Highly Cited Papers

Our group was founded in 1988 by Klaus Schulten. His publications have been cited over 25,000 times as of August 2009. The most ( ~100 times or more) highly cited publications are listed below with descriptions of the associated scientific discoveries or technology developments. The list will be extended as citation statistics are updated.

VMD - Visual Molecular Dynamics. William Humphrey, Andrew Dalke, and Klaus Schulten. Journal of Molecular Graphics, 14:33-38, 1996.
(Citations: 3942)

VMD is a molecular graphics program designed for the display and analysis of molecular assemblies, in particular biopolymers such as proteins and nucleic acids. VMD can simultaneously display any number of structures using a wide variety of rendering styles and coloring methods. Molecules are displayed as one or more `representations', where each representation embodies a particular rendering method and coloring scheme for a selected subset of atoms. The atoms displayed in each representation are chosen using an extensive atom selection syntax, which includes boolean operators and regular expressions. VMD provides a complete graphical user interface for program control, as well as a text interface using the Tcl embeddable parser to allow for complex scripts with variable substitution, control loops, and function calls. Full session logging is supported, which produces a VMD command script for later playback. High-resolution raster images of displayed molecules may be produced by generating input scripts for use by a number of photorealistic image processing applications. VMD has also been expressly designed with the ability to animate molecular dynamics (MD) simulation trajectories, imported either from files or from a direct connection to a running MD simulation. VMD is the visualization component of MDScope, a set of tools for interactive problem solving in structural biology, which also includes the parallel MD program NAMD, and the MDCOMM software used to connect the visualization and simulation programs. VMD is written in C++, using an object-oriented design; the program, including source code and extensive documentation, is freely available via anonymous ftp and through the World Wide Web.



NAMD2: Greater scalability for parallel molecular dynamics. Laxmikant Kalé, Robert Skeel, Milind Bhandarkar, Robert Brunner, Attila Gursoy, Neal Krawetz, James Phillips, Aritomo Shinozaki, Krishnan Varadarajan, and Klaus Schulten. Journal of Computational Physics, 151:283-312, 1999.
(Citations: 821)

Molecular dynamics programs simulate the behavior of biomolecular systems, leading to insights and understanding of their functions. However, the computational complexity of such simulations is enormous. Parallel machines provide the potential to meet this computational challenge. To harness this potential, it is necessary to develop a scalable program. It is also necessary that the program be easily modified by application-domain programmers. The NAMD2 program presented in this paper seeks to provide these desirable features. It uses spatial decomposition combined with force decomposition to enhance scalability. It uses intelligent periodic load balancing, so as to maximally utilize the available compute power. It is modularly organized, and implemented using a parallel C++ dialect, so as to enhance its modifiability. It uses a combination of numerical techniques and algorithms to ensure that energy drifts are minimized, ensuring accuracy in long running calculations. NAMD2 uses a portable run-time framework that also supports interoperability among multiple parallel paradigms. As a result, different components of applications can be written in the most appropriate parallel paradigms. NAMD2 runs on most parallel machines including workstation clusters. This paper also describes the performance obtained on some benchmark applications.



Scalable molecular dynamics with NAMD. James C. Phillips, Rosemary Braun, Wei Wang, James Gumbart, Emad Tajkhorshid, Elizabeth Villa, Christophe Chipot, Robert D. Skeel, Laxmikant Kale, and Klaus Schulten. Journal of Computational Chemistry, 26:1781-1802, 2005.
(Citations: 825)

NAMD is a parallel molecular dynamics code designed for high-performance simulation of large biomolecular systems. NAMD scales to hundreds of processors on high-end parallel platforms, as well as tens of processors on low-cost commodity clusters, and also runs on individual desktop and laptop computers. NAMD works with AMBER and CHARMM potential functions, parameters, and file formats. This article, directed to novices as well as experts, first introduces concepts and methods used in the NAMD program, describing the classical molecular dynamics force field, equations of motion, and integration methods along with the efficient electrostatics evaluation algorithms employed and temperature and pressure controls used. Features for steering the simulation across barriers and for calculating both alchemical and conformational free energy differences are presented. The motivations for and a roadmap to the internal design of NAMD, implemented in C++ and based on Charm++ parallel objects, are outlined. The factors affecting the serial and parallel performance of a simulation are discussed. Finally, typical NAMD use is illustrated with representative applications to a small, a medium, and a large biomolecular system, highlighting particular features of NAMD, for example, the Tcl scripting language. The article also provides a list of the key features of NAMD and discusses the benefits of combining NAMD with the molecular graphics/sequence analysis software VMD and the grid computing/collaboratory software BioCoRE. NAMD is distributed free of charge with source code at www.ks.uiuc.edu.



Textbook: Neural Computation and Self-Organizing Maps: An Introduction. Helge Ritter, Thomas Martinetz, and Klaus Schulten. Addison-Wesley, New York, revised English edition, 1992.
(Citations: 669)

This book is a comprehensive introduction to neural networks and neural information processing. It describes the most important models of neural networks and how they contribute to our understanding of information and organization processes in the brain. One of the few generally recognized organizational principles of the nervous system, the development of cortical feature maps (brain maps), is described in detail, and the reader is introduced to the biological background and the mathematical properties of self-organizing maps as important functional building blocks of the brain. Examples show how neural networks can solve important information processing tasks, including the development of sensory maps, the traveling salesman problem, and visuomotor control of robots.



The crystal structure of the light harvesting complex II (B800-850) from Rhodospirillum molischianum. Juergen Koepke, Xiche Hu, Cornelia Muenke, Klaus Schulten, and Hartmut Michel. Structure, 4:581-597, 1996.
(Citations: 595)

In photosynthesis light is absorbed by light-harvesting antenna complexes (LHs) and its energy is transferred to the photosynthetic reaction center. In purple photosynthetic bacteria and higher plants the LHs are integral membrane protein/pigment complexes. LH-II from the purple bacterium Rhodospirillum molischianum is an octamer of heterodimers, the later consisting of two polypeptides, the $\alpha$ and $\beta$-apoproteins, noncovalently binding three bacteriochlorophyll-a (BChl-a) molecules and at least one lycopene molecule as an additional chromophore. LH-II absorbs light and converts it into a BChl-a exciton, which is then transferred to the photosynthetic reaction center through the core light harvesting complex LH-I. The crystal structure of LH-II from Rhodospirillum molischianum has been determined by molecular replacement at 2.4 Åresolution using X-ray diffraction. The search model for molecular replacement was a computationally modelled octamer of $\alpha\beta$ heterodimer of a nonameric LH-II from Rps. acidophila. The crystal structure displays two concentric cylinders of membrane-spanning helical protein subunits with the $\alpha$-apoprotein at the inner and the $\beta$-apoprotein at the outer side. Sixteen BChl-a molecules absorbing maximally at 846 nm (B850), oriented perpendicular to the plane of the membrane and sandwiched between the helical apoproteins, form a ring of radius 23.0 Å. The other eight BChl-a molecules absorbing maximally at 800nm (B800) situated between the $\beta$-apoproteins and bound through their central Mg atoms to an aspartate ($\alpha$-Asp6), form a concentric ring of radius 28.8 Å. Eight membrane spanning lycopene pigments, held in place through aromatic side groups, stretch out between the B800 and B850 BChl-a's. The light-harvesting complexes from different bacteria assume various ring sizes. In LH-II of Rs. molischianum, the $Q_{y}$ transition dipole moments of neighboring B850 and B800 BChl-a's are nearly parallel to each other, i.e., are optimally aligned for Forster exciton transfer; Dexter energy transfer is possible through B850 BChl-a's are in van der Waals distance to a lycopene, such that singlet and triplet energy transfer between lycopene and the BChl-a's is optimal for light energy transfer in that it samples all spatial absorption and emission characteristics as well as places all oscillator strength into energetically low lying, thermally accessible exciton states.



Linear polyene electronic structure and potential surfaces. Bruce S. Hudson, Bryan E. Kohler, and Klaus Schulten. In Edward C. Lim, editor, Excited States, volume 6, pp. 1-95. Academic Press, New York, 1982.
(Citations: 533)

Polyenes are linear conjugated chains of carbon atoms joined by alternating double and single bonds and are deservedly the objects of a good deal of experimental and theoretical attention. There are many reasons for this, including the historical importance of these systems in the development of molecular quantum theory, the fundamental importance of cis-trans photoisomerization, which is the distinctive photochemistry of these molecules, and the fact that polyene chromophores play starring roles in biologically important photoprocesses, such as vision and energy production in the purple-membrane Halobacterium halobium. In both of these biological examples, the electronic structure of teh polyene and how it changes upon excitation is of key importance. Until relatively recently, this structure was thought to be rather simple, similar to that of other conjugated systems such as the polyacenes, and well described by approximate molecular orbital ideas. Recent experiments and theoretical discoveries have revealed that this is not the case: Polyene electronic structure is both more complicated and more interesting than was previously thought.



"Neural gas" for vector quantization and its application to time-series prediction. Thomas M. Martinetz, Stanislav G. Berkovich, and Klaus Schulten. IEEE Transactions on Neural Networks, 4:558-569, 1993.
(Citations: 443)

As a data compression technique, vector quantization requires the minimalization of a cost function - the distortion error - which, in general, has many local minima. In this paper, a neural network algorithm based on a "soft-max" adaptation rule is presented that exhibits good performance in reaching the optimum, or at least coming close. The soft-max rule employed is an extension of the standard K-means clustering procedure and takes into account a "neighborhood ranking" of the reference (weight) vectors. It is shown that the dynamics of the reference (weight) vectors during the input-driven adaptation procedure 1) is determined by the gradient of an energy function whose shape can be modulated through a neighborhood determining parameter, and 2) resembles the dynamics of Brownian particles moving in a potential determined by the data point density. The network is employed to represent the attractor of the Mackey-Glass equation and to predict the Mackey-Glass time series, with additional local linear mappings for generating output values. The results obtained for the time-series prediction compare very favorably with the results achieved by back-propagation and radial basis function networks.



On the origin of a low-lying forbidden transition in polyenes and related molecules. Klaus Schulten and Martin Karplus. Chemical Physics Letters, 14:305-309, 1972.
(Citations: 376)

It is demonstrated that the inclusion of double-excited configurations in semi-empirical and a priori calculations of polyenes leads to a significant alteration of the spectrum. In agreement with the recent experiment of Hudson and Kohler, a forbidden ($^{1}A_{g}$) state appears below the strongly allowed ($^{1}B_{u}$) state.



Correlation effects in the spectra of polyenes. Klaus Schulten, I. Ohmine, and Martin Karplus. Journal of Chemical Physics, 64:4422-4441, 1976.
(Citations: 356)

A Hamiltonian of the Pariser-Parr-Pople form is employed to investigate the effect of correlation on the $\pi$-electron spectrum of polyenes. Two limiting cases for the electron-electron interaction (short- and long-range limit) are considered, and it is shown that they yield descriptions corresponding to the standard valence-bond (Dirac-Heisenberg) and molecular-orbital models, respectively. The intermediate, chemically most interesting, range is examined in detail by means of a full configuration-interaction treatment with an exponential model potential that includes a variable effective range parameter. It is shown that correlation effects become more important as the effective range of the interaction decreases. The states of polyenes are classified as covalent or noncovalent, and it is found that the former are much more sensitive to correlation than the latter. Configuration interaction through double excitations yields a qualitatively correct ordering for all states in the chemical range, but triple and quadruple excitations are required for quantitative results. Applications to butadiene, hexatriene, and benzene demonstrate that correlation effects in these molecules lead to an important lowering in energy of the manifold of covalent states relative to that of the noncovalent states; most important, the first covalent ($^{1}A^{-}_{g}$) state of the polyenes is found to be near degenerate with the strongly allowed noncovalent ($^{1}B^{+}_{u}$) state. Density correlation functions and the fluctuation potential are obtained for the polyenes and used to clarify the nature of the correlation correction. Configuration interaction including double excitations is performed for polyenes through $C_{12}H_{14}$ to exhibit the length dependence of the correlation effects. It is shown that with increasing chain length, an increasing number of covalent states appears in the energy range of the two usually observed excited $^{1}B^{+}_{u}$ and $^{1}A^{-}_{g}$ (cis peak) states.



First passage time approach to diffusion controlled reactions. Attila Szabo, Klaus Schulten, and Zan Schulten. Journal of Chemical Physics, 72:4350-4357, 1980.
(Citations: 359)

Association reactions involving diffusion in one, two, and three-dimensional finite domains governed by Smoluchowski-type equations (e.g., interchain reaction of macromolecules, ligand binding to receptors, repressor-operator association of DNA strand) are shown to be often well described by first-order kinetics and characterized by an average reaction (passage) time $\tau$. An inhomogeneous differential equation is derived which,for problems with high symmetry, yields $\tau$ by simple quadrature without taking recourse to detailed cumbersome time-dependent solutions of the original Smoluchowski equation. The cases of diffusion and nondiffusion controlled processes are included in the treatment. For reaction processes involving free diffusion and intramolecular chain motion, the validity of the passage time approximation is analyzed.



Molecular biomimetics: nanotechnology through biology. Mehmet Sarikaya, Candan Tamerler, Alex K. -Y. Jen, Klaus Schulten, and François Baneyx. Nature Materials, 2:577-585, 2003.
(Citations: 362)

Proteins, through their unique and specific interactions with other macromolecules and inorganics, control structures and functions of all biological hard and soft tissues in organisms. Molecular biomimetics is an emerging field in which hybrid technologies are developed by using the tools of molecular biology and nanotechnology. Taking lessons from biology, polypeptides can now be genetically engineered to specifically bind to selected inorganic compounds for applications in nano- and biotechnology. This review discusses combinatorial biological protocols, that is, bacterial cell surface and phage-display technologies, in the selection of short sequences that have affinity to (noble) metals, semiconducting oxides and other technological compounds. These genetically engineered proteins for inorganics (GEPIs) can be used in the assembly of functional nanostructures. Based on the three fundamental principles of molecular recognition, self-assembly and DNA manipulation, we highlight successful uses of GEPI in nanotechnology.



Mechanical unfolding intermediates in titin modules. Piotr E. Marszalek, Hui Lu, Hongbin Li, Mariano Carrion-Vazquez, Andres F. Oberhauser, Klaus Schulten, and Julio M. Fernandez. Nature, 402:100-103, 1999.
(Citations: 345)

The modular protein titin, which is responsible for the passive elasticity of muscle, is subjected to stretching forces. Previous work on the experimental elongation of single titin molecules has suggested that force causes consecutive unfolding of each domain in an all-or-none fashion$^{1-6}$. To avoid problems associated with the heterogeneity of the modular, naturally occurring titin, we engineered single proteins to have multiple copies of single immunoglobulin domains of human cardiac titin$^{7}$ Here we report the elongation of these molecules using the atomic force microscope. We find an abrupt extension of each domain by $\sim 7\AA$ before the first unfolding event. This fast initial extension before a full unfolding event produces a reversible `unfolding intermediate'. Steered molecular dynamics $^{8,9}$ simulations show that the rupture of a pair of hydrogen bonds near the amino terminus of the protein domain causes an extension of about $6 \AA$, which is in good agreement with our observations. Disruption of these hydrogen bonds by site-directed mutagenesis eliminates the unfolding intermediate. The unfolding intermediate extends titin domains by $\sim 15$ percent of their slack length, and is therefore likely to be an important previously unrecognized component of titin elasticity.



Molecular dynamics simulation of a bilayer of 200 lipids in the gel and in the liquid crystal-phases. Helmut Heller, Michael Schaefer, and Klaus Schulten. Journal of Physical Chemistry, 97:8343-8360, 1993.
(Citations: 303)

We have constructed and simulated a membrane-water system which consists of 200 molecules of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine forming a rectangular patch of a bilayer and of 5483 water molecules covering the head groups on each side of the bilayer. The total number of atoms is approximately 27 000. The lateral dimensions of the bilayer are 85 A $\times$ 100 A, and the distance between the bilayer surfaces as given by the average phosphorus to phosphorus distance is 35 A. The thickness of each water layer is up to 15 A. In all, we simulated 263 ps of the dynamics of the system. To prevent system disintegration, atoms within 5 A from the surface were harmonically restrained and treated by Langevin dynamics, forming a stochastic boundary. Interior lipids and water molecules were unrestrained. The first 120 ps of the dynamics calculation were used to equilibrate the system and to achieve a low internal pressure. We performed two simulations for analysis: simulation I of the system that resulted from the equilibration: simulation II of the system after an increase of the area per head group from 46 to 70 $A^2$. The decrease of the lateral lipid density was achieved by scaling the atomic x-, y-, and z-coordinates independently, leaving the volume of the system constant. For both simulations, I and II, we determined the internal pressure, the lipid self-diffusion coefficients, the order parameter profile, the distribution of molecular groups, and other properties. The parameters extracted from simulation II are in good agreement with observations on bilayers in the liquid-crystal phase. We provide evidence that the bilayer of simulation I corresponds to the gel phase. The membrane structures resulting from this work can be used for molecular dynamics investigations of membrane proteins, e.g., for the study of lipid-protein interactions or for the equilibration of structural models.



Control of the selectivity of the aquaporin water channel family by global orientational tuning. Emad Tajkhorshid, Peter Nollert, Morten Ø. Jensen, Larry J. W. Miercke, Joseph O'Connell, Robert M. Stroud, and Klaus Schulten. Science, 296:525-530, 2002.
(Citations: 293)

Aquaporins are transmembrane channels found in cell membranes of all life forms. We examine their apparently paradoxical property, facilitation of efficient permeation of water while excluding protons, which is of critical importance to preserving the electrochemical potential across the cell membrane. We have determined the structure of the Escherichia coli aquaglyceroporin GlpF with bound water, in native (2.7 angstroms) and in W48F/F200T mutant (2.1 angstroms) forms, and carried out 12-nanosecond molecular dynamics simulations that define the spatial and temporal probability distribution and orientation of a single file of seven to nine water molecules inside the channel. Two conserved asparagines force a central water molecule to serve strictly as a hydrogen bond donor to its neighboring water molecules. Assisted by the electrostatic potential generated by two half-membrane spanning loops, this dictates opposite orientations of water molecules in the two halves of the channel, and thus prevents the formation of a "proton wire," while permitting rapid water diffusion. Both simulations and observations revealed a more regular distribution of channel water and an increased water permeability for the W48F/F200T mutant.



Electronic excitations in finite and infinite polyenes. Paul Tavan and Klaus Schulten. Physical Review B, 36:4337-4358, 1987.
(Citations: 282)

We study electronic excitations in long polyenes, i.e., in one-dimensional strongly correlated electron systems which are neither infinite nor small. The excitations are described within Hubbard and Pariser-Parr-Pople (PPP) models by means of a multiple-reference double-excitation expansion [P. Tavan and K. Schulten, J. Chem. Phys. 85, 6602 (1986)]. We find that quantized "transition" momenta can be assigned to electronic excitations in finite chains. These momenta link excitation energies of finite chains to dispersion relations of infinite chains, i.e., they bridge the gap between finite and infinite systems. A key result is the following: Excitation energies E in polyenes with N carbon atoms are described very accurately by the formula $E^\beta=\Delta E^\beta_0 + \alpha^\beta k(N)q, q = 1,2,...$, where $\beta$ denotes the excitation class, $\Delta^\beta_0$ the energy gap in the infinite system [ $\alpha^\beta k(N)>0$], and k(N) the elementary transition momentum. The parameters $\Delta E^\beta_0$ and $\alpha^\beta$ are determined for covalent and ionic excitations in alternating and nonalternating polyenes. The covalent excitations are combinations of triplet excitations T, i.e., T, TT, TTT, ... . The lowest singlet excitations in the infinite polyene, e.g., in polyacetylene or polydiacetylene, are TT states. Available evidence proves that these states can dissociate into separate triplets. The bond structure of TT states is that of a neutral soliton-antisoliton pair. The level density of TT states in long polyenes is high enough to allow dissociation into separate solitons.



Magnetic field dependence of the geminate recombination of radical ion pairs in polar solvents. Klaus Schulten, H. Staerk, Albert Weller, Hans-Joachim Werner, and B. Nickel. Zeitschrift für Physikalische Chemie, NF101:371-390, 1976.
(Citations: 274)

Pairs of radical ions are generated in polar solvents by nanosecond laser flashes in a singlet electron spin state via photoinduced electron transfer. The recombination monitored spectroscopically with a time resolution $\sim$ 3 ns can be resolved into a fast geminate ($\sim$10 ns) and a slow homogeneous ( $\sim$1000 ns) process. It has been observed for the system pyrene + 3,5 dimethoxy-dimethyl-aniline in methanol that triplet products appear already during the geminate phase of the recombination. The yield of these fast triplet products is reduced by an external magnetic field of 500 Gauss to about 80% of its zero field value. The magnetic field dependence of this effect in the range 0-500 Gauss has been measured under stationary conditions. The results are found to be in agreement with a theoretical model based on the assumption that the change of spin multiplicity of the initial radical pairs originates from the hyperfine coupling between unpaired electron spins and nuclear spins within each radical.



Unfolding of titin immunoglobulin domains by steered molecular dynamics simulation. Hui Lu, Barry Isralewitz, André Krammer, Viola Vogel, and Klaus Schulten. Biophysical Journal, 75:662-671, 1998.
(Citations: 278)

Titin, a 1 $\mu$m long protein found in striated muscle myofibrils, possesses unique elastic and extensibility properties in its I-band region, which is largely composed of 7-strand $\beta$-sandwich immunoglobulin-like (Ig) domains. The behavior of titin as a hysteretical, multi-stage entropic spring has been shown in atomic force microscope and optical tweezer experiments to depend on the reversible unfolding of individual Ig domains. We performed steered molecular dynamics simulations to stretch single titin Ig domains in solution with pulling speeds of 0.5 and 1.0 $\AA$/ps. Resulting force-extension profiles exhibit a single dominant peak for each Ig domain unfolding, consistent with the experimentally observed sequential, as opposed to concerted, unfolding of Ig domains. This force peak can be attributed to an initial burst of backbone hydrogen bonds, which takes place between $\beta$-strands A and B in between $\beta$-strands A' and G. Additional features of the simulations, including the position of the force peak and relative unfolding resistance of different Ig domains, can be related to experimental observations.



Topology representing networks. Thomas Martinetz and Klaus Schulten. Neural Networks, 7:507-522, 1994.
(Citations: 238)

A Hebbian adaptation rule with winner-take-all like competition is introduced. It is shown that this competitive Hebbian rule forms so-called Delaunay triangulations, which play an important role in computational geometry for efficiently solving proximity problems. Given a set of neural units i, i = 1, ..., N, the synaptic weights of which can be interpreted as pointers $w_{i}$, i = 1, ..., N in $\Re$$_{D}$, the competitive Hebbian rule leads to a connectivity structure between the units i that corresponds to the Delaunay triangulation of the set of pointers w$_{i}$. Such competitive Hebbian rule develops connections ($C_{ij}$ > 0) between neural units i, j with neighboring receptive fields (Voronoi polygons) $V_{i} $,$V_{j}$, whereas between all other units i, j no connections evolve ($C_{ij}$ = 0). Combined with a procedure that distributes the pointers $w_{i}$ over a given feature manifold M, for example, a submanifold M $\subset$ $\Re$$_{D}$, the competitive Hebbian rule provides a novel approach to the problem of constructing topology preserving feature maps and representing intricately structured manifolds. The competitive Hebbian rule connects only neural units, the receptive fields (Voronoi polygons) $V_{i} $, $V_{j}$ of which are adjacent on the given manifold M. This leads to a connectivity structure that defines a perfectly topology preserving map and forms a discrete, path preserving representation of M, also in cases where M has an intricate topology. This makes this novel approach particularly useful in all applications where neighborhood relations have to be exploited or the shape and topology of submanifolds have to be taken into account.



Steered molecular dynamics and mechanical functions of proteins. Barry Isralewitz, Mu Gao, and Klaus Schulten. Current Opinion in Structural Biology, 11:224-230, 2001.
(Citations: 218)

Atomic force microscopy of single molecules, steered molecular dynamics, and the theory of stochastic processes have established a new field that investigates mechanical functions of proteins such as ligand - receptor binding/unbinding and elasticity of muscle proteins during stretching. The combination of these methods yields information on the energy landscape that controls mechanical function and on the force bearing components of proteins, as well as on the underlying physical mechanisms.



The low-lying electronic excitations in long polyenes: A PPP-MRD-CI study. Paul Tavan and Klaus Schulten. Journal of Chemical Physics, 85:6602-6609, 1986.
(Citations: 206)

A correct description of the electronic excitations in polyenes demands that electron correlation is accounted for correctly. Very large expansions are necessary including many electron configurations with at least one, two, three, and four electrons promoted from the Hartree-Fock ground state. The enormous size of such expansions had prohibited accurate computations of the spectra for polyenes with more than ten $\pi$ electrons. We present a multireference double excitation configuration interaction method (MRD-CI) which allows such computations for polyenes with up to $16\pi$ electrons. We employ a Pariser-Parr-Pople (PPP) model Hamiltonian. For short polyenes with up to ten electrons our calculations reproduce the excitation energies resulting from full-CI calculations. We extend our calculations to study the low-lying electronic excitations of the longer polyenes, in particular, the gap between the first optically forbidden and the first optically allowed excited singlet state. The size of this gap is shown to depend strongly on the degree of bond alternation and on the dielectric shielding of the Coulomb repulsion between the $\pi$ electrons.



A mechanism for the light-driven proton pump of Halobacterium halobium. Klaus Schulten and Paul Tavan. Nature, 272:85-86, 1978.
(Citations: 201)

Mitchell's hypothesis of chemiosmotic coupling between redox reactions and ATP synthesis in membranes is supported by the finding of a light-driven proton pump in the purple membrane of Halobacterium halobium. The purple membrane contains the protein bacteriorhodopsin in a crystalline array, with retinal as a chromophore. We propose here, on the basis of quantumchemical arguments and experimental observations, that the H. halobium proton pump may involve proton translocation through photoisomerisation of retinal about its 14-15 single bond.



Semiclassical description of electron spin motion in radicals including the effect of electron hopping. Klaus Schulten and Peter G. Wolynes. Journal of Chemical Physics, 68:3292-3297, 1978.
(Citations: 190)

The coherent electron spin motion in radicals induced by the hyperfine coupling to nuclear spins is described semiclassically. The nuclear spins are treated as constant classical vectors around which the electron spin precesses. The ensemble average over all nuclear spin configurations is taken yielding the electron spin correlation tensor . Borrowing from the theory of rotational diffusion the effect of electron hopping between molecules on the spin correlation tensor is described. The treatment is applied to the time evolution of the electron spin state of a radical pair initially prepared in a singlet state.



Renormalized configuration interaction method for electron correlation in the excited states of polyenes. I. Ohmine, Martin Karplus, and Klaus Schulten. Journal of Chemical Physics, 68:2298-2318, 1978.
(Citations: 168)

Extensive configuration interaction (CI) is needed to achieve satisfactory descriptions of the optical spectra and photochemical properties of the $\pi$-electron systems of polyenes. Although a basis of single and double excitations with respect to the SCF ground state yields a qualitatively correct energy level scheme, such a treatment introduces an imbalance in the ground state correlation (well described) relative to that of the excited states (poorly described). The result is a divergence in the excitation energies with increasing size of the $\pi$-electron system. A renormalized configuration interaction method is developed to account correctly for the excited state correlation. The method is based on the finding that the main contribution to the correlation energy in the excited states is from the electrons not directly involved in the excitation, so that the correlation correction closely resembles that in the ground state. A detailed analysis of the excited state energy permits one to isolate the dominant ground-state correlation term and to determine the smaller, but not negligible, rearrangement correction. The former does not contribute to the excitation energy. The fact that the latter is approximately constant, independent of chain length, provides an explanation for the success achieved by appropriately parametrized single excitation calculations in the assignment of the optically allowed states of polyenes. To implement the renormalized CI method a localized SCF orbital set is employed and the basis functions used for the CI expansion are expressed in terms of single and double excitations with respect to the correlated ground state. It is demonstrated that for the $^{1}B_{u}^{+}$, $^{3} B_{u}^{+}$, and $^{3}A_{g}^{+}$ states, which can be characterized in terms of " elementary " single excitations, this approach gives excellent agreement with the results of more extended CI calculations. Further, it is shown that the correlation energy of the excited state can be estimated using the results of a single-excitation calculation and the ground-state double excitation coefficients.



A model for photoreceptor-based magnetoreception in birds. Thorsten Ritz, Salih Adem, and Klaus Schulten. Biophysical Journal, 78:707-718, 2000.
(Citations: 172)

A large variety of animals has the ability to sense the geomagnetic field and utilize it as a source of directional (compass) information. It is not known by which biophysical mechanism this magnetoreception is achieved. We investigate the possibility that magnetoreception involves radical pair processes which are governed by anisotropic hyperfine coupling between (unpaired) electron and nuclear spins. We will show theoretically that fields of geomagnetic field strength and weaker can produce significantly different reaction yields for different alignments of the radical pairs with the magnetic field. As a model for a magnetic sensory organ we propose a system of radical pairs being (1) orientationally ordered in a molecular substrate and (2) exhibiting changes in the reaction yields that affect the visual transduction pathway. We evaluate three-dimensional visual modulation patterns that can arise from the influence of the geomagnetic field on radical pair systems. The variations of these patterns with orientation and field strength can furnish the magnetic compass ability of birds with the same characteristics as observed in behavioral experiments. We propose that the recently discovered photoreceptor cryptochrome is part of the magnetoreception system and suggest further studies to prove or disprove this hypothesis.



Architecture and Mechanism of the light harvesting apparatus of purple bacteria. Xiche Hu, Ana Damjanovic, Thorsten Ritz, and Klaus Schulten. Proceedings of the National Academy of Sciences, USA, 95:5935-5941, 1998. (Citations: 165)


Photosynthetic organisms fuel their metabolism with light energy and have developed for this purpose an efficient apparatus for harvesting sunlight. The atomic structure of the apparatus, as it evolved in purple bacteria, has been constructed through a combination of X-ray crystallography, electron microscopy, and modeling. The detailed structure and overall architecture reveals a hierarchical aggregate of pigments that utilizes, as shown through femtosecond spectroscopy and quantum physics, elegant and efficient mechanisms for primary light absorption and transfer of electronic excitation towards the photosynthetic reaction center.



Pigment organization and transfer of electronic excitation in the purple bacteria. Xiche Hu, Thorsten Ritz, Ana Damjanovic, and Klaus Schulten. Journal of Physical Chemistry B, 101:3854-3871, 1997.
(Citations: 155)

Absorption of light by light harvesting complexes and transfer of electronic excitation to the photosynthetic reaction center (RC) constitutes the primary light harvesting process of photosynthesis. This process is investigated on the basis of an atomic level structure of the so-called photosynthetic unit of the photosynthetic bacterium Rb. sphaeroides. The photosynthetic unit combines in the intracytoplasmic membrane a nanometric assembly of three protein complexes: (i) the photosynthetic reaction center, (ii) a ring-shaped light harvesting complex LH-I, and (iii) multiple copies of a similar complex, LH-II. The unit has been modeled using the known structure of (i) and for (ii) a model, recently obtained and complexed appropriately with (i); for (iii) the structure of LH-II of Rs. molischianum is substituted. The model describes in detail the organization of chromophores involved in primary light absorption and excitation transfer: a hierarchy of ring-shaped chlorophyll aggregates which surround four centrally located chlorophylls of the photosynthetic reaction center. The chlorophylls involved in the overall transfer are found in a co-planar arrangements. On the basis of the modeled structure a quantum-mechanical description of the entire light harvesting process is developed. For this purpose an effective Hamiltonian is established a priori and then employed to describe the LH-II $\rightarrow$ LH-II $\rightarrow$ LH-I $\rightarrow$ RC cascade of excitation transfer. The transfer times calculated are in agreement with measured transfer times. The results suggest that excitons are the key carriers of the excitation transferred, i.e., electronic excitations are coherently delocalized in the photosynthetic unit. This suggestion is corroborated by an investigation of the effect of inhomogeneous broadening on the predicted excitons in LH-II and LH-I, and effect, which is found to be significant, but small. A particularly important role is played by the lowest energy excitons to which the circular arrangement of chlorophylls imparts vanishing oscillator strength. Despite the lack of oscillator strength the low energy excitons are well suited for exciton transfer on a sub-picosecond and picosecond time scale. The accessory chlorophylls of the photosynthetic reaction center are found to be critical for the LH-I $\rightarrow$ RC transfer which would take several hundred picoseconds without these chlorophylls.



Theory of the magnetic field modulated geminate recombination of radical ion pairs in polar solvents: Application to the pyrene-N,N-dimethylaniline system. Hans-Joachim Werner, Zan Schulten, and Klaus Schulten. Journal of Chemical Physics, 67:646-663, 1977.
(Citations: 152)

Pairs of radical ions generated in polar solvents by photoinduced electron transfer either recombine within a few nanoseconds or separate. The (geminate) recombination process is governed by a hyperfine-coupling-induced coherent motion of the unpaired electron spins which can be modulated by weak external magnetic fields. The process which also generates the well-known CIDNP and CIDEP effects is described theoretically by a stochastic Liouville equation comprising for realistic systems a large set of coupled diffusion equations. For the integration of these equations a finite-difference algorithm with space and time discretization is developed. By comparison with exact solutions of the Liouville equation for model systems, it is demonstrated that an approximate Liouville equation which entails only two coupled diffusion equations for singlet and triplet radical pairs, respectively, suffices to predict the geminate recombination yields accurately. The approximate Liouville equation is employed then to study on the basis of known hyperfine coupling constants, second-order recombination rate constants, diffusion coefficients, and dielectric constants, the solvent, temperature, concentration, and magnetic field dependence of the geminate (singlet and triplet) recombination yields for the system pyrene-N,N-dimethylaniline. The effect of deuteration upon the recombination yield and its magnetic field dependence is also studied. Furthermore, the influence of an exchange interaction acting at small separations of the radicals is investigated for a model system.



Convergence properties of Kohonen's topology conserving maps: Fluctuations, stability and dimension selection. Helge Ritter and Klaus Schulten. Biological Cybernetics, 60:59-71, 1988.
(Citations: 148)

We analyse a Markovian algorithm for the formation of topologically correct feature maps proposed earlier by Kohonen. The maps from a space of input signals onto an array of formal neurons are generated by a learning scheme driven by a random sequence of input samples. The learning is described by an equivalent Fokker-Planck equation. Convergence to an equilibrium map can be ensured by a criterion for the time dependence of the learning step size. We investigate the stability of the equilibrium map and calculate the fluctuations around it. We also study an instability responsible for a phenomenon termed by Kohonen "automatic selection of feature dimensions".



Oxygen and proton pathways in cytochrome c oxidase. Ivo Hofacker and Klaus Schulten. PROTEINS: Structure, Function, and Genetics, 30:100-107, 1998.
(Citations: 143)

Background: Cytochrome c oxidase is a redox-driven proton pump, which couples the reduction of oxygen to water to the translocation of protons across the membrane. The recently solved x-ray structures of cytochrome c oxidase permit molecular dynamics simulations of the underlying transport processes. To eventually establish the proton pump mechanism we investigate the transport of the substrates, oxygen and protons, through the enzyme.
Results: Molecular dynamics simulations of oxygen diffusion through the protein reveal a pathway to the oxygen binding site starting at a hydrophobic cavity near the membrane exposed surface of subunit I, close to the interface to subunit III. A large number of water sites is predicted within the protein. The water molecules form two channels along which protons can enter from the cytoplasmic (matrix) side of the protein and reach the binuclear center.
Conclusions: Oxygen is channeled to the catalytic center of the enzyme along a well defined path. Hydrophobic cavities at the start of the path could serve as reservoirs for oxygen. Water might play an essential role for the transfer of protons in cytochrome c oxidase. A possible pumping mechanism is proposed that involves a shuttling motion of a glutamic acid side chain, which could then transfer a proton to a propionate group of heme $a_{3}$.



On the stationary state of Kohonen's self-organizing sensory mapping. Helge Ritter and Klaus Schulten. Biological Cybernetics, 54:99-106, 1986.
(Citations: 142)

The stationary state of the self-organizing sensory mappings of Kohonen is investigated. For this purpose the equation for the stationary state is derived for the case of one-dimensional and two-dimensional mappings. The equation can be solved for special cases, including the general one-dimensional case, to yield an explicit expression for the local magnification factor of the map.



Calculating potentials of mean force from steered molecular dynamics simulations. Sanghyun Park and Klaus Schulten. Journal of Chemical Physics, 120:5946-5961, 2004.
(Citations: 148)

Steered molecular dynamics (SMD) permits efficient investigations of molecular processes by focusing on selected degrees of freedom. We explain how one can, in the framework of SMD, employ Jarzynski's equality (also known as the nonequilibrium work relation) to calculate potentials of mean force (PMF). We outline the theory that serves this purpose and connects nonequilibrium processes (such as SMD simulations) with equilibrium properties (such as the PMF). We review the derivation of Jarzynski's equality, generalize it to isobaric-isothermal processes, and discuss its implications in relation to the second law of thermodynamics and computer simulations. In the relevant regime of steering by means of stiff springs, we demonstrate that the work on the system is Gaussian-distributed regardless of the speed of the process simulated. In this case, the cumulant expansion of Jarzynski's equality can be safely terminated at second order. We illustrate the PMF calculation method for an exemplary simulation and demonstrate the Gaussian nature of the resulting work distribution.



Self-organizing maps: Ordering, convergence properties and energy functions. Edgar Erwin, Klaus Obermayer, and Klaus Schulten. Biological Cybernetics, 67:47-55, 1992.
(Citations: 141)

We investigate the convergence properties of the self-organizing feature map algorithm for a simple, but very instructive case: the formation of a topographic representation of the unit interval [0,1] by a linear chain of neurons. We extend the proofs of convergence of Kohonen and of Cottrell and Fort to hold in any case where the neighborhood function, which is used to scale the change in the weight values at each neuron, is a monotonically decreasing function of distance from the winner neuron. We prove that the learning dynamics cannot be described by a gradient descent on a single energy function, but may be described using a set of potential functions, one for each neuron, which are independently minimized following a stochastic gradient descent. We derive the correct potential functions for the one- and multi-dimensional case, and show that the energy functions given by Tolat (1990) are an approximation which is no longer valid in the case of highly disordered maps or steep neighborhood functions.



The 21Ag - 11Bu energy gap in the polyenes: An extended configuration interaction study. Paul Tavan and Klaus Schulten. Journal of Chemical Physics, 70:5407-5413, 1979.
(Citations: 139)

For a correct account of the ordering of excited states in the polyenes, in particular the low-lying $2^{1}A_{g}$ and $1^{1}B_{u}$ states, single as well as double excited configurations must be included in a CI expansion. However, for longer $\pi$ systems such expansion shows, in contrast to spectroscopic observations, a divergence of excitation energies and a reversal of the $2^{1}A_{g}$ and $1^{1}B_{u}$ state ordering. We have therefore extended the CI expansion to include all triple and quadruple excitations for the polyene series $C_{n}H_{n+2}$, n = 4,6,8,10,12 and n = 4,6,8,10, respectively. In our calculations we employed a PPP model Hamiltonian. The extended CI expansions correct the faults of previous treatments and predict an increase of the $2^{1}A_{g}$ - $1^{1}B_{u}$ energy gap with increasing polyene length in agreement with recent spectroscopic observations. The effect of higher excitations is mainly due to triple excitations involving three simultaneous spin flips of ground state electrons.



The generation, diffusion, spin motion, and recombination of radical pairs in solution in the nanosecond time domain. Zan Schulten and Klaus Schulten. Journal of Chemical Physics, 66:4616-4634, 1977.
(Citations: 138)

Pairs of radical ions generated in polar solvents by photoinduced electron transfer either recombine within a few nanoseconds to singlet and triplet products or separate. On the basis of recent time-resolved observations of a magnetic field dependence of the pair recombination a theoretical description of this process is provided. The description, similar to the radical pair theory of CIDNP and CIDEP, is founded on a coherent spin motion superimposed on the diffusive motion of the radicals. The spin motion is induced by the hyperfine coupling between electron and nuclear spins and can be modulated by low (0-200 G) magnetic fields. The spin-selective recombination of radicals is accounted for by a Feshbach optical potential. The diffusion process described by a Smoluchowski operator depends sensitively on the solvent properties. For the case of free Brownian motion, simple analytical expressions for the time and magnetic-field-dependent recombination yields are derived. For the Brownian motion of oppositely charged radical ions a differential-difference approximation is used to demonstrate the dependence of the recombination yields on the viscosity and polarity of the solvent medium as well as on the strength of the hyperfine coupling and on the rate of the electron back transfer.



The key event in force-induced unfolding of titin's immunoglobulin domains. Hui Lu and Klaus Schulten. Biophysical Journal, 79:51-65, 2000.
(Citations: 132)

Steered molecular dynamics simulation of force-induced titin immunoglobulin domain I27 unfolding led to the discovery of a significant potential energy barrier at an extension of about 14 Å on the unfolding pathway that protects the domain against stretching. Previous simulations showed that this barrier is due to the concurrent breaking of six interstrand hydrogen bonds (H-bonds) between $\beta$-strands A$^\prime$ and G that is preceded by the breaking of two to three hydrogen bonds between strands A and B leading to an unfolding intermediate. The simulation results is supported by -resolution atomic force microscopy data. Here we perform a structural and energetic analysis of the H-bonds breaking. It is confirmed that H-bonds between strands A and B break rapidly. However, the breaking of the H-bonds between strands A' and G needs to be assisted by fluctuations of water molecules. In nanosecond simulations, water molecules are found to repeatedly interact with the protein backbone atoms, weakening individual interstrand H-bonds until all six A$^\prime$-G H-bonds break simultaneously under the influence of external stretching forces. Only when those bonds are broken can the generic unfolding take place, which involves hydrophobic interactions of the protein core and exerts weaker resistance against stretching than the key event.



Energetics of glycerol conduction through aquaglyceroporin GlpF. Morten Ø. Jensen, Sanghyun Park, Emad Tajkhorshid, and Klaus Schulten. Proceedings of the National Academy of Sciences, USA, 99:6731-6736, 2002.
(Citations: 130)

Aquaglyceroporin GlpF selectively conducts water and linear polyalcohols, such as glycerol, across the inner membrane of Escherichia coli. We report steered molecular dynamics simulations of glycerol conduction through GlpF, in which external forces accelerate the transchannel conduction in a manner that preserves the intrinsic conduction mechanism. The simulations reveal channel-glycerol hydrogen bonding interactions and the stereoselectivity of the channel. Employing Jarzynski's identity between free energy and irreversible work, we reconstruct the potential of mean force along the conduction pathway through a time series analysis of molecular dynamics trajectories. This potential locates binding sites and barriers inside the channel; it also reveals a low energy periplasmic vestibule suited for efficient uptake of glycerol from the environment.



Protein domain movements: Detection of rigid domains and visualization of hinges in comparisons of atomic coordinates. Willy Wriggers and Klaus Schulten. PROTEINS: Structure, Function, and Genetics, 29:1-14, 1997.
(Citations: 129)

The activity of many proteins induces conformational transitions by hinge-bending, which involves the movement of relatively rigid parts of a protein about flexible joints. We present an algorithm to identify and visualize the movements of rigid domains about common hinges in proteins. In comparing two structures, the method partitions a protein into domains of preserved geometry. The domains are extracted by an adaptive selection procedure using least-squares fitting. The user can maintain the spatial connectivity of the domains and filter significant structural differences (domain movements) from noise in the compared sets of atomic coordinates. The algorithm subsequently characterizes the relative movements of the found domains by effective rotation-axes (hinges). The method is applied to several known instances of domain movements in protein structures, namely, in lactoferrin, hexokinase, actin, the extracellular domains of human tissue factor, and of the receptor of human growth factor. The results are visualized with the molecular graphics package VMD (Humphrey et al., J. Mol. Graphics 14(1):33-38, 1996). Applications of the algorithm to the analysis of conformational changes in proteins and to biomolecular docking are discussed.



How nature harvests sunlight. Xiche Hu and Klaus Schulten. Physics Today, 50:28-34, 1997.
(Citations: 126)

Specialized molecular aggregates in purple bacteria exploit subtle quantum physics to collect and convert light energy for photosynthesis.



Forced unfolding of the fibronectin type III module reveals a tensile molecular recognition switch. André Krammer, Hui Lu, Barry Isralewitz, Klaus Schulten, and Viola Vogel. Proceedings of the National Academy of Sciences, USA, 96:1351-1356, 1999.
(Citations: 125)

The tenth type III module of fibronectin, $FnIII_{10}$, mediates cell adhesion to surfaces. It posesses a $\beta$-sandwich structure containing seven $\beta$-strands (A through G) that are arranged in two anti-parallel sheets where the cell binding motif, Arg78-Gly79-Asp80 (RGD), is placed at the apex of the loop connecting $\beta$-strands F and G. Steered molecular dynamics (SMD) simulations in which tension is applied to the protein's terminal ends reveal that the module can act as a tensile molecular recognition switch. Analysis of the forced unfolding process of $FnIII_{10}$ shows that the G-strand is the first to be released while the remaining module maintains its structural integrity. This leads to a gradual shortening of the distance between the apex of the RGD-containing loop and th surface of the remaining module followed by a straightening of this loop from a $\beta$-turn into a linear conformation. Experimental data have previously shown that shortening the RGD-containing loop reduces its accessibility to membrane-bound integrins, and that the loop's affinity and selectivity to integrins decreases when linearized.



Dynamics of reactions involving diffusive barrier crossing. Klaus Schulten, Zan Schulten, and Attila Szabo. Journal of Chemical Physics, 74:4426-4432, 1981.
(Citations: 121)

We develop a first passage time description for the kinetics of reactions involving diffusive barrier crossing in a bistable (and also in a more general) potential, a situation realized, for example, in some photoisomerization processes. In case the reactant is in thermal equilibrium, the first passage times account well for the reaction dynamics as shown by comparison with exact numerical calculations. A simple integral expression for the rate constants is presented. For a case involving a reactant initially far off equilibrium, a two relaxation time description for the particle number ${\it N (t )}$ is derived and compared with the results of an " exact " calculation. This description results from a knowledge of ${\it N (t =0 )}$, ${\it N' (t =0 )}$, $\int_{o}^{\infty}{\it dt N (t )} $ , i.e., the first passage time, and $\int_{o}^{\infty}{\it dt t N (t )}$.



The effect of protonation and electrical interactions on the stereochemistry of retinal Schiff bases. Paul Tavan, Klaus Schulten, and Dieter Oesterhelt. Biophysical Journal, 47:415-430, 1985.
(Citations: 120)

Based on quantumchemical MNDOC calculations it is shown that the ground-state properties of a retinal Schiff base depend sensitively on its protonation state and charge environment. This is exemplified for the equilibrium geometry, for the distribution of partial charges and, in particular, for the thermal isomerization barriers around the $\pi$-bonds. It is demonstrated that a protein, by protonating the retinal Schiff base and by providing one or two negative ions in its environment, can reduce double-bond isomerization barriers from 50 kcal/mol for the unprotonated compound to $\sim$ 5 kcal/mol and can increase single bond barriers from 5 kcal/mol to $\sim$ 20 kcal/mol. Thereby, the specific location of the ions relative to the polyene chain of the protonated retinal Schiff base determines the barrier heights. The results explain the ground-state isomerization reactions of retinal observed in bacteriorhodopsin and in squid retinochrome.



Principal component analysis and long time protein dynamics. Manel A. Balsera, Willy Wriggers, Yoshitsugu Oono, and Klaus Schulten. Journal of Physical Chemistry, 100:2567-2572, 1996.
(Citations: 121)

It has been suggested that principal component analysis can identify slow modes in proteins and, thereby, facilitate the study of long time dynamics. However, sampling errors due to finite simulation times preclude the identification of slow modes that can be used for this purpose. This is demonstrated numerically with the aid of G-actin simulations, and analytically with the aid of a model which is exactly recoverable by Principal Component Analysis.



Accelerated molecular dynamics simulation with the parallel fast multipole algorithm. John A. Board, Jr., J. W. Causey, James F. Leathrum, Jr., Andreas Windemuth, and Klaus Schulten. Chemical Physics Letters, 198:89-94, 1992.
(Citations: 117)

We have implemented the fast multipole algorithm (FMA) of Greengard and Rokhlin and incorporated it into the molecular dynamics program MD of Windemuth and Schulten, allowing rapid computation of the non-bonded forces acting in dynamical protein systems without truncation or other corruption of the Coulomb force. The resulting program speeds up simulations of protein systems with approximately 24000 atoms by up to an order of magnitude on a single workstation. Additionally, we have implemented a parallel version of the three-dimensional FMA code on a loosely coupled network of workstations, further reducing simulation times. Large (in both size of system and length of simulated time) protein molecular dynamics simulations are now possible on workstations rather than supercomputers, and very large protein computations are possible on clusters of workstations and parallel machines.



Free energy calculation from steered molecular dynamics simulations using Jarzynski's equality.Sanghyun Park, Fatemeh Khalili-Araghi, Emad Tajkhorshid, and Klaus Schulten. Journal of Chemical Physics, 119:3559-3566, 2003.
(Citations: 115)

Jarzynski's equality is applied to free energy calculations from steered molecular dynamics simulations of biomolecules. The helix-coil transition of deca-alanine in vacuum is used as an example. With about ten trajectories sampled, the second order cumulant expansion, among the various averaging schemes examined, yields the most accurate estimates. We compare umbrella sampling and the present method, and find that their efficiencies are comparable.



Electron transfer and spin exchange contributing to the magnetic field dependence of the primary photochemical reaction of bacterial photosynthesis. Hans-Joachim Werner, Klaus Schulten, and Albert Weller. Biochimica et Biophysica Acta, 502:255-268, 1978.
(Citations: 112)

The yield $\varphi_{T}$ of triplet products "P${}_{R}$" generated in reaction centers of Rhodopseudomonas sphaeroides in which the "primary" acceptor is reduced had been found to depend on external magnetic fields. The magnetic field dependence varies, however, between different reaction center preparations. By means of a theoretical description of the primary electron transfer processes and hyperfine coupling-induced electron spin motion the factors influencing the magnetic field behaviour of the triplet products are studied. The following quantities characteristic of the primary electron transfer in photosynthesis have a strong effect on $\varphi_{T}$: (1) the rate constants of reversible electron transfer between the initially excited singlet state of the reaction center and an intermediate radical ion pair state; (2) the rate constants of irreversible electron transfer of the radical pair to the ground and excited triplet state of the reaction center; (3) the electron exchange interactions between the radical pair and the "primary" acceptor. From the observed magnetic field dependence of $\varphi_{T}$ estimates for these quantities are obtained. A temperature dependence of the magnetic field behaviour of $\varphi_{T}$ and a magnetic field effect on the fluorescence quantum yield of the reaction center are predicted.



Models of orientation and ocular dominance columns in the visual cortex: A critical comparison. Edgar Erwin, Klaus Obermayer, and Klaus Schulten. Neural Computation, 7:425-468, 1995.
(Citations: 115)

Orientation and ocular dominance maps in the primary visual cortex of mammals are among the most thoroughly investigated of the patterns in the cerebral cortex. A considerable amount of work has been dedicated to unraveling both their detailed structure and the neural mechanisms that underlie their formation and development. Many schemes have been proposed, some of which are in competition. Some models focus on development of receptive fields while others focus on the structure of cortical maps, i.e., the arrangement of receptive field properties across the cortex. Each model used different means to determine its success at reproducing experimental map patterns, often relying principally on visual comparison. Experimental data are becoming available that allow a more careful evaluation of models. In this contribution more than 10 of the most prominent models of cortical map formation and structure are critically evaluated and compared with the most recent experimental findings from macaque striate cortex. Comparisons are based on properties of the predicted or measured cortical map patterns. We introduce several new measures for comparing experimental and model map data that reveal important differences between models. We expect that the use of these measures will improve current models by helping determine parameters to match model maps to experimental data now becoming available from a variety of species. Our study reveals that (1) despite apparent differences, many models are based on similar principles and consequently make similar predictions, (2) several models produce orientation map patterns that are not consistent with the experimental data from macaques, regardless of the plausibility of the models' suggested physiological implementations, and (3) no models have yet fully accounted for both the local and the global relationships between orientation and ocular dominance map patterns.



Photosynthetic apparatus of purple bacteria. Xiche Hu, Thorsten Ritz, Ana Damjanovic, Felix Autenrieth, and Klaus Schulten. Quarterly Reviews of Biophysics, 35:1-62, 2002.
(Citations: 108)

This article reviews work accomplished during the past decade on the structure and function of the photosynthetic unit of purple bacteria with a main focus on the light harvesting component. The photosynthetic unit exists as aggregates of proteins in the intracellular membranes of these bacteria; the units absorb sun light and utilize its energy for the synthesis of ATP. The light harvesting component involves ring-shape proteins that surround directly in the form of satellite rings the so-called reaction center. The structure of the proteins as established through a combination of experimental and computational methods is reviewed. The proteins provide a scaffold for a hierarchical aggregate of chlorophylls and carotenoids that funnel electronic excitation towards the reaction center. The physics of this process is reviewed in detail. Finally, the genomic level organization of the light harvesting system is summarized.



Steered molecular dynamics simulations of force-induced protein domain unfolding. Hui Lu and Klaus Schulten. PROTEINS: Structure, Function, and Genetics, 35:453-463, 1999.
(Citations: 107)

Steered molecular dynamics (SMD), a computer simulation method for studying force-induced reactions in biopolymers, has been applied to investigate the response of protein domains to stretching apart of their terminal ends. The simulations mimic atomic force microscopy and optical tweezer experiments, but proceed on much shorter time scales. The simulations on different domains for 0.6 nanosecond each reveal two types of protein responses: the first type, arising in certain $\beta$-sandwich domains, exhibits nanasecond (ns) unfolding only after a force above 1500 pN is applied; the second type, arising in a wider class of protein domain structures, requires significantly weaker forces for ns unfolding. In the first case, strong forces are needed to concertedly break a set of interstrand hydrogen bonds which protect the domains against unfolding through stretching; in the second case, stretching breaks backbone hydrogen bonds stabilizing the structure one by one, and does not require strong forces for this purpose. Stretching of $\beta$-sandwich (immunoglobulin) domains has been investigated further revealing a specific relationship between response to mechanical strain and the architecture of $\beta$-sandwich domains.



Molecular dynamics study of bacteriorhodopsin and artificial pigments. William Humphrey, Ilya Logunov, Klaus Schulten, and Mordechai Sheves. Biochemistry, 33:3668-3678, 1994.
(Citations: 103)

The structure of bacteriorhodopsin, as provided by the so-called Henderson model, is refined using molecular dynamics simulations. The work is based on a previously refined structure which had added the interhelical loops to the Henderson model. The present study applies an all-atom description to this structure and constraints to the original Henderson model, albeit with helix D shifted. Sixteen waters are then added to the protein, six in the retinal Schiff base region, four in the retinal-Asp-96 interstitial space, and six near the extracellular side. The root mean square deviation between the resulting structure and the Henderson model measures only 1.8 A. Further simulations of retinal analogues for substitutions at the 2-and 4-positions of retinal and an analogue without a $\beta$-ionone ring agree well with observed spectra. The refined structure is characterized in view of bacteriorhodopsin's function; key features are (1) a retinal Schiff base-counterion complex which is formed by a hydrogen bridge network involving six water molecules, Asp-85, Asp-212, Tyr-185, Tyr-57, Arg-82, and Thr-89, and which exhibits Schiff base nitrogen-Asp-85 and -Asp-212 distances of 6 and 4.6 A; (2) retinal assumes a corkscrew twist as one views retinal along its backbone; and (3) a destabilization of the cytoplasmic side of helix G.



Statistical-mechanical analysis of self-organization and pattern formation during the development of visual maps. Klaus Obermayer, Gary G. Blasdel, and Klaus Schulten. Physical Review A, 45:7568-7589, 1992.
(Citations: 101)

We report a detailed analytical and numerical model study of pattern formation during the development of visual maps, namely, the formation of topographic maps and orientation and ocular dominance columns in the striate cortex. Pattern formation is described by a stimulus-driven Markovian process, the self-organizing feature map. This algorithm generates topologically correct maps between a space of (visual) input signals and an array of formal "neurons," which in our model represents the cortex. We define order parameters that are a function of the set of visual stimuli an animal perceives, and we demonstrate that the formation of orientation and ocular dominance columns is the result of a global instability of the retinoptic projection above a critical value of these order parameters. We characterize the spatial structure of the emerging patterns by power spectra, correlation functions, and Gabor transforms, and we compare model predictions with experimental data obtained from the striate cortex of the macaque monkey with optical imaging. Above the critical value of the order parameters the model predicts a lateral segregation of the striate cortex into (i) binocular regions with linear changes in orientation preference, where iso-orientation slabs run perpendicular to the ocular dominance bands, and (ii) monocular regions with low orientation specificity, which contain the singularities of the orientation map. Some of these predictions have already been verified by experiments.



Coupling of protein motion to electron transfer: Molecular dynamics and stochastic quantum mechanics study of photosynthetic reaction centers. Klaus Schulten and Markus Tesch. Chemical Physics, 158:421-446, 1991.
(Citations: 97)

We investigate how electron transfer is controlled by protein motion in photosynthetic reaction centers. Our study is based on molecular dynamics (MD) simulations of two electron transfer steps in the reaction center of Rps. viridis at physiological and at lower temperatures. The classical simulations of protein nuclear motions are complemented by a quantum mechanical description for the electron transfer, incorporating in a two-state model a coupling to the classical protein motion through a fluctuating diagonal contribution which is determined as the energy difference $\Delta$E(t), the distribution p($\Delta$E) and correlation function <$\Delta$E(t+$\tau$)$\Delta$E($\tau$)>, are investigated and a stochastic quantum mechanical model for electron transfer is introduced that incorporates three characteristics of $\Delta$E(t), namely its mean value, its rms-deviations from the mean, and the mean relaxation time of its correlation function. The calculations which go beyond second-order perturbation theory predict a bell-shaped dependence of the electron transfer rate on redox energies with a so-called inverted region and with a width of about 20 kcal/mol (about 10 kcal/mol for the stochastic model). Rapid (0.05 ps) dielectric relaxation after electron transfer induces a shift of the mean <$\Delta$E> which causes reactant and product states to become sufficiently out of resonance and which, thereby, prevents electron back-transfer. It is shown that all components of photosynthetic reaction centers contribute rather evenly to the coupling between electron transfer and medium.



A principle for the formation of the spatial structure of cortical feature maps. Klaus Obermayer, Helge Ritter, and Klaus Schulten. Proceedings of the National Academy of Sciences, USA, 87:8345-8349, 1990.
(Citations: 95)

Orientation-selective cells in the striate cortex of higher animals are organized as a hierarchical topographic map of two stimulus features: (em i) position in visual space and (em ii) orientation. We show that the observed structure of the topographic map can arise from a principle of continuous mapping. For the realization of this principle we use a mathematical model that can be interpreted as an adaptive process changing a set of synaptic weights, or synaptic connection strengths, between two layers of cells. The patterns of orientation preference and selectivity generated by the model are similar to the patterns seen in the visual cortex of macaque monkey and cat and correspond to a neural projection that maps a more than two-dimensional feature space onto a two-dimensional cortical surface under the constraint that shape and position of the receptive fields of the neurons vary smoothly over the cortical surface.



Chromophore-protein interactions and the function of the photosynthetic reaction center: A molecular dynamics study. Herbert Treutlein, Klaus Schulten, Axel Brünger, Martin Karplus, J. Deisenhofer, and H. Michel.Proceedings of the National Academy of Sciences, USA, 89:75-79, 1992.
(Citations: 95)

The coupling between electron transfer and protein structure and dynamics in the photosynthetic reaction center of em Rhodopseudomonas viridis is investigated. For this purpose molecular dynamics simulations of the essential portions (a segment of 5797 atoms) of this protein complex have been carried out. Electron transfer in the primary event is modeled by altering the charge distributions of the chromophores according to quantum chemical calculations. The simulations show (em i) that fluctuations of the protein matrix, which are coupled electrostatically to electron transfer, play an important role in controlling the electron transfer rates and (em ii) that the protein matrix stabilizes the separated electron pair state through rapid (200fs) and temperature-independent dielectric relaxation. The photosynthetic reaction center resembles a polar liquid in that the internal motions of the whole protein complex, rather than only those of specific side groups, contribute to em i and em ii. The solvent reorganization energy is about 4.5 kcal/mol. The simulations indicate that rather small structural rearrangements and changes in motional amplitudes accompany the primary electron transfer.



Molecular dynamics study of a membrane-water interface. Feng Zhou and Klaus Schulten. Journal of Physical Chemistry, 99:2194-2208, 1995.
(Citations: 91)

A 200 ps molecular dynamics simulation of a membrane bilayer consisting of 202 dilauroylphosphatidyl-ethanolamine molecules and 8108 water molecules at 315 K is conducted. Distribution functions of lipid groups, order parameters, and other properties of the lipid bilayer are calculated and compared with experimental measurements. A detailed analysis is conducted for the structure at the membrane-water interface. Water polarization profile, membrane dipole potential profile, and susceptibility profile are calculated. Simulation results suggest that the polarization of water is determined mainly by the distribution of lipid head groups in the interfacial region. The membrane dipole potential is mainly due to the ester groups linked to the glycerol backbone, while the contribution due to the phosphatidylethanolamine head groups is almost completely cancelled by the contribution due to oriented water molecules. The susceptibility profile suggests a dielectric constant around 30 for the head group-water interface and a dielectric constant around 10 for the ester group region. The ammonium groups of the DLPE membrane are found to form hydrogen bonds with water molecules, while no orientational preference is observed for water molecules around the choline groups of a previously simulated POPC (1-palmitoyl-2-oleoyl-sn-glycerol-3-phosphatidylcholine) membrane. Correlations of the membrane surface charge density are also analyzed. The simulations which involved 32 808 atoms included Coulombic forces between all atom pairs evaluated by means of the fast multipole algorithm. Effects of cutting off Coulombic forces at a distance of 8 angstroms are discussed.



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