accounts for dispersion and enhances βCT as the fundamental frequency (ω) and first harmonic (2ω) approach the energy of the charge-transfer function. In eqs 2 and 3, W is the energy of transition, pω is the energy of a laser photon, f is the oscillator strength, and ∆µg,e is the difference between the excited-state dipole moment and that of the ground state: ∆µg,e ) e(〈1|r|1〉 - 〈0|r|0〉). Using the transition energy and the dipole transition moment, the oscillator strength can be written as f ) (2m/p2)W|〈0|r|1〉|2. In a formal sense, the external field dependence of the nonlinear process can be described by the electric-field dependence of the βZZZ component of the second-order polarizability tensor. Zyss22 and Bouevitch1 have expressed the βZZZ component of the hyperpolarizability, taking voltage-dependent alterations in ∆µg,e into consideration, as

β = βCT )

3e2p2 F(ω)f[∆µe0 + (Pe - Pg)E] 2m

(4)

They argued that the voltage-dependent alteration in ∆µg,e probably results from the fact that the linear molecular polarizability in the excited state, Pe, is as a rule larger than that in the ground state, Pg, for molecules with extended conjugated π-electron systems. As a result, one obtains a linear response of the second-order polarizability to the electric field E, in our case the transmembrane potential (TMP). This leads to the relationship for the SHG signal intensity (ISHG) with an additional quadratic contribution to ISHG.23

ISHG ∝ P2(2ω) ∝ βZZZ2 ∝ ∆µe02 + 2∆µe0(Pe - Pg)E + (Pe - Pg)2E2 (5) In this way, ∆ISHG can be computed as a function of TMP changes. In this paper, we will use this method to monitor an AP in a heart muscle cell optically. Biological membranes are vital components of living organisms. They play an active part in the life of all cells. Among other functions, semipermeability allows them to control the flow of information and movement of substances between cells. This is accomplished either by recognizing signal molecules received from other cells or by sending chemical or electrical signals to other cells. Therefore, since they are involved in such critical processes of living matter, membranes are essential for all living things. Their study is thus important, in both biology and other related fields. The structure of biomembranes can be represented as a twodimensional, fluid mosaic of lipids, proteins, and sugars that adopts a two-layer pattern. Apart from very fast conformational movements on the picosecond time scale, most of the lipids and proteins also show dynamics over longer time scales. They can move along or, more rarely, across the membrane, giving lateral diffusion coefficients up to several µm2 s-1 and flipflop diffusion coefficients on the order of one event every several hours.24 Information about this complex dynamic situation can be obtained from spectroscopic25-27 and microscopic28-33 techniques. However, as the structures involved become smaller than the wavelength of the visible light, crucial details about dynamic processes are lost. Electron microscopy provides some insight at this level, but only gives statistical snapshots of what are, in reality, ever-changing systems. In the past two decades, atomic-level computer simulation has developed into a complementary technique34 for studying bilayers. Lipid bilayer models of relevant biological lipids have been investigated computationally, giving insight into previously hidden phenomena. As the available computing power increases, theoretical studies on larger bilayer systems and for larger simulation times become possible. Techniques that reveal the motion of molecules and fluids on length scales of nano- to micrometers and time scales of nano- to microseconds are being developed.35-39 Despite the intense interest and extensive experimental work on following and interpreting fast biophysiological processes in living cells using these relatively new methods, few theoretical investigations are available. Thus, we now report on a method to simulate the hyperpolarizability modulation of a di-8ANEPPS molecule as a function of membrane-potential changes and its application. The technique is essentially an ensemble model40 that involves computing the hyperpolarizability of an embedded dye molecule for many snapshots at different times. A model for a biological membrane is used as a matrix for the dye molecule. External electrical fields of different strengths are applied across the bilayer, at different times and thus for different conformational geometries, to simulate the AP of a cardiac cell. Finally, the hyperpolarizability values are used to compute the percentage change in the intensity of the secondharmonic signal (ISHG). By doing this, we compute ∆ISHG as a function of an AP. These values can be then compared directly with experiment. Methods Molecular dynamics (MD) calculations used the GROMACS41,42 package to perform two 10 ns (MD) simulations of

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Figure 1. Di-8-ANEPPS (left) and DPPC (right) molecules. Sulfur atoms are shown in yellow, oxygen in red, nitrogen in blue, phosphorus in tan, and methyl and methylene groups in light blue.

two similar bilayer/dye systems, each consisting of one di-8ANEPPS and 63 DPPC solvated lipid molecules. These were equilibrated in a first preproduction step of 10 ns. The production phase was started using the last snapshots of the preproduction phase, but with newly generated forces and velocities for all atoms. Snapshots taken from the production phases of the simulations gave “hot” geometries of the dye molecule that where used further for combined quantum mechanical/molecular mechanical (QM/MM) single-point configuration interaction (CI) calculations, performed with the semiempirical program package VAMP 9.0.43,44 The relevant chromophores were embedded in the bilayer environment and the solvent using a QM/MM CI approach in which the environment is represented by a classical force field. The point charges MM environment is allowed to polarize the QM wave function via an additional one-electron term in the Fock matrix.45-47 The QM/MM CI treatment was that described in our recent work on FRET in proteins.40 The semiempirical CI calculations provide all variables required to compute the absorption wavelengths and intensities and thus second-order hyperpolarizabilities, used to approximate the second-harmonic intensity (ISHG), as described in ref 40 and shown in the Supporting Information. In our simulation, we analyzed the temporal evolution of a single-dye test system to mimic the properties of the bulk system, which in reality, consists of a large number of molecules. This is know as the ensemble model.40 Geometry snapshots taken every 10 ps for the last 7 ns production phase of the two systems (700 snapshots per simulation) were used for the analysis. Molecular Dynamics Simulations. Initial Structures. As starting structures for building the two bilayer/dye systems, we used a 64 DPPC lipid bilayer, made available for download by the courtesy of P. Tieleman48 (www.moose.bio.ucalgary.ca) and a di-8-ANEPPS molecule, whose structure file was generated with the PRODRG49 program. The dye was then embedded in the membranes, at different locations for each system, by replacing one phospholipid per bilayer with an ANEP molecule. The details of preparation of the bilayer/dye structures are given in the Supporting Information. Both lipid and dye are amphiphilic molecules (Figure 1). The dye molecule is therefore expected to stabilize between the lipid molecules during the MD simulation. Details are given below. The first system consisted of 63 DPPC lipid, one dye, and 3837 water molecules. The second system had four additional solvent molecules. The different numbers of water molecules arise from the system

Figure 2. Starting system for the production phase of the first MD simulation. Water molecules are shown in blue, the dye in red, and the lipids in green. The polar heads of the lipid molecules are shown in bold.

preparation phase, in which the dye is inserted into the membrane. The large relative amount of water (the water:lipid molecules ratio was 60:1) leaves ample room to investigate a possible case in which the dye molecule might stabilize at a position half embedded, half in the surrounding solvent. Generally, a bilayer with a large amount of water is also expected to mimic a biological membrane better than a system with little water.50 Experimental and theoretical results suggest that DPPC bilayers in the LR phase are fully hydrated at water weight fractions of c ) 0.36,51,52 c ) 0.40,53 or c ) 43;50 all these values lie below our value of c ) 0.59. Each of the two starting systems was subjected to a 10 ns preproduction MD in order to allow the chromophore to accommodate and equilibrate in the new environment. This was followed by a 10 ns production phase. The starting system for the production phase of the first MD simulation is shown in Figure 2. MD Simulations. For the simulations, we used the force field reported previously by Marrink et al.,48 i.e., optimized hydrocarbon Lennard-Jones (LJ) parameters54 together with the GROMOS parameter for angles and dihedrals.55 The optimized hydrocarbon LJ parameters give better agreement with the experimental membrane density in the liquid-crystalline phase than standard parameters. For lipids, we used the fractional charges derived by Chiu et al.,56 and for the dye molecule, we used the charges generated by PRODRG using the “full charges” option. Additional information is given in the Supporting Information. The simulations used the particle-mesh Ewald method57 (PME) for the electrostatics with a Coulomb cutoff of 1 nm, a spacing of 0.12 nm for the Fourier transform grid, and a fourthorder interpolation. The van der Waals cutoff was 1.4 nm. The

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Figure 3. Root-mean-square deviation (RMSD) for lipid (left) and dye (right) molecules. Blue, first simulation; red, second simulation. Only the part of the trajectories for which the system is equilibrated (i.e., after the dashed line at 3 ns) was used for further analysis. The last geometry of the MD preproduction phase was used as the reference structure for the RMSD calculations. The initial rise in the RMSD is observed because the production simulations were started with new velocities.

integration time step was 2 fs, with the neighbor list being updated every 10th step by using the grid option and a cutoff distance of 1 nm. The default dielectric constant of 1 was used during the calculations. The water molecules were modeled as simple point charges (SPC).58 Full periodic boundary conditions were applied, so that an infinite multilamellar system was generated during the calculation. A constant number of particles in the systems, constant pressure, and constant temperature simulation conditions (NPT) were imposed. For this, the systems were coupled to an external constant temperature (325 K, τ ) 0.1 ps) and pressure bath (1 atm, τ ) 1.0 ps).59 Finally, all bond lengths were kept constant using the LINCS60 algorithm. Analysis of the MD Trajectories. The root-mean-square deviation (RMSD) for the production phase of the two systems was calculated for the 10 ns production simulations using the last geometry of the preproduction phase as the reference structure (Figure 3). The RMSD for the embedded dye molecules was also calculated to verify coarsely that the chromophore was stable in the membrane (Figure 3). We present a detailed analysis of the position and stability of the di-8ANEPPS molecule in the membrane below. Only the parts of the trajectories in which the system was found to be equilibrated, i.e., the last 7 ns production MD for each system, were used for analysis and for QM/MM CI calculations. The 700 snapshots used for the QM/MM CI calculations were taken as outlined above. Both trajectories were processed and analyzed within the GROMACS package using the available tools. The bilayer thickness, area per lipid head group, electrostatic potential shape across the bilayer, diffusion coefficients for DPPC and the dye, and the embedding depth of the chromophore in the membrane were calculated in order to validate the simulations and provide a detailed description of the results. Semiempirical CI Calculations. The 700 snapshots per system collected during the production phase were used for semiempirical CI single-point calculations, performed with the VAMP43,44 program package using the AM1 Hamiltonian.61 The QM part comprised the di-8-ANEPPS dye molecule and the MM part the rest of the system, i.e., the lipid and water molecules from one periodic box. For the QM calculations, explicit hydrogen atoms were added to the di-8-ANEPPS potential sensitive probe, using the PRODRG49 program. This yielded a system consisting of 94 QM atoms, surrounded by 14 661 MM atoms for the first system and 14 673 MM atoms for the second system. The MM environment was represented by a rigid point-charge environment.46,47 The polarization of

the QM wave function by the point charge in the MM environment was taken into account via an additional oneelectron term in the Fock matrix.45-47 No back-polarization was included. The MM charges were those used in the MD. After preliminary tests, a dielectric constant of 4.0 for interactions between the QM and MM parts was used as in our previous work on the tetracycline repressor.40 Using an active space of 18 electrons in 18 orbitals, symmetrically distributed around the HOMO-LUMO frontier, and including all single and pair double excitations in the CI43 led to results that agree well with experiment and that do not change significantly when the active space for the chromophore is increased. The wavelengths and oscillator strengths of electronic absorptions in the organic dye molecule were calculated, and the results were used in a sum-over-states62 treatment to compute the polarizabilities of the ground and excited states, Pg and Pe, and the second-order hyperpolarizabilities, β. The calculated values of the polarizabilities were obtained as Pg ) 70.8 × 10-30 m3 and Pe ) 71.2 × 10-30 m3. It has been shown62 that the sumover-states method suffers from severe resonances at excitation energies close to half the vertical absorption energies. To find the appropriate excitation energy in our case, we computed the frequency dependence of the hyperpolarizability for different conformations of the di-8-ANEPPS molecule for 15 snapshots taken from each of the two production phase MD trajectories. The results for only one of the trajectories are shown in Figure 4 for clarity, but the other is very similar. The positions of the resonances confirmed that we can use the default excitation energy of 1.17 eV (1060 nm). The two-photon absorption of di-4-ANEPPS in a model phospholipid membrane was observed to exhibit an energy of 1.29-1.26 eV (960-980 nm).10 This suggests a two-photon resonance2,10,63,64 that corresponds to the 465 nm one-photon-absorption maximum of di-4-ANEPPS in lipid environment.10 The structurally similar di-8-ANEPPS also has a one-photon-absorption transition of 465 nm when embedded in a lipid membrane and is therefore expected to exhibit two-photon absorption at the double energy around 1.28 eV. Nearly all experiments that use ANEP dyes for nonlinear optical imaging of the membrane potential are conducted at excitation energies in the range 1.72-1.27 eV (720-980 nm). Irradiation with visible light would damage the cells, since it is absorbed by the tissue. The femtosecond Ti:sapphire laser imposes the upper limit on the wavelength range. Nevertheless, we computed hyperpolarizability values for a 1.17 eV (1060 nm) excitation which are in excellent agreement with experimental values for the same65 or similar66 classes of compounds.

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Figure 5. Bilayer thickness. Red, first simulation; blue, second simulation.

Figure 4. Frequency dependence of the calculated hyperpolarizability for 15 different conformations of the embedded di-8-ANEPPS using the sum-over-states method.62 The vertical solid bar indicates the excitation energy (1.17 eV) used, which is lower than the energies at which resonance occurs. The two dotted vertical lines indicate the excitation energy interval (1.27-1.72 eV) used experimentally for studies with di-8-ANEPPS12,14 or di-4ANEPPS.6,10,14

Transmembrane Potential. The membrane normal in our simulations corresponded to the z-axis of the Cartesian coordinate system. The dye molecule was embedded in the lower leaflet of the phospholipid bilayer. Since our main goal was to simulate and analyze the dependence of the second-harmonicintensity signal (ISHG) on changes in the transmembrane potential (TMP), we applied an external electrical field across the simulation box in order to mimic the TMP. Three cases were considered: one where no external electrical field was present, one with the vector of the applied field parallel to the z-axis of the system, and one with an antiparallel alignment. It is wellknown that the TMP of a cardiomyocyte experiences a total change of about 0.13 V during an action potential (AP), with a resting potential of -0.09 V and a maximal depolarization phase value of +0.04 V. We chose the applied field intensity so that, for each of the two cases investigated in which a field was present, a field equivalent to a potential of 0.1 V was felt at the site of the dye molecule. This was accomplished by first computing the dipole moment of di-8-ANEPPS in a vacuum at an electrical field strength equivalent to a potential of 0.1 V. Subsequently, QM/MM CI calculations were performed for the whole system and the applied electrical field across the simulation box was varied until the induced dipole moment of the dye molecule corresponded to that found in vacuum. More details are given in the Supporting Information. With this procedure, we were able to compute the variation of ISHG for a total change in the TMP of 0.1 and 0.2 V. For these relatively small TMP changes, the ISHG and the TMP are linearly dependent. Thus, we can easily compare all our results with experimental results, which are mostly given as percentage ISHG changes for a 0.1 V TMP variation. Results and Discussion Validation of the Simulated Bilayer. Bilayer Thickness. Defining the bilayer thickness as the averaged P-P distance of the phosphorus atoms in the two bilayer leaflets gives values of 3.77 ( 0.03 and 3.79 ( 0.03 nm for the two simulations. These values are in very good agreement with the experimental value of 3.75 nm found by X-ray experiments.67 Our results

are also comparable with values obtained from MD simulations of DPPC bilayers, under similar conditions: 4.055 and 3.6 nm.50 The small variation occurs because of differences in the treatment of the electrostatics, the charges used, the water:DPPC molecule ratio, and the parameters used for the thermal coupling. The bilayer thickness for the two bilayers investigated in our study is plotted in Figure 5 as a function of simulation time. AVerage Area per Lipid Head Group. The average area occupied by each individual lipid (〈A〉) is frequently used to describe simulated lipid bilayers and compare simulation results to living matter. Experimental values for this quantity can be obtained from X-ray diffraction and nuclear magnetic resonance (NMR) experiments. A broad spectrum of 〈A〉 values has been found by these methods, mainly because of the uncertainty in determining the relevant amount of water in the lipid/water system or to different data interpretation, as Tieleman and Berendsen50 have shown. However, Nagle has argued that the average area per DPPC in a fully hydrated bilayer is 0.62 ( 0.02 nm2 at 323 K.68 The same author reported a value of 0.64 nm2 a few years later69 after introducing a new structural correction based on fluctuations of the bilayers. Another method to determine 〈A〉 experimentally was used by Clarke.70 He worked with lipid vesicles composed of a range of saturated and unsaturated phosphatidylcholines and used fluorescent dye molecules to stain the vesicles. 〈A〉 could be calculated by determining the fluorescence excitation ratio of embedded chromophores. For vesicles consisting of unsaturated lipids with tails of 15 carbon atoms, as in our system, 〈A〉 was found to be 0.63 nm2 at 313 K. A wide range of average areas per lipid head group has also been reported from computer simulations. Tieleman and Berendsen’s50 NPT simulations led to values of 0.60 and 0.63 nm2 for DPPC bilayers using the SPC water model and Lennard-Jones parameters different from those used in our simulations. Pandit et al.71 reported a value of 0.627 nm2 working under conditions similar to those in our study but with a lipid system consisting of 128 DPPC molecules. It has been generally shown that 〈A〉 depends strongly on the treatment of the electrostatics for a given system. Values of 0.645, 0.615, and 0.555 nm2 were obtained from three distinct simulations by Patra et al.72 on a system composed of 128 DPPC lipids and in which the electrostatic interactions were treated with the PME method and with cutoffs of 2.0 and 1.8 nm, respectively. Lindahl and Edholm35 performed very long simulations on three different systems, consisting of 64, 256, and 1024 DPPC lipids with additional water. They showed that the average area per lipid head group undergoes strong fluctuations and that its equilibrium value depends clearly on the system size. Thus, values of 0.611,

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Figure 6. Average area per lipid (〈A〉). Red, first simulation; blue, second simulation.

0.630, and 0.633 nm2 were reported for 〈A〉 for the small, medium, and large systems, respectively. We found the same average values of 0.618 ( 0.001 nm2 for 〈A〉 for the two systems investigated. The last 7 ns of the production phase MD were used for the analysis. Considering the various factors that may influence the value obtained for 〈A〉, such as the force field or MD parameters, simulation length, bilayer size, and water:lipid ratio, we conclude that our values fit well those reported by other authors and those found in experiments, indicating equilibrated systems. Figure 6 shows the computed area per lipid head group as a function of simulation time for each of the two systems studied in this work. Electrostatic Potential Shape across the Bilayer. We calculated the electrostatic potential for the membrane/water system along the bilayer normal (z) by integrating the charge density along z twice. The charge density was computed by dividing the whole box into slabs parallel to the xy-plane and counting the number of charges in each slab. The zero of the potential was chosen to be at the center of the simulation box (z0). Since the two leaflets of the bilayer are not equivalent in our study, with one of them containing a dye molecule instead of a lipid molecule, different shapes of the total potential were obtained for the two interfaces. However, by defining lipid groups for the two membrane leaflets, it is possible to compute the electrostatic potential for the pure lipid/water system. These results are shown in Figure 7 and can be compared with similar results of other authors and with experimental findings. The general behavior agrees well with previous simulations of DPPC bilayers.50,71-75 The lipid molecules contribute with a large positive potential, which is overcompensated by the contribution of the solvent. Consequently, a positive potential with respect to the water region is found in the bilayer. The total potential for the two systems was found to be -579 and -590 mV, respectively. These values were obtained for the membrane leaflet that did not contain an embedded dye molecule by averaging over the last 7 ns simulation time. Only the contributions of lipid and water molecules were considered. The positive potential in the bilayer interior indicates that permeation of anions across the membrane is easier than that of cations. This is consistent with experimental observations of permeation of hydrophobic ions through lipid bilayers.76 For comparison, the experimental values for the potential range from -200 to -575 mV for different phosphatidylcholine/water interfaces. We also investigated how the presence of the embedded dye molecule influences the electrostatic potential shape of the DPPC/water interface. The results obtained from the first MD are shown in Figure 8. The sulfonate head group and the naphthalene have a negative contribution to the total interface

Figure 7. Lipid/water electrostatic potential contribution across one membrane leaflet of the first system investigated. The center of the bilayer was defined as having zero potential. These values were obtained by averaging over 7 ns MD simulation time. The structural diagram above the graph is included for clarity.

Figure 8. Influence of presence of di-8-ANEPPS on electrostatic potential profile across the membrane. Red, di-8-ANEPPS; blue, pure lipid/water system; green, lipid/water/di-8-ANEPPS. The center of the bilayer was defined as having zero potential. These values were obtained by averaging over 7 ns MD simulation time. The structural diagram above the graph is included for better orientation.

electrostatic profile, whereas the pyridine exhibits a positive contribution. Mass-Density Profiles. The mass-density profiles along the bilayer normal were calculated for both MD trajectories. There is a good correspondence between mass profiles and the contribution of the different contributors to the total electrostatic potential. The results also confirm the spatial location and alignment of the embedded dye molecule during the simulation. The dye is expected to float in the membrane with the molecular axis orientated orthogonal to the membrane surface9,10 and to exhibit a linear molecular geometry with the polar sulfonate head in the polar solvent environment and the alkyl tails in the bilayer interior.9,10 The mass-density profiles for water, lipid

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Figure 9. Average mass-density profiles for lipids (green), water (blue), and di-8-ANEPPS (red). These results were obtained for the second MD by averaging over the last 7 ns of the trajectory. The center of the bilayer occurs at 5 nm. The vertical solid and dotted lines indicate the location of the maximum mass density for the sulfur atom and amine nitrogen atom of the dye molecule, respectively. Figure 11. Trajectories of the centers of mass for di-8-ANEPPS (red) and four neighboring lipid molecules (green) during the last 7 ns of the production run of the first system.

TABLE 1: Lateral Diffusion Coefficients (cm2 s-1) of Dye and Lipid Molecules lipids (LR) di-8-ANEPPS

Figure 10. Positions of selected atoms in the bilayer. The z-coordinates of some polar atoms (N, P, and O) of the polar head of the lipid molecules are plotted over time (green). Both membrane leaflets were considered. The coordinates of the nitrogen atoms and the sulfur atom of the dye molecule are also plotted over time (red) and indicated by the red arrows. The structural diagram on the right is shown for clarity.

and dye molecules are shown in Figure 9. For water and lipids, the results agree well with other published theoretical studies.72,77 Embedding Depth of the Chromophore in the Membrane. Our goal is to compute the percent change in the intensity of the SHG as a function of membrane-potential variations. The dye must therefore be sensitive to these variations. The sensitivity is highest when the molecular axis of the dye is oriented orthogonal to the membrane surface. On the other hand, because of the structural resemblance between the amphiphilic dye and lipid molecules, the dye is found by experiment to float in the membrane and to exhibit a linear molecular geometry with the polar sulfonate head anchored in the polar solvent environment and the alkyl tails in the bilayer interior.7,9,17 To investigate our simulated system for these conditions, we plotted the z-coordinates of some relevant DPPC and dye atoms versus time. The resulting plot is shown in Figure 10. The analysis suggests that the dye molecule adopts a linear geometry and is aligned parallel to the membrane normal throughout the simulation. Moreover, the dye molecule undergoes small smooth movements along the z-axis of the simulation box. However,

simulation 1

simulation 2

experiment25,27-33,78

5.5 × 10 4.9 × 10-5

5.3 × 10 5.2 × 10-5

10-7-10-9

-5

-5

the polar head is always situated between the polar heads of the lipid molecules, in a region where the computed water and lipid mass densities are equal, as can be seen from the massdensity-profile plots. The chromophore and the lipophilic tails are located in the membrane interior. The distance between the z-coordinates of the two nitrogen atoms of the dye molecule during the whole analysis corresponds to an orthogonal orientation of the molecule with respect to the membrane. The embedded potential-sensitive probe is thus best oriented in the membrane in order to monitor membrane potential changes. Diffusion Coefficients for DPPC and Dye. Lateral diffusion constants D of DPPC and dye molecules were calculated from mean-square displacements in two dimensions (x,y) of the centers of mass of the molecules. Table 1 shows the values of the lateral diffusion constants of DPPC and dye molecules computed for the two trajectories, together with the range of experimental values determined by pulsed NMR25,27,78 and fluorescence-recovery-after-photobleaching28-30,32,33 experiments. Experimentally, diffusion coefficients are often determined at temperatures close to the phase-transition temperature, 315 K, whereas our values are computed for a system at 325 K. Experimental results31 indicate that the diffusion constant approximately doubles every 25 K in this region. As can be seen from the experimental values, the reliability of the calculated diffusion coefficients is questionable; most of the reported experimental results are 2 orders of magnitude smaller than the MD results. This is also the case for other theoretical studies.55 The dye and lipid molecules have similar diffusion coefficients. The similarities in the kinetic behavior of these two types of molecules can also be observed by comparing the trajectories of the centers of mass of the molecules along the trajectory. Figure 11 shows the positions of the centers of mass of the dye and four neighboring lipid molecules projected onto the xy-plane. CI QM/MM Calculations: Hyperpolarizability. Nonlinear optical experimental techniques for monitoring AP in living tissues are novel1-4,9,23,79,80 and still under development.2,5,10,12,81-86 Several authors have reported applications in the past few

2452 J. Phys. Chem. B, Vol. 112, No. 8, 2008

Rusu et al.

Figure 12. Histogram of percentage change in the intensity of the SHG signal as a function of a 0.1 V membrane-potential variation. Red, first simulation; blue, second simulation.

Figure 13. Histogram of percentage change in the intensity of the SHG signal as a function of a 0.2 V membrane-potential change. Red, first simulation; blue, second simulation.

years.9,12,23,82,83,85,86 As the design of potential-sensitive probes,1,5,6,9,84-86 the optoelectronic equipment,81,86 and the know-how improve, research activities have been increasingly focused on these techniques, so that experimental results are constantly improving.10,12 The latest experiments suggest values of about 18% change6 in the intensity of the SHG signal, or more,10,12 for TMP variations of 0.1 V, when working with ANEP probes. Our calculations give an average percentage ∆ISHG of 20.0 ( 5.1 and 20.3 ( 6.5 for a TMP change of 0.1 V, which was chosen to simulate the living matter physiological parameters as closely as possible. These results were obtained for the case where the electrical field vector points from the exterior toward the interior of the cell and are plotted in Figure 12. The membrane leaflet containing the embedded dye molecule is considered to face the extracellular space. Details are available in the Supporting Information. The percentage ∆ISHG for a TMP variation of 0.2 V was also computed and the results are plotted in Figure 13. According to eqs 2, 4, and 5, ∆ISHG should scale linearly with ∆TMP for small TMP variations. Indeed, we calculated an average percentual ∆ISHG of 41.6 ( 11.1 and 43.2 ( 13.0 for the two systems investigated, respectively, when the TMP changed by a value of 0.2 V. These values are about twice as large as those obtained for a TMP change of only 0.1 V. We also investigated the change in dipole moment between the excited and ground states (∆µg,e, eq 2), since it correlates with the ISHG of the dye molecules. An analysis of 270 frames of the trajectory of the first system for the case in which the electrical field vector points from the exterior toward the interior

of the cell yielded an average ∆µg,e of 11.4 ( 1.7 D with an average ground-state dipole moment (µg) of 36.7 ( 2.7 D and an average excited-state dipole moment (µe) of 48.3 ( 2.6 D. The large value for the dipole moment difference between the excited and ground states obtained for di-8-ANEPPS is characteristic for this type of specially engineered chromophores, but is somewhat smaller than that estimated by Loew9 for this class of dyes. The ground- and excited-state dipole moments in our calculations are generally larger than those expected from theoretical studies in the gas phase, as expected for “condensedphase” systems. We also performed gas-phase QM CI computations for some snapshots taken from the first trajectory. This allowed us to estimate the influence of the presence of the membrane on the calculated spectroscopic data. The comparison of the groundstate dipole moments for seven geometries investigated with this method is plotted in Figure 14. As Figure 14 shows, the membrane influence on the ground-state dipole moment of different conformations of di-8-ANEPPS is small. The computed ground-state dipole moment is slightly larger for the gas-phase computations than for the one conducted in the MM environment. The change in the dipole moment upon excitation to the first excited state has also been computed for the same snapshots as above, and the results are plotted in Figure 15. We found that the change in the dipole moment is enhanced when the dye molecule is embedded in the membrane. According to eqs 2 and 5, this would also enhance ISHG.

Biological Membrane-Potential Changes

Figure 14. Ground-state dipole moments for snapshots taken at seven different simulation times for a simulation in which no external electrical field was applied across the membrane. Blue, dye in the membrane; red, dye in the gas phase.

J. Phys. Chem. B, Vol. 112, No. 8, 2008 2453 membrane surface throughout the simulation. This implies that the embedded dye exhibits structural dynamics similar to that of the surrounding lipid molecules. The computed hyperpolarizability values of the dye molecule for different simulated TMPs in the physiological range agree with experimental results. Our computations confirm the spectroscopically observed ability of hemicyanines to exhibit sufficiently large nonlinearities for good experimental resolution. The change in the intensity of the SHG signal as a function of the TMP determines the absolute potential sensitivity of the hemicyanine dyes and is the ultimate target quantity. Our computations suggest values for this quantity that support and confirm the most recent experimental results. The laser excitation energy plays an important role in calculations of this parameter. Even though the excitation energy used in this study is outside the range used experimentally (because of resonance), we expect our results to mimic the experimental ones obtained for an excitation wavelength of about 860 nm, which indeed is the case. Furthermore, the experimental tendency to obtain larger values of the ISHG for the polarized state of heart muscle cells (resting potential) is also confirmed by our calculations. Acknowledgment. Financial support for this research was provided by two stipends from the Deutsche Forschungsgemeinschaft (GRK 312 and 1161). Supporting Information Available: Details of the following aspects of this work: (1) preparation of the bilayer/dye systems; (2) MD simulations; (3) semiempirical CI calculations; simulating the action potential (AP) of cardiomyocytes; (4) computing the intensity and the change in intensity of the second-harmonicgeneration signal (ISHG/∆ISHG); (5) example of input file where a positive electrical field is used to simulate the TMP; (6) part of the ITP file containing the charges used in the MD for the dye molecule; (7) references. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes

Figure 15. Change in dipole moment upon excitation to the first excited state for snapshots taken at seven different simulation times for a simulation in which no external electrical field was applied across the membrane. Blue, dye in the membrane; red, dye in the gas phase.

Summary and Conclusions Recently, optical second-harmonic generation and its sensitivity to an externally applied electric field were shown experimentally to constitute a promising tool for probing fast physiological processes in living matter. Our theoretical results presented in this paper are in good agreement with the experiments conducted so far. Thus, the computational procedures performed here provide a complementary tool for optimizing the experimental components and conditions. The technique exploits the link between the electrophysiological and optical processes using specificially engineered potential-sensitive dye probes embedded in the cell membrane. The orientation of these molecular probes within the membrane is critical for SHG and for its sensitivity to TMP. Ideally, dye probes should be aligned parallel to the membrane surface normal, with the polar head anchoring the dye in the polar water phase and the lipophilic tails fixing it between the phospholipid molecules. Starting from an initial configuration with one dye molecule embedded in nearly ideal orientation, our 20 ns MD simulations show that the dye molecule accommodates and maintains its alignment normal to the

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