Cardiac energy metabolism: models of cellular respiration.

The heart requires a large amount of energy to sustain both ionic homeostasis and contraction. Under normal conditions, adenosine triphosphate (ATP) production meets this demand. Hence, there is a complex regulatory system that adjusts energy production to meet this demand. However, the mechanisms for this control are a topic of active debate. Energy metabolism can be divided into three main stages: substrate delivery to the tricarboxylic acid (TCA) cycle, the TCA cycle, and oxidative phosphorylation. Each of these processes has multiple control points and exerts control over the other stages. This review discusses the basic stages of energy metabolism, mechanisms of control, and the mathematical and computational models that have been used to study these mechanisms.

In situ Ca2+ imaging reveals neurotransmitter receptors for glutamate in taste receptor cells.

The neurotransmitters at synapses in taste buds are not yet known with confidence. Here we report a new calcium-imaging technique for taste buds that allowed us to test for the presence of glutamate receptors (GluRs) in living isolated tissue preparations. Taste cells of rat foliate papillae were loaded with calcium green dextran (CaGD). Lingual slices containing CaGD-labeled taste cells were imaged with a scanning confocal microscope and superfused with glutamate (30 micromter to 1 mm), kainate (30 and 100 micrometer), AMPA (30 and 100 micrometer), or NMDA (100 micrometer). Responses were observed in 26% of the cells that were tested with 300 micrometer glutamate. Responses to glutamate were localized to the basal processes and cell bodies, which are synaptic regions of taste cells. Glutamate responses were dose-dependent and were induced by concentrations as low as 30 microm. The non-NMDA receptor antagonists CNQX and GYKI 52466 reversibly blocked responses to glutamate. Kainate, but not AMPA, also elicited Ca(2+) responses. NMDA stimulated increases in [Ca(2+)](i) when the bath medium was modified to optimize for NMDA receptor activation. The subset of cells that responded to glutamate was either NMDA-unresponsive (54%) or NMDA-responsive (46%), suggesting that there are presumably two populations of glutamate-sensitive taste cells-one with NMDA receptors and the other without NMDA receptors. The function of GluRs in taste buds is not yet known, but the data suggest that glutamate is a neurotransmitter there. GluRs in taste cells might be presynaptic autoreceptors or postsynaptic receptors at afferent or efferent synapses.

Modeling short-term interval-force relations in cardiac muscle.

This study employs two modeling approaches to investigate short-term interval-force relations. The first approach is to develop a low-order, discrete-time model of excitation-contraction coupling to determine which parameter combinations produce the degree of postextrasystolic potentiation seen experimentally. Potentiation is found to increase 1) for low recirculation fraction, 2) for high releasable fraction, i.e., the maximum fraction of Ca(2+) released from the sarcoplasmic reticulum (SR) given full restitution, and 3) for strong negative feedback of the SR release on sarcolemmal Ca(2+) influx. The second modeling approach is to develop a more detailed single ventricular cell model that simulates action potentials, Ca(2+)-handling mechanisms, and isometric force generation by the myofilaments. A slow transition from the adapted state of the ryanodine receptor produces a gradual recovery of the SR release and restitution behavior. For potentiation, a small extrasystolic release leaves more Ca(2+) in the SR but also increases the SR loading by two mechanisms: 1) less Ca(2+)-induced inactivation of L-type channels and 2) reduction of action potential height by residual activation of the time-dependent delayed rectifier K(+) current, which increases Ca(2+) influx. The cooperativity of the myofilaments amplifies the relatively small changes in the Ca(2+) transient amplitude to produce larger changes in isometric force. These findings suggest that short-term interval-force relations result mainly from the interplay of the ryanodine receptor adaptation and the SR Ca(2+) loading, with additional contributions from membrane currents and myofilament activation.

Electrophysiological modeling of cardiac ventricular function: from cell to organ.

Three topics of importance to modeling the integrative function of the heart are reviewed. The first is modeling of the ventricular myocyte. Emphasis is placed on excitation-contraction coupling and intracellular Ca2+ handling, and the interpretation of experimental data regarding interval-force relationships. Second, data on use of diffusion tensor magnetic resonance (DTMR) imaging for measuring the anatomical structure of the cardiac ventricles are presented. A method for the semi-automated reconstruction of the ventricles using a combination of gradient recalled acquisition in the steady state (GRASS) and DTMR images is described. Third, we describe how these anatomically and biophysically based models of the cardiac ventricles can be implemented on parallel computers.

Modeling gain and gradedness of Ca2+ release in the functional unit of the cardiac diadic space.

A model of the functional release unit (FRU) in rat cardiac muscle consisting of one dihydropyridine receptor (DHPR) and eight ryanodine receptor (RyR) channels, and the volume surrounding them, is formulated. It is assumed that no spatial [Ca2+] gradients exist in this volume, and that each FRU acts independently. The model is amenable to systematic parameter studies in which FRU dynamics are simulated at the channel level using Monte Carlo methods with Ca2+ concentrations simulated by numerical integration of a coupled system of differential equations. Using stochastic methods, Ca(2+)-induced Ca2+ release (CICR) shows both high gain and graded Ca2+ release that is robust when parameters are varied. For a single DHPR opening, the resulting RyR Ca2+ release flux is insensitive to the DHPR open duration, and is determined principally by local sarcoplasmic reticulum (SR) Ca2+ load, consistent with experimental data on Ca2+ sparks. In addition, single RyR openings are effective in triggering Ca2+ release from adjacent RyRs only when open duration is long and SR Ca2+ load is high. This indicates relatively low coupling between RyRs, and suggests a mechanism that limits the regenerative spread of RyR openings. The results also suggest that adaptation plays an important modulatory role in shaping Ca2+ release duration and magnitude, but is not solely responsible for terminating Ca2+ release. Results obtained with the stochastic model suggest that high gain and gradedness can occur by the recruitment of independent FRUs without requiring spatial [Ca2+] gradients within a functional unit or cross-coupling between adjacent functional units.

Subcellular [Ca2+]i gradients during excitation-contraction coupling in newborn rabbit ventricular myocytes.

The central role of T-tubule and sarcoplasmic reticulum (SR) diadic junctions in excitation-contraction (EC) coupling in adult (AD) ventricular myocytes suggests that their absence in newborn (NB) cells may manifest as an altered EC coupling phenotype. We used confocal microscopy to compare fluo-3 [Ca2+]i transients in the subsarcolemmal space and cell center of field-stimulated NB and AD rabbit ventricular myocytes. Peak systolic [Ca2+]i occurred sooner and was higher in the subsarcolemmal space compared with the cell center in NB myocytes. In AD myocytes, [Ca2+]i rose and declined with similar profiles at the cell center and subsarcolemmal space. Disabling the SR (10 micromol/L thapsigargin) slowed the rate of rise and decline of Ca2+ in AD myocytes but did not alter Ca2+ transient kinetics in NB myocytes. In contrast to adults, localized SR Ca2+ release events ("Ca2+ sparks") occurred predominantly at the cell periphery of NB myocytes. Immunolabeling experiments demonstrated overlapping distributions of the Na(+)-Ca2+ exchanger and ryanodine receptors (RyR2) in AD myocytes. In contrast, RyR2s were spatially separated from the sarcolemma in NB myocytes. Confocal sarcolemmal imaging of di-8-ANEPPS-treated myocytes confirmed an extensive T-tubule network in AD cells, and that T-tubules are absent in NB myocytes. A mathematical model of subcellular Ca2+ dynamics predicts that Ca2+ flux via the Na(+)-Ca2+ exchanger during an action potential can account for the subsarcolemmal Ca2+ gradients in NB myocytes. Spatial separation of sarcolemmal Ca2+ entry from SR Ca2+ release channels may minimize the role of SR Ca2+ release during normal EC coupling in NB ventricular myocytes.

Cardiac sodium channel Markov model with temperature dependence and recovery from inactivation.

A Markov model of the cardiac sodium channel is presented. The model is similar to the CA1 hippocampal neuron sodium channel model developed by Kuo and Bean (1994. Neuron. 12:819-829) with the following modifications: 1) an additional open state is added; 2) open-inactivated transitions are made voltage-dependent; and 3) channel rate constants are exponential functions of enthalpy, entropy, and voltage and have explicit temperature dependence. Model parameters are determined using a simulated annealing algorithm to minimize the error between model responses and various experimental data sets. The model reproduces a wide range of experimental data including ionic currents, gating currents, tail currents, steady-state inactivation, recovery from inactivation, and open time distributions over a temperature range of 10 degrees C to 25 degrees C. The model also predicts measures of single channel activity such as first latency, probability of a null sweep, and probability of reopening.

Substance P regulates Ih via a NK-1 receptor in vagal sensory neurons of the ferret.

Substance P (SP) hyperpolarizes approximately 80% of ferret vagal sensory neurons (nodose ganglion neurons) via NK-1 receptor-mediated activation of a potassium current (IK). A depolarizing current activated by membrane hyperpolarization could minimize the SP-induced hyperpolarization. Such a current exists in 65% of the nodose neurons (n = 264). In this study, we examine this current and how it can interact with SP-induced membrane hyperpolarizations. This slowly developing, noninactivating inward current, designated Ih, was activated maximally at about -120 mV and had a reversal potential value of -23 +/- 4.4 mV (n = 4). The time course of activation followed voltage-dependent, monoexponential kinetics. Steady-state activation curves derived from tail current analysis were well fit by a Boltzmann equation yielding a half-activation potential (V1/2) of-77 +/- 1.5 mV and a ks value of 18 +/- 0.5 (n = 8). In the presence of 1 mM cesium, the current was completely abolished. These parameters are consistent with those derived for Ih in other neurons. Substance P (200 nM) reduced the magnitude of Ih elicited by membrane hyperpolarizations to about -110 mV but did not affect the magnitude of Ih elicited by hyperpolarizations to more negative potentials. Tail current analysis revealed that this effect was the result of a SP-induced shift of the Ih activation curve to more negative membrane potentials. The V1/2 value for Ih was shifted by -20 +/- 1.4 mV in the presence of SP with no change in ks (18 +/- 0.7; n = 5). The SP effect on Ih, like its effect on IK, was blocked reversibly by 10 nM CP99,994, a NK-1 antagonist, and was mimicked by the NK-1 agonist Ac-[Arg6, Sar9, Met(O2)11]SP(6-11) (ASMSP; 200 nM). Ih was not affected by NK-2 or NK-3 selective agonists (n = 4 for each) nor was the effect of SP on Ih reduced by an NK-2 antagonist (n = 4). These results show that SP activates a NK-1 receptor coupled to the Ih channel. Thus NK-1 receptor activation in ferret vagal afferents not only leads to membrane hyperpolarization but it also can enhance synergistically this inhibitory effect by decreasing Ih.

Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load.

We construct a detailed mathematical model for Ca2+ regulation in the ventricular myocyte that includes novel descriptions of subcellular mechanisms based on recent experimental findings: 1) the Keizer-Levine model for the ryanodine receptor (RyR), which displays adaptation at elevated Ca2+; 2) a model for the L-type Ca2+ channel that inactivates by mode switching; and 3) a restricted subspace into which the RyRs and L-type Ca2+ channels empty and interact via Ca2+. We add membrane currents from the Luo-Rudy Phase II ventricular cell model to our description of Ca2+ handling to formulate a new model for ventricular action potentials and Ca2+ regulation. The model can simulate Ca2+ transients during an action potential similar to those seen experimentally. The subspace [Ca2+] rises more rapidly and reaches a higher level (10-30 microM) than the bulk myoplasmic Ca2+ (peak [Ca2+]i approximately 1 microM). Termination of sarcoplasmic reticulum (SR) Ca2+ release is predominately due to emptying of the SR, but is influenced by RyR adaptation. Because force generation is roughly proportional to peak myoplasmic Ca2+, we use [Ca2+]i in the model to explore the effects of pacing rate on force generation. The model reproduces transitions seen in force generation due to changes in pacing that cannot be simulated by previous models. Simulation of such complex phenomena requires an interplay of both RyR adaptation and the degree of SR Ca2+ loading. This model, therefore, shows improved behavior over existing models that lack detailed descriptions of subcellular Ca2+ regulatory mechanisms.

Agonist-induced calcium waves in oscillatory cells: a biological example of Burgers' equation.

Oscillations in cytosolic Ca2+ concentrations in living cells are often a manifestation of propagating waves of Ca2+. Numerical simulations with a realistic model of inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ wave trains lead to wave speeds that increase linearly at long times when (a) IP3 levels are in the range for Ca2+ oscillations, (b) a gradient of phase is established by either an initial ramp or pulse of IP3, and (c) IP3 concentrations asymptotically become uniform. We explore this phenomenon with analytical and numerical methods using a simple two-variable reduction of the De Young-Keizer model of the IP3 receptor that includes the influence of Ca2+ buffers. For concentrations of IP3 in the oscillatory regime, numerical solution of the resulting reaction diffusion equations produces nonlinear wave trains that shows the same asymptotic growth of wave speed. Due to buffering, diffusion of Ca2+ is quite slow and, as previously noted, these waves occur without appreciable bulk movement of Ca2+. Thus, following Neu and Murray, we explore the behavior of these waves using an asymptotic expansion based on the small size of the buffered diffusion constant for Ca2+. We find that the gradient in phase of the wave obeys Burgers' equation asymptotically in time. This result is used to explain the linear increase of the wave speed observed in the simulations.

Histamine H1 receptor activation blocks two classes of potassium current, IK(rest) and IAHP, to excite ferret vagal afferents.

1. Intracellular recordings were made in intact and acutely dissociated vagal afferent neurones (nodose ganglion cells) of the ferret to investigate the membrane effects of histamine. 2. In current-clamp or voltage-clamp recordings, histamine (10 microM) depolarized the membrane potential (10 +/- 0.8 mV; mean +/- S.E.M.; n = 27) or produced an inward current of 1.6 +/- 0.35 nA (n = 27) in approximately 80% of the neurones. 3. Histamine (10 microM) also blocked the post-spike slow after-hyperpolarization (AHP slow) present in 80% of these neurones (95 +/- 3.2%; n = 5). All neurones possessing AHPslow in ferret nodose were C fibre neurones; all AHPslow neurones had conduction velocities < or = 1 m s-1 (n = 7). 4. Both the histamine-induced inward current and the block of AHPslow were concentration dependent and each had an estimated EC50 value of 2 microM. These histamine-induced effects were mimicked by the histamine H1 receptor agonist 2-(2-aminoethyl) thiazole dihydrochloride (10 microM) and blocked by the H1 antagonists pyrilamine (100 nM) or diphenhydramine (100 nM). Schild plot analysis of the effect of pyrilamine on the histamine-induced inward current revealed a pA2 value of 9.7, consistent with that expected for an H1 receptor. Neither impromidine (10 microM) nor R(-)-alpha-methylhistamine (10 microM), selective H2 or H3 agonists, respectively, significantly affected the membrane potential, input resistance or AHPslow. 5. The reversal potential (Vrev) for the histamine-induced inward current was -84 +/- 2.1 mV (n = 4). The Vrev for the histamine response shifted in a Nernstian manner with changes in the extracellular potassium concentration. Alterations in the extracellular chloride concentration had no significant effect on the Vrev of the histamine response (n = 3). The Vrev for the AHPslow was -85 +/- 1.7 mV (n = 4). 6. These results indicate that histamine increases the excitability of ferret vagal afferent somata by interfering with two classes of potassium current: the resting or 'leak' potassium current (IK(rest)) and the potassium current underlying a post-spike slow after-hyperpolarization (IAHP). Both these effects can occur in the same neurone and involve activation of the same histamine receptor subtype, the histamine H1 receptor.

Ca(2+)-induced Ca2+ release mediates Ca2+ transients evoked by single action potentials in rabbit vagal afferent neurones.

1. Standard intracellular recording techniques with 'sharp' micropipettes were used to evoke action potentials (APs) in acutely dissociated adult nodose neurones. 2. APs induced a transient increase in [Ca2+]i (a calcium transient), recorded with fura-2, that was dependent upon [Ca2+]o and the number of APs. Over the range of one to sixty-five APs, the relation between the amplitude of the calcium transient and the number of APs was well fitted by a rectangular hyperbola (chi 2 = 3.53, r = 0.968). From one to four APs, the calcium transient-AP relation can be described by a line with a slope of 9.6 nM AP-1 (r = 0.999). 3. Charge movement corresponding to Ca2+ influx evoked by a single AP was 39 +/- 2.8 pC (mean +/- S.E.M.) and did not change significantly during trains of one to thirty-one APs (P < 0.05). 4. Caffeine (10 mM), a known agonist of the ryanodine receptor, produced an increase in [Ca2+]i. The caffeine-induced rise in [Ca2+]i was attenuated (by > 90%) by lowering [Ca2+]o, and by ryanodine (10 microM), 2,5-di(t-butyl)hydroquinone (DBHQ, 10 microM), or thapsigargin (100 nM). 5. Neurones incubated with ryanodine, DBHQ or thapsigargin required at least eight APs to evoke a detectable calcium transient. These reagents did not significantly affect Ca2+ influx (P < 0.05). In the presence of these inhibitors, the calcium transient-AP relation exhibited slopes of 1.2, 1.1 and 1.9 nM AP-1 for ryanodine, DBHQ and thapsigargin, respectively. When compared with the slope of 9.6 nM AP-1 in non-treated neurones, it appears that Ca2+ influx produced by a single AP is amplified by ca 5- to 10-fold.

Substance P hyperpolarizes vagal sensory neurones of the ferret.

1. Intracellular recordings were made in intact and in acutely dissociated vagal afferent neurones (nodose ganglion cells) of the ferret to investigate the effects of substance P(SP). 2. In current-clamp recordings, SP (100 nM) applied by superfusion hyperpolarized the membrane potential (7 +/- 0.7 mV; mean +/- S.E.M.; n = 105) and decreased the input resistance in 80% of the neurones. With voltage-clamp recording, SP produced an outward current of 3 +/- 0.2 nA (n = 10). 3. The SP current was concentration dependent with an estimated EC50 of 68 nM. The SP-induced hyperpolarization or current was mimicked by the tachykinin receptor NK1 agonist Ac-[Arg6, Sar9, Met(O2)11]SP(6-11) (ASM-SP; 100 nM; n = 10) and blocked by the NK1 antagonist CP-96,345 (10 nM; n = 6), but not by the NK2 antagonist SR48968 (100 nM; n = 4). No measurable change in membrane potential or input resistance was observed with application of either [beta-Ala8]neurokinin A or senktide, selective NK2 and NK3 receptor agonists, respectively (100 nM; n = 3 for each agonist). 4. The reversal potential (Erev) for the SP outward current was -85 +/- 2.5 mV (n = 4). The Erev for the SP response shifted in a Nernstian manner with changes in extracellular potassium concentration. Alterations in extracellular sodium or chloride concentrations had no significant effect on the Erev for the SP response (n = 3 for each ion). 5. Nominally Ca(2+)-free external solution abolished the SP response. Removal of magnesium from the extracellular solution had no effect on the response. 6. Caesium (100 microM), barium (1 mM), tetraethylammonium (TEA; 5 mM), apamin (10 nM) and 4-aminopyridine (4-AP; 4 mM) each completely prevented the SP response (n > or = 3 for each). 7. These results indicate that SP, via an NK1 receptor, can induce a Ca(2+)-dependent outward potassium current which hyperpolarizes the resting membrane potential of vagal afferent somata.

On the roles of Ca2+ diffusion, Ca2+ buffers, and the endoplasmic reticulum in IP3-induced Ca2+ waves.

We have investigated the effects of Ca2+ diffusion, mobile and stationary Ca2+ buffers in the cytosol, and Ca2+ handling by the endoplasmic reticulum on inositol 1,4,5-trisphosphate-induced Ca2+ wave propagation. Rapid equilibration of free and bound Ca2+ is used to describe Ca2+ sequestration by buffers in both the cytosol and endoplasmic reticulum (ER) lumen. Cytosolic Ca2+ regulation is based on a kinetic model of the inositol 1,4,5-trisphosphate (IP3) receptor of De Young and Keizer that includes activation and inhibition of the IP3 receptor Ca2+ channel in the ER membrane and SERCA Ca2+ pumps in the ER. Diffusion of Ca2+ in the cytosol and the ER and the breakdown and diffusion of IP3 are also included in our calculations. Although Ca2+ diffusion is severely limited because of buffering, when conditions are chosen just below the threshold for Ca2+ oscillations, a pulse of IP3 or Ca2+ results in a solitary trigger wave that requires diffusion of Ca2+ for its propagation. In the oscillatory regime repetitive wave trains are observed, but for this type of wave neither the wave shape nor the speed is strongly dependent on the diffusion of Ca2+. Local phase differences lead to waves that are predominately kinematic in nature, so that the wave speed (c) is related to the wavelength (lambda) and the period of the oscillations (tau) approximately by the formula c = lambda/tau. The period is determined by features that control the oscillations, including [IP3] and pump activity, which are related to recent experiments. Both solitary waves and wave trains are accompanied by a Ca2+ depletion wave in the ER lumen, similar to that observed in cortical preparations from sea urchin eggs. We explore the effect of endogenous and exogenous Ca2+ buffers on wave speed and wave shape, which can be explained in terms of three distinct effects of buffering, and show that exogenous buffers or Ca2+ dyes can have considerable influence on the amplitude and width of the waves.

A theoretical study of cytosolic calcium waves in Xenopus oocytes.

Calcium waves have been observed in Xenopus oocytes through the use of confocal fluorescence microscopy. The waves assume many patterns including plane waves, target patterns and spirals. Modification of the membrane model (Jafri et al., 1992) to include diffusion terms in two dimensions reproduces planar and circular propagating waves. A one-dimensional model is used to study the properties of these waves. The model predicts that increasing the buffering capacity of the cytosol decreases the wave amplitude and speed. A decrease in the wave speed is also observed when the net calcium entry rate into the cytosol decreases. The model also predicts that as the diffusion constant is increased, the wave speed increases.

Diffusion of inositol 1,4,5-trisphosphate but not Ca2+ is necessary for a class of inositol 1,4,5-trisphosphate-induced Ca2+ waves.

Combining a realistic model of inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ oscillations with the diffusion of IP3 and buffered diffusion of Ca2+, we have found that diffusion of Ca2+ plays only a minor role in a class of agonist-induced Ca2+ wave trains. These waves are primarily kinematic in nature, with variable wavelengths and speeds that depend primarily on the phase differences between oscillators at different spatial points. The period is set by the steady-state value of IP3, while the wave speed approximately equals the wavelength/period. Ca2+ diffusion, which is much slower than that of IP3 because of endogenous buffers, is shown to have only a small effect on the wave trains and not to be necessary for the apparent wave propagation. Diffusion of IP3 sets the phase gradient responsible for these wave trains, which consist primarily of localized cycles of Ca2+ uptake and release. Our results imply a possible previously undisclosed role for IP3 in cell signaling.

A membrane potential model with counterions for cytosolic calcium oscillations.

An initial model has been proposed to describe a mechanism for cytosolic calcium oscillations [Jafri MS. Vajda S. Pasik P. Gillo B. (1992) A membrane model for cytosolic calcium oscillations: a study using Xenopus oocytes. Biophys. J., 63, 235-246]. In this paper we extend our original model to include the effects of counterion movement into the ER in response to calcium release. This produces smoother oscillations over a wider parameter range. We have lowered the endoplasmic reticulum (ER) intraluminal free calcium concentration and shown that the oscillations can occur at lower ER membrane potentials, consistent with physiological values. The improved model is then tested with two representative paradigms that are currently under investigation by many researchers. The model predicts that the reduction of the ER calcium pump (Ca-ATPase) rate can cause the termination of cytosolic calcium oscillations in an active cell, and induce oscillations in a resting cell. This result is consistent with experiments with thapsigargin, a Ca-ATPase activity inhibitor. In addition, we simulate the latency period for the response to the application of agonist and offer a plausible explanation for it. Our mathematical model is currently the only model that formulates the contributions of calcium binding proteins, ER membrane potential, ER counterion movements, and distinct calcium pump populations, and describes their effects on cytosolic calcium oscillations.

Necessary conditions for avoiding incorrect polypeptide folds in conformational search by energy minimization.

Low energy conformations have been generated for melittin, pancreatic polypeptide, and ribonuclease S-peptide, both in the vicinity of x-ray structures by energy refinement and by an unconstrained search over the entire conformational space. Since the structural polymorphism of these medium-sized peptides in crystal and solution is moderate, comparing the calculated conformation to x-ray and nmr data provides information on local and global behavior of potential functions. Local analysis includes standardization calculations, which show that models with standard geometry can approximate good resolution x-ray data with less than 0.5 A rms deviation (RMSD). However, the atomic coordinates are shifted up to 2 A RMSD by local energy minimization, and thus 2 A is generally the smallest RMSD value one can target in a conformational search using the same energy evaluation models. The unconstrained search was performed by a buildup-type method based on dynamic programming. To accelerate the generation of structures in the conformational search, we used the ECEPP potential, defined in terms of standard polypeptide geometry. A number of low energy conformations were further refined by relaxing the assumption of standard bond lengths and bond angles through the use of the CHARMM potential, and the hydrophobic folding energies of Eisenberg and McLachlan were calculated. Each conformation is described in terms of the RMSD from the native, hydrogen-bonding structure, solvent-accessible surface area, and the ratio of surfaces corresponding to nonpolar and polar residues. The unconstrained search finds conformations that are different from the native, sometimes substantially, and in addition, have lower conformational energies than the native. The origin of deviations is different for each of the three peptides, but in all examples the refined x-ray structures have lower energies than the calculated incorrect folds when (1) the assumption of standard bond lengths and bond angles is relaxed; (2) a small and constant effective dielectric permittivity (epsilon < 10) is used; and (3) the hydrophobic folding energy is incorporated into the potential.

Disulfide bonding patterns and protein topologies.

This paper examines the topological properties of protein disulfide bonding patterns. First, a description of these patterns in terms of partially directed graphs is developed. The topologically distinct disulfide bonding patterns available to a polypeptide chain containing n disulfide bonds are enumerated, and their symmetry and reducibility properties are examined. The theoretical probabilities are calculated that a randomly chosen pattern of n bonds will have any combination of symmetry and reducibility properties, given that all patterns have equal probability of being chosen. Next, the National Biomedical Research Foundation protein sequence and Brookhaven National Laboratories protein structure (PDB) databases are examined, and the occurrences of disulfide bonding patterns in them are determined. The frequencies of symmetric and/or reducible patterns are found to exceed theoretical predictions based on equiprobable pattern selection. Kauzmann's model, in which disulfide bonds form during random encounters as the chain assumes random coil conformations, finds that bonds are more likely to form with near neighbor cysteines than with remote cysteines. The observed frequencies of occurrence of disulfide patterns are found here to be virtually uncorrelated with the predictions of this alternative random bonding model. These results strongly suggest that disulfide bond pattern formation is not the result of random factors, but instead is a directed process. Finally, the PDB structure database is examined to determine the extrinsic topologies of polypeptides containing disulfide bonds. A complete survey of all structures in the database found no instances in which two loops formed by disulfide bonds within the same polypeptide chain are topologically linked. Similarly, no instances are found in which two loops present on different polypeptide chains in a structure are catenated. Further, no examples of topologically knotted loops occur. In contrast, pseudolinking has been found to be a relatively frequent event. These results show a complete avoidance of nontrivial topological entanglements that is unlikely to be the result of chance events. A hypothesis is presented to account for some of these observations.