Testing the applicability of Nernst-Planck theory in ion channels: comparisons with Brownian dynamics simulations

The macroscopic Nernst-Planck (NP) theory has often been used for predicting ion channel currents in recent years, but the validity of this theory at the microscopic scale has not been tested. In this study we systematically tested the ability of the NP theory to accurately predict channel currents by combining and comparing the results with those of Brownian dynamics (BD) simulations. To thoroughly test the theory in a range of situations, calculations were made in a series of simplified cylindrical channels with radii ranging from 3 to 15 Å, in a more complex ‘catenary’ channel, and in a realistic model of the mechanosensitive channel MscS. The extensive tests indicate that the NP equation is applicable in narrow ion channels provided that accurate concentrations and potentials can be input as the currents obtained from the combination of BD and NP match well with those obtained directly from BD simulations, although some discrepancies are seen when the ion concentrations are not radially uniform. This finding opens a door to utilising the results of microscopic simulations in continuum theory, something that is likely to be useful in the investigation of a range of biophysical and nano-scale applications and should stimulate further studies in this direction.     

Brownian dynamics simulations of ion transport through the VDAC

It is important to gain a physical understanding of ion transport through the voltage-dependent anion channel (VDAC) because this channel provides primary permeation pathways for metabolites and electrolytes between the cytosol and mitochondria. We performed grand canonical Monte Carlo/Brownian dynamics (GCMC/BD) simulations to explore the ion transport properties of human VDAC isoform 1 (hVDAC1; PDB:2K4T) embedded in an implicit membrane. When the MD-derived, space-dependent diffusion constant was used in the GCMC/BD simulations, the current-voltage characteristics and ion number profiles inside the pore showed excellent agreement with those calculated from all-atom molecular-dynamics (MD) simulations, thereby validating the GCMC/BD approach. Of the 20 NMR models of hVDAC1 currently available, the third one (NMR03) best reproduces both experimental single-channel conductance and ion selectivity (i.e., the reversal potential). In addition, detailed analyses of the ion trajectories, one-dimensional multi-ion potential of mean force, and protein charge distribution reveal that electrostatic interactions play an important role in the channel structure and ion transport relationship. Finally, the GCMC/BD simulations of various mutants based on NMR03 show good agreement with experimental ion selectivity. The difference in ion selectivity between the wild-type and the mutants is the result of altered potential of mean force profiles that are dominated by the electrostatic interactions.     

Brownian dynamics simulation of ion flow through porin channels

Bacterial porins, which allow the passage of solutes across the outer bacterial membrane, are structurally well characterized. They therefore lend themselves to detailed studies of the determinants of ion flow through transmembraneous channels. In a comparative study, we have performed Brownian dynamics simulations to obtain statistically significant transfer efficiencies for cations and anions through matrix porin OmpF, osmoporin OmpK36, phosphoporin PhoE and two OmpF charge mutants.The simulations show that the electrostatic potential at the highly charged channel constriction serves to enhance ion permeability of either cations or anions, dependent on the type of porin. At the same time translocation of counterions is not severely impeded. At the constriction, cations and anions follow distinct trajectories, due to the segregation of basic and acidic protein residues.Simulated ion selectivity and relative conductance agree well with experimental values, and are dependent crucially on the charge constellation at the pore constriction. The experimentally observed decrease in ion selectivity and single channel conductance with increasing ionic strength is well reproduced and can be attributed to electrostatic shielding of the pore lining.     

Brownian dynamics simulations of the recognition of the scorpion toxin maurotoxin with the voltage-gated potassium ion channels

The recognition of the scorpion toxin maurotoxin (MTX) by the voltage-gated potassium (Kv1) channels, Kv1.1, Kv1.2, and Kv1.3, has been studied by means of Brownian dynamics (BD) simulations. All of the 35 available structures of MTX in the Protein Data Bank (http://www.rcsb.org/pdb) determined by nuclear magnetic resonance were considered during the simulations, which indicated that the conformation of MTX significantly affected both the recognition and the binding between MTX and the Kv1 channels. Comparing the top five highest-frequency structures of MTX binding to the Kv1 channels, we found that the Kv1.2 channel, with the highest docking frequencies and the lowest electrostatic interaction energies, was the most favorable for MTX binding, whereas Kv1.1 was intermediate, and Kv1.3 was the least favorable one. Among the 35 structures of MTX, the 10th structure docked into the binding site of the Kv1.2 channel with the highest probability and the most favorable electrostatic interactions. From the MTX-Kv1.2 binding model, we identified the critical residues for the recognition of these two proteins through triplet contact analyses. MTX locates around the extracellular mouth of the Kv1 channels, making contacts with its beta-sheets. Lys23, a conserved amino acid in the scorpion toxins, protrudes into the pore of the Kv1.2 channel and forms two hydrogen bonds with the conserved residues Gly401(D) and Tyr400(C) and one hydrophobic contact with Gly401(C) of the Kv1.2 channel. The critical triplet contacts for recognition between MTX and the Kv1.2 channel are Lys23(MTX)-Asp402(C)(Kv1), Lys27(MTX)-Asp378(D)(Kv1), and Lys30(MTX)-Asp402(A)(Kv1). In addition, six hydrogen-bonding interactions are formed between residues Lys23, Lys27, Lys30, and Tyr32 of MTX and residues Gly401, Tyr400, Asp402, Asp378, and Thr406 of Kv1.2. Many of them are formed by side chains of residues of MTX and backbone atoms of the Kv1.2 channel. Five hydrophobic contacts exist between residues Pro20, Lys23, Lys30 and Tyr32 of MTX and residues Asp402, Val404, Gly401, and Arg377 of the Kv1.2 channel. The simulation results are in agreement with the previous molecular biology experiments and explain the binding phenomena between MTX and Kv1 channels at the molecular level. The consistency between the results of the BD simulations and the experimental data indicated that our three-dimensional model of the MTX-Kv1.2 channel complex is reasonable and can be used in additional biological studies, such as rational design of novel therapeutic agents blocking the voltage-gated channels and in mutagenesis studies in both the toxins and the Kv1 channels. In particular, both the BD simulations and the molecular mechanics refinements indicate that residue Asp378 of the Kv1.2 channel is critical for its recognition and binding functionality toward MTX. This phenomenon has not been appreciated in the previous mutagenesis experiments, indicating this might be a new clue for additional functional study of Kv1 channels.     

Brownian dynamics study of ion transport in the vestibule of membrane channels.

Brownian dynamics simulations have been carried out to study the transport of ions in a vestibular geometry, which offers a more realistic shape for membrane channels than cylindrical tubes. Specifically, we consider a torus-shaped channel, for which the analytical solution of Poisson's equation is possible. The system is composed of the toroidal channel, with length and radius of the constricted region of 80 A and 4 A, respectively, and two reservoirs containing 50 sodium ions and 50 chloride ions. The positions of each of these ions executing Brownian motion under the influence of a stochastic force and a systematic electric force are determined at discrete time steps of 50 fs for up to 2.5 ns. All of the systematic forces acting on an ion due to the other ions, an external electric field, fixed charges in the channel protein, and the image charges induced at the water-protein boundary are explicitly included in the calculations. We find that the repulsive dielectric force arising from the induced surface charges plays a dominant role in channel dynamics. It expels an ion from the vestibule when it is deliberately put in it. Even in the presence of an applied electric potential of 100 mV, an ion cannot overcome this repulsive force and permeate the channel. Only when dipoles of a favorable orientation are placed along the sides of the transmembrane segment can an ion traverse the channel under the influence of a membrane potential. When the strength of the dipoles is further increased, an ion becomes detained in a potential well, and the driving force provided by the applied field is not sufficient to drive the ion out of the well. The trajectory of an ion navigating across the channel mostly remains close to the central axis of the pore lumen. Finally, we discuss the implications of these findings for the transport of ions across the membrane.     

Molecular dynamics simulations of water within models of ion channels.

The transbilayer pores formed by ion channel proteins contain extended columns of water molecules. The dynamic properties of such waters have been suggested to differ from those of water in its bulk state. Molecular dynamics simulations of ion channel models solvated within and at the mouths of their pores are used to investigate the dynamics and structure of intra-pore water. Three classes of channel model are investigated: a) parallel bundles of hydrophobic (Ala20) alpha-helices; b) eight-stranded hydrophobic (Ala10) antiparallel beta-barrels; and c) parallel bundles of amphipathic alpha-helices (namely, delta-toxin, alamethicin, and nicotinic acetylcholine receptor M2 helix). The self-diffusion coefficients of water molecules within the pores are reduced significantly relative to bulk water in all of the models. Water rotational reorientation rates are also reduced within the pores, particularly in those pores formed by alpha-helix bundles. In the narrowest pore (that of the Ala20 pentameric helix bundle) self-diffusion coefficients and reorientation rates of intra-pore waters are reduced by approximately an order of magnitude relative to bulk solvent. In Ala20 helix bundles the water dipoles orient antiparallel to the helix dipoles. Such dipole/dipole interaction between water and pore may explain how water-filled ion channels may be formed by hydrophobic helices. In the bundles of amphipathic helices the orientation of water dipoles is modulated by the presence of charged side chains. No preferential orientation of water dipoles relative to the pore axis is observed in the hydrophobic beta-barrel models.     

Brownian dynamics simulations of fluorescence fluctuation spectroscopy

We have developed a program for the simulation of the fluorescence fluctuations as detected from highly diluted samples of (bio)molecules. The model is applied to translational diffusion and takes into account the hydrodynamic interactions. The solution concentration is kept constant by assuming periodic boundary conditions and spans here the range 0.5< C < 10 nM. We show that the fluorescence correlation functions can be accurately computed on systems of limited size (a few molecules per simulation box) by simulating for a total time approximately 100-300 times the diffusion relaxation time of the fluorescence autocorrelation function. The model is applied also to the simulation of the scanning fluorescence correlation spectroscopy (FCS) and of the photon counting histograms for the confocal collection configuration. Scanning FCS simulations of highly diluted samples (C approximately equals 0.5 nM) show anticorrelation effects in the autocorrelation functions of the fluorescence signal that are less evident for higher concentrations. We suggest here that this effect may be due to the non-uniform occupancy of the scanning area by the fluorophores.     

Brownian dynamics simulations of interaction between scorpion toxin Lq2 and potassium ion channel

The association of the scorpion toxin Lq2 and a potassium ion (K(+)) channel has been studied using the Brownian dynamics (BD) simulation method. All of the 22 available structures of Lq2 in the Brookhaven Protein Data Bank (PDB) determined by NMR were considered during the simulation, which indicated that the conformation of Lq2 affects the binding between the two proteins significantly. Among the 22 structures of Lq2, only 4 structures dock in the binding site of the K(+) channel with a high probability and favorable electrostatic interactions. From the 4 candidates of the Lq2-K(+) channel binding models, we identified a good three-dimensional model of Lq2-K(+) channel complex through triplet contact analysis, electrostatic interaction energy estimation by BD simulation and structural refinement by molecular mechanics. Lq2 locates around the extracellular mouth of the K(+) channel and contacts the K(+) channel using its beta-sheet rather than its alpha-helix. Lys27, a conserved amino acid in the scorpion toxins, plugs the pore of the K(+) channel and forms three hydrogen bonds with the conserved residues Tyr78(A-C) and two hydrophobic contacts with Gly79 of the K(+) channel. In addition, eight hydrogen-bonds are formed between residues Arg25, Cys28, Lys31, Arg34 and Tyr36 of Lq2 and residues Pro55, Tyr78, Gly79, Asp80, and Tyr82 of K(+) channel. Many of them are formed by side chains of residues of Lq2 and backbone atoms of the K(+) channel. Thirteen hydrophobic contacts exist between residues Met29, Asn30, Lys31 and Tyr36 of Lq2 and residues Pro55, Ala58, Gly79, Asp80 and Tyr82 of the K(+) channel. These favorable interactions stabilize the association between the two proteins. These observations are in good agreement with the experimental results and can explain the binding phenomena between scorpion toxins and K(+) channels at the level of molecular structure. The consistency between the BD simulation and the experimental data indicates that our three-dimensional model of Lq2-K(+) channel complex is reasonable and can be used in further biological studies such as rational design of blocking agents of K(+) channels and mutagenesis in both toxins and K(+) channels.     

Brownian dynamics simulations of intramolecular energy transfer

A novel technique for modelling intramolecular energy transfer is presented. Brownian dynamics calculations are used to compute the trajectories of donor and acceptor species, and the instantaneous orientation factor is calculated during each temporal iteration. In this work, several model systems are considered. Trajectories were computed for energy transfer between a flexible donor and a rigidly fixed acceptor. We have considered configurations where the donor is, (1) tethered to a fixed point in space, but free to diffuse rotationally, and (2) constrained to wobble in a cone. The luminescence decay of the donor is ‘measured’, and a non-single-exponential decay is observed for configurations of efficient energy transfer. Luminescence anisotropy measurements of constrained and unconstrained donors reflect the contribution of both energy transfer and rotational diffusion to the shape of the anisotropy decay curve.     

Brownian dynamics simulations of ion atmospheres around polyalanine and B-DNA: effects of biomolecular dielectric

We have extended an earlier Brownian dynamics simulation algorithm for simulating the structural dynamics of ions around biomolecules to accommodate dielectric inhomogeneity. The electrostatic environment of a biomolecule immersed in water was obtained by numerically solving the Poisson equation with the biomolecule treated as a low dielectric region and the solvent treated as a high dielectric region. Instead of using the mean-field type approximations of ion interactions as in the Poisson-Boltzmann model, the ions were treated explicitly by allowing them to evolve dynamically under the electrostatic field of the biomolecule. This model thus accounts for ion-ion correlations and the finite-size effects of the ions. For a 13-residue alpha-helical polyalanine and a 12-base-pair bp B-form DNA, we found that the choice of the dielectric constant of the biomolecule has much larger effects on the mean ionic structure around the biomolecule than on the fluctuational and dynamical properties of the ions surrounding the biomolecule.     

Brownian dynamics study of flux ratios in sodium channels

Measurements of unidirectional fluxes in ion channels provide one of the experimental methods for studying the steps involved in ion permeation in biological pores. Conventionally, the number of ions in the pore is inferred by fitting the ratio of inward and outward currents to an exponential function with an adjustable parameter known as the flux ratio exponent. Here we investigate the relationship between the number of ions in the pore and the flux ratio exponent in a model sodium channel under a range of conditions. Brownian dynamics simulations enable us to count the precise number of ions in the channel and at the same time measure the currents flowing across the pore in both directions. We show here that the values of the flux ratio exponent n′ ranges between 1 and 3 and is highly dependent on the ionic concentrations in which measurements are made. This is a consequence of the fact that both inward and outward currents are susceptible to saturation with increasing concentration. These results indicate that measurements of the flux ratio exponent cannot be directly related to the number of ions in the pore and that interpretation of such experimental measurements requires careful consideration of the conditions in which the study is made.     

Molecular dynamics simulations of ion channels formed by bundles of amphipathic α-helical peptides

Ion channels may be formed by self-assembly of amphipathic α-helical peptides into parallel helix bundles. The transbilayer pores formed by such peptides contain extended columns of water molecules, the properties of which may differ from those of water in its bulk state. The de novo designed peptides of DeGrado et al., which contain only leucine and serine residues, are considered as a simple example of such channels. Molecular dynamics simulations of peptide helix bundles with water molecules within and at the mouths of their pores are used to refine such models and to investigate the properties of intra-pore water. The translational and rotational mobility of water molecules within the pores are reduced relative to bulk water. Furthermore, intra-pore waters orient themselves with their dipoles anti-parallel to the helix dipoles, as do the hydroxyl groups of serine residues. Comparison of approximate predictions of ionic conductances with experimental values provides support for the validity of these models. Received: 23 April 1996 / Accepted: 7 August 1996     

Brownian dynamics simulations of glycolytic enzyme subsets with F-actin

Previous Brownian dynamics (BD) simulations identified specific basic residues on fructose-1,6-bisphophate aldolase (aldolase) (I. V. Ouporov et al., Biophysical Journal, 1999, Vol. 76, pp. 17-27) and glyceraldehyde-3-phosphate dehydrogenase (GAPDH) (I. V. Ouporov et al., Journal of Molecular Recognition, 2001, Vol. 14, pp. 29-41) involved in binding F-actin, and suggested that the quaternary structure of the enzymes may be important. Herein, BD simulations of F-actin binding by enzyme dimers or peptides matching particular sequences of the enzyme and the intact enzyme triose phosphate isomerase (TIM) are compared. BD confirms the experimental observation that TIM has little affinity for F-actin. For aldolase, the critical residues identified by BD are found in surface grooves, formed by subunits A/D and B/C, where they face like residues of the neighboring subunit enhancing their electrostatic potentials. BD simulations between F-actin and aldolase A/D dimers give results similar to the native tetramer. Aldolase A/B dimers form complexes involving residues that are buried in the native structure and are energetically weaker; these results support the importance of quaternary structure for aldolase. GAPDH, however, placed the critical residues on the corners of the tetramer so there is no enhancement of the electrostatic potential between the subunits. Simulations using GAPDH dimers composed of either S/H or G/H subunits show reduced binding energetics compared to the tetramer, but for both dimers, the sets of residues involved in binding are similar to those found for the native tetramer. BD simulations using either aldolase or GAPDH peptides that bind F-actin experimentally show complex formation. The GAPDH peptide bound to the same F-actin domain as did the intact tetramer; however, unlike the tetramer, the aldolase peptide lacked specificity for binding a single F-actin domain.     

Brownian dynamics simulations of the recognition of the scorpion toxin P05 with the small-conductance calcium-activated potassium channels

The recognition of the scorpion toxin P05 and the small-conductance, calcium-activated potassium (SK) channels, rsk1, rsk2, and rsk3, has been studied by means of the Brownian dynamics (BD) method. All of the 25 available structures of P05 in the RCSB Protein Data Bank determined by NMR were considered during the simulation, which indicated that the conformation of P05 affects both the recognition and binding between the two proteins significantly. Comparing the top four high-frequency structures of P05 binding to the SK channels, we found that the rsk2 channel, with high frequencies and lowest electrostatic interaction energies (E (int)(ES)), is the most favorable for P05 binding, while rsk3 is intermediate, and rsk1 is the least favorable. Among the 25 structures of P05, the 13th structure docks into the binding site of the rsk2 channel with the highest probability and most favorable electrostatic interactions. From the P05-rsk2 channel binding model, we identified the residues critical for the recognition of these two proteins through triplet contact analyses. P05 locates around the extracellular mouth of the SK channels and contacts the SK channels using its alpha-helix rather than beta-sheets. The critical triplet contacts for recognition between P05 and the rsk2 channel are Arg6 (P05)-Asp364 (SK), Arg7 (P05)-Asn368 (SK), and Arg13 (P05)-Asp341 (SK). The structure of the P05-rsk2 complex with the most favorable electrostatic interaction energy was further refined by molecular mechanics, showing that six hydrogen bonding interactions exist between P05 and the rsk2 channel: one hydrogen bond is formed between Arg6 (P05) and Asp364(D) (rsk2); Arg7 (P05) forms three hydrogen bonds with Asp341(B) (rsk2)) and Asp364(C) (rsk2); two hydrogen bonds are formed by Arg13 (P05) with Asp341(A) (rsk2) and Asp364(B) (rsk2). The simulation results are in good agreement with the previous molecular biological experiments and can explain the binding phenomena between P05 and SK channels at the level of molecular structure. The consistency between the results of the BD simulations and the experimental data indicated that our 3D model of the P05-rsk2 channel complex is reasonable and can be employed in further biological studies, such as rational design of the novel therapeutic agents blocking the small-conductance, calcium-activated and apamin-sensitive potassium channels, and for mutagenesis studies in both toxins and SK channels. In particular, both the BD simulations and the molecular mechanics refinements indicate that residue Asp364 of the rsk2 channel is critical for its recognition and binding functionality towards P05. This phenomenon has not been appreciated in the previous mutagenesis experiments, indicating that this might be a new clue for further functional study of SK channels.     

Weighted-ensemble Brownian dynamics simulations for protein association reactions.

A new method, weighted-ensemble Brownian dynamics, is proposed for the simulation of protein-association reactions and other events whose frequencies of outcomes are constricted by free energy barriers. The method features a weighted ensemble of trajectories in configuration space with energy levels dictating the proper correspondence between "particles" and probability. Instead of waiting a very long time for an unlikely event to occur, the probability packets are split, and small packets of probability are allowed to diffuse almost immediately into regions of configuration space that are less likely to be sampled. The method has been applied to the Northrup and Erickson (1992) model of docking-type diffusion-limited reactions and yields reaction rate constants in agreement with those obtained by direct Brownian simulation, but at a fraction of the CPU time (10(-4) to 10(-3), depending on the model). Because the method is essentially a variant of standard Brownian dynamics algorithms, it is anticipated that weighted-ensemble Brownian dynamics, in conjunction with biophysical force models, can be applied to a large class of association reactions of interest to the biophysics community.     

Brownian dynamics simulations of molecular recognition in an antibody-antigen system.

The crystal structure for an antibody-antigen system, that of the anti-hen egg lysozyme monoclonal antibody HyHEL-5 complexed to lysozyme, is used as the starting point for computer simulations of diffusional encounters between the two proteins. The investigation consists of two parts: first, the linearized Poisson-Boltzmann equation is solved to determine the long-range electrostatic forces between antibody and antigen, and then, the relative motion as influenced by these forces is modeled within Brownian motion theory. The effects of various point mutations on the calculated reaction rate are considered. It is found that charged residues close to the binding site exert the greatest influence in steering the proteins into a configuration favorable for their binding, while more distant mutations are qualitatively described by the Smoluchowski model for the mutual diffusion of two uniformly charged spheres. The antibody residues involved in forming salt links with the lysozyme, Glu-H35 and Glu-H50, appear to be particularly important in electrostatic steering, as neutralization of both of them yields reaction rates that are two to three orders of magnitude below those of wild-type rates. The relative rates obtained from the simulations can be tested through kinetic measurements on mutant protein complexes. Kinetically efficient partners can also be designed and constructed through directed mutagenesis.     

Modeling diverse range of potassium channels with Brownian dynamics

Using the experimentally determined KcsA structure as a template, we propose a plausible explanation for the diversity of potassium channels seen in nature. A simplified model of KcsA is constructed from its atomic resolution structure by smoothing out the protein-water boundary and representing the atoms forming the channel protein as a homogeneous, low dielectric medium. The properties of the simplified and atomic-detail models, deduced from electrostatic calculations and Brownian dynamics simulations, are shown to be qualitatively similar. We then study how the current flowing across the simplified model channel changes as the shape of the intrapore region is modified. This is achieved by increasing the radius of the intracellular pore systematically from 1.5 to 5 A while leaving the dimensions of the selectivity filter and inner chamber unaltered. The strengths of the dipoles located near the entrances of the channel, the carbonyl groups lining the selectivity filter, and the helix macrodipoles are kept constant. The channel conductance increases steadily as the radius of the intracellular pore is increased. The rate-limiting step for both the outward and inward current is the time it takes for an ion to cross the residual energy barrier located in the intrapore region. The current-voltage relationship obtained with symmetrical solutions is linear when the applied potential is less than approximately 100 mV but deviates slightly from Ohm's law at higher applied potentials. The nonlinearity in the current-voltage curve becomes less pronounced as the radius of the intracellular pore is increased. When the strengths of the dipoles near the intracellular entrance are reduced, the channel shows a pronounced inward rectification. Finally, the conductance exhibits the saturation property observed experimentally. We discuss the implications of these findings on the transport of ions across the potassium channels and membrane channels in general.     

Tests of continuum theories as models of ion channels. I. Poisson-Boltzmann theory versus Brownian dynamics

Continuum theories of electrolytes are widely used to describe physical processes in various biological systems. Although these are well-established theories in macroscopic situations, it is not clear from the outset that they should work in small systems whose dimensions are comparable to or smaller than the Debye length. Here, we test the validity of the mean-field approximation in Poisson-Boltzmann theory by comparing its predictions with those of Brownian dynamics simulations. For this purpose we use spherical and cylindrical boundaries and a catenary shape similar to that of the acetylcholine receptor channel. The interior region filled with electrolyte is assumed to have a high dielectric constant, and the exterior region representing protein a low one. Comparisons of the force on a test ion obtained with the two methods show that the shielding effect due to counterions is overestimated in Poisson-Boltzmann theory when the ion is within a Debye length of the boundary. As the ion gets closer to the boundary, the discrepancy in force grows rapidly. The implication for membrane channels, whose radii are typically smaller than the Debye length, is that Poisson-Boltzmann theory cannot be used to obtain reliable estimates of the electrostatic potential energy and force on an ion in the channel environment.     

Normal mode theory for the Brownian dynamics of a weakly bending rod: Comparison with Brownian dynamics simulations

A normal mode theory is developed for the Brownian dynamics of weakly bending rods with preset hydrodynamic interactions. The rod is replaced by a chain of contiguous spheres whose radius is chosen to yield the appropriate uniform translational and rotational diffusion coefficients. Despite the inclusion of preset hydrodynamic interactions in the dynamical operator, its normal modes are not coupled by the potential energy, so their amplitudes remain pairwise “orthogonal” under equilibrium averaging. The uniform translational and rotational diffusion coefficients obtained from Langevin theory are shown to be identical to those obtained from the Kirkwood algorithm, despite their rather different appearance. An expression is given for the mean squared angular displacement 〈Δ_xm(t)²〉 of the mth bond vector around the instantaneous x axis (perpendicular to the end-to-end vector z). Necessary algorithms are presented for the numerical evaluation of all quantities. The normal mode theory is compared with Brownian dynamics simulations for the same model by examining 3〈Δ_xm(t)²〉 for the central bond vector of rods comprising 10 and 30 subunits with various persistence lengths. The normal mode theory works very well for all times for L/P ? 0.6, where P = κ/k_BT is the persistence length and κ is the bending rigidity. With increasing flexibility, the domain of validity of the normal mode theory is restricted to shorter times, where violations of the weak bending approximation are less severe. However, increasing the length of the rod from 10 to 30 subunits yields improved agreement with the simulations for the same and even longer times. This latter effect is tentatively attributed to the greater fluctuating tension in the longer chains, which acts to retard the rotational relaxation in the simulations, but is not taken into account in the present normal mode theory.