The Influence of Look-Ahead on the Error Rate of Transcription.

In this paper we study the error rate of RNA synthesis in the look-ahead model for the random walk of RNA polymerase along DNA during transcription. The model's central assumption is the existence of a window of activity in which ribonucleoside triphosphates (rNTPs) bind reversibly to the template DNA strand before being hydrolyzed and linked covalently to the nascent RNA chain. An unknown, but important, integer parameter of this model is the window size w. Here, we use mathematical analysis and computer simulation to study the rate at which transcriptional errors occur as a function of w. We find dramatic reduction in the error rate of transcription as w increases, especially for small values of w. The error reduction method provided by look-ahead occurs before hydrolysis and covalent linkage of rNTP to the nascent RNA chain, and is therefore distinct from error correction mechanisms that have previously been considered.

Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions.

Blood flow in the large systemic arteries is modeled using one-dimensional equations derived from the axisymmetric Navier-Stokes equations for flow in compliant and tapering vessels. The arterial tree is truncated after the first few generations of large arteries with the remaining small arteries and arterioles providing outflow boundary conditions for the large arteries. By modeling the small arteries and arterioles as a structured tree, a semi-analytical approach based on a linearized version of the governing equations can be used to derive an expression for the root impedance of the structured tree in the frequency domain. In the time domain, this provides the proper outflow boundary condition. The structured tree is a binary asymmetric tree in which the radii of the daughter vessels are scaled linearly with the radius of the parent vessel. Blood flow and pressure in the large vessels are computed as functions of time and axial distance within each of the arteries. Comparison between the simulations and magnetic resonance measurements in the ascending aorta and nine peripheral locations in one individual shows excellent agreement between the two.

Recovery of cable properties through active and passive modeling of subthreshold membrane responses from laterodorsal tegmental neurons.

The laterodorsal tegmental nucleus (LDT) is located in the dorsolateral pontine reticular formation. Cholinergic neurons in the LDT and the adjacent pedunculopontine tegmental nucleus (PPT) are hypothesized to play a critical role in the generation of the electroencephalographic-desynchronized states of wakefulness and rapid eye movement sleep. A quantitative analysis of the cable properties of these cells was undertaken to provide a more detailed understanding of their integrative behavior. The data used in this analysis were the morphologies of intracellularly labeled guinea pig LDT neurons and the voltage responses of these cells to somatic current injection. Initial attempts to model the membrane behavior near resting potential and in the presence of tetrodotoxin (TTX, 1 microM) as purely passive produced fits that did not capture many features of the experimental data. Moreover, the recovered values of membrane conductance or intracellular resistivity were often very far from those reported for other neurons, suggesting that a passive description of cell behavior near rest was not adequate. An active membrane model that included a subthreshold A-type K+ current and/or a hyperpolarization-activated cation current (H-current) then was used to model cell behavior. The voltage traces calculated using this model were better able to reproduce the experimental data, and the cable parameters determined using this methodology were more consistent with those reported for other cells. Additionally, the use of the active model parameter extraction methodology eliminated a problem encountered with the passive model in which parameter sets with widely varying values, sometimes spanning an order of magnitude or more, would produce effectively indistinguishable fits to the data. The use of an active model to directly fit the experimentally measured voltage responses to both long and short current pulses is a novel approach that is of general utility.

Simulating the role of microtubules in depolymerization-driven transport: a Monte Carlo approach.

In this paper we present a model that simulates the role of microtubules in depolymerization-driven transport. The model simulates a system that consists of a 13-protofilament microtubule with "five-start" helical structure and a motor protein-coated bead that moves along one of the protofilaments of the microtubule, as in in vitro experiments. The microtubule is simulated using the lateral cap model, with substantial generalizations. For the new terminal configurations in the presence of the bead, rate constants for association and dissociation events of tubulin molecules are calculated by exploring the geometric similarities between different patterns of terminal configurations and by decomposing complex patterns into simpler patterns whose corresponding rate constants are known. In comparison with a previous model, in which simplifications are made about the structure of the microtubule and in which the microtubule can only depolymerize, the detailed structure of the microtubule is taken into account in the present model. Furthermore, the microtubule can be either polymerizing or depolymerizing. Force-velocity curves are obtained for both zero and non-zero tubulin guanosine 5'-triphosphate (GTP) concentrations. By analyzing the trajectory of the bead under different parameters, the condition for "run and pause" is analyzed, and the time scale of "run" and "pause" is found to be different for different motor proteins. We also suggest experiments that can be used to examine the results predicted by the model.

Quantitative morphology of physiologically identified and intracellularly labeled neurons from the guinea-pig laterodorsal tegmental nucleus in vitro.

Mesopontine cholinergic neurons have been implicated in the initiation and maintenance of rapid eye movement sleep via their efferent connections to the thalamus and the medial pontine reticular formation. As a first step toward understanding how these modulatory neurons integrate synaptic input, we have investigated the dendritic architecture of laterodorsal tegmental nucleus neurons. The principal cells of the guinea-pig laterodorsal tegmental nucleus were identified electrophysiologically in a brain slice preparation, then were intracellularly injected with biocytin and reconstructed using a computer-aided tracing system. The somata were large (27 +/- 3 microns; n = 11) and gave rise to an average of 4.8 primary dendrites which, in most cases, emerged from the soma in a pattern that was radially symmetric in the plane of the slice. Primary dendrites had an average of 3.7 endings. A single axon arose from either the soma or a proximal dendrite and exited the nucleus with a medial and/or lateral trajectory. Some axons also gave rise to a local terminal plexus composed of fine fibers bearing numerous punctate swellings that ramified profusely within the neuron's dendritic field. Total dendritic area averaged about 10(5) microns2, and therefore the average contribution of the soma to the total surface area (20%) was significantly larger than the values reported for many other cell types. Dendritic diameters were non-uniform in three respects. Some processes were sparsely spiny. Most processes were varicose, with the degree of varicosity increasing substantially in secondary and tertiary dendritic segments. There was also a large degree of taper in dendritic processes; those processes with a non-negative taper had an average diameter decrease of 40 +/- 25%. Dendritic processes deviated from the criteria necessary for a Rall equivalent cylinder approximation due to non-uniformity in morphotonic path length, failure to conform to the Rall 3/2 branching rule and non-uniformity of dendritic diameter. An analysis was done to assess the impact of dendritic varicosities on the extraction of cable parameters for these cells. Voltage traces were simulated by solving the cable equation for a varicose dendrite and then membrane parameters were recovered using an equivalent cylinder model. Errors in the extracted values of specific membrane conductance and specific membrane capacitance were quite small (< or = 5%), while larger errors were seen for electrotonic length (< or = 21%) and intracellular resistivity (< or = 5%). These data indicate that the principal cells of the laterodorsal tegmental nucleus, while possessing a relatively simple dendritic structure in terms of number and branchiness of dendrites, display a heterogeneity of dendritic process types. Processes range from smooth to markedly varicose, and can be aspiny or sparsely spiny. The possibility that the dendritic varicosities function as sites of either electrical or chemical compartmentalization is discussed. The degree of error resulting from a Rall equivalent cylinder approximation in light of these varicosities indicated that a generalized cable model approach may prove more effective in estimating their cable parameters.

Force production by depolymerizing microtubules: load-velocity curves and run-pause statistics.

Experiments indicate that depolymerization of microtubules generates sufficient force to produce the minus-end-directed transport of chromosomes during mitosis (Koshland et al., 1988). In vitro, analogous transport of kinesin-coated microspheres exhibits a paradoxical effect. Minus-end-directed transport of the microspheres driven by depolymerization is enhanced by the presence of ATP, which fuels the motor action of kinesin driving the microspheres in the opposite direction, toward the plus end of the microtubule. Here we present a mathematical model to explain this behavior. We postulate that a microsphere at the plus end of the microtubule facilitates depolymerization and hence enhances minus-end-directed transport. The force-velocity curve of the model is derived; it has the peculiar feature that velocity is maximal at some positive load (opposing the motion) rather than at zero load. The model is used to simulate the stochastic process of microsphere-facilitated depolymerization-driven transport. Simulated trajectories at low load show distinctive runs and pauses, the statistics of which are calculated from the model. The statistics of the process provide sufficient information to determine all of the model's parameters.

Coordinated hydrolysis explains the mechanical behavior of kinesin.

The two-headed motor protein kinesin hydrolyzes nucleotide to move unidirectionally along its microtubule track at speeds up to 1000 nm/s (Saxton et al., 1988) and develops forces in excess of 5 pN (Hunt et al., 1994; Svoboda et al., 1994a). Individual kinesin molecules have been studied recently in vitro, and their behavior has been characterized in terms of force-velocity curves and variance measurements (Svoboda and Block, 1994a; Svoboda et al., 1994b). We present a model for force generation in kinesin in which the ATP hydrolysis reactions are coordinated with the relative positions of the two heads. The model explains the experimental data and permits us to study the relative roles of Brownian motion and elastic deformation in the motor mechanism of kinesin.

A general method for the computer simulation of biological systems interacting with fluids.

At this Symposium on Biological Fluid Dynamics, it is appropriate to ask whether there is any common theme that unites the diverse problems that arise in the study of living systems interacting with fluids. The answer that immediately comes to mind is this: biological fluid dynamics invariably involves the interaction of elastic flexible tissue with viscous incompressible fluid. (In many cases the tissue is not only elastic, it is also active, i.e. capable of doing work on the fluid). This paper describes the immersed boundary method, which is a general framework for the computer simulation of biofluid dynamic systems. This method has already been applied to blood flow in the heart (including the computer-assisted design of prosthetic cardiac valves), platelet aggregation during blood clotting, aquatic animal locomotion, wave propagation along the basilar membrane of the inner ear, and flow in collapsible tubes. In the immersed boundary method, the elastic (and possibly active) biological tissue is treated as a part of the fluid in which additional forces (derived from the tissue stresses) are applied. Because the tissue is represented in terms of its force field, the method remains straightforward, even when the geometry of the biological tissue is complicated, dynamic and not known in advance.

Mechanical equilibrium determines the fractal fiber architecture of aortic heart valve leaflets.

In this work, the structure of the aortic valve is derived from its function, which (in the closed-valve configuration) is to support a uniform pressure load. It is assumed that this load is transferred to the aortic wall by a one-parameter family of fibers under tension. The equation of equilibrium for this fiber structure turns out to be equivalent to the equation of motion of vortex lines in the self-induction approximation. The method of Buttke (J. Comput. Phys. 76:301-326, 1988) is used to solve these equations and, hence, to determine the fiber architecture of the aortic leaflets. Because of a singularity at the center of the aortic valve, the computed fiber architecture has a fractal character with increasing complexity at progressively smaller scales. The computed fiber architecture resembles the branching braided structure of the collagen fibers that support the real aortic valve.

Cellular motions and thermal fluctuations: the Brownian ratchet.

We present here a model for how chemical reactions generate protrusive forces by rectifying Brownian motion. This sort of energy transduction drives a number of intracellular processes, including filopodial protrusion, propulsion of the bacterium Listeria, and protein translocation.

What drives the translocation of proteins?

We propose that protein translocation across membranes is driven by biased random thermal motion. This "Brownian ratchet" mechanism depends on chemical asymmetries between the cis and trans sides of the membrane. Several mechanisms could contribute to rectifying the thermal motion of the protein, such as binding and dissociation of chaperonins to the translocating chain, chain coiling induced by pH and/or ionic gradients, glycosylation, and disulfide bond formation. This helps explain the robustness and promiscuity of these transport systems.

Cardiac fluid dynamics.

The heart is modeled as a system of elastic and/or contractile fibers immersed in a viscous incompressible fluid. Simulated heart walls and valves are constructed by arranging the fibers according to an idealized version of the actual distribution of muscle fibers in the heart walls and collagen fibers in the valve leaflets. Then the combined motion of the fluid-fiber system is predicted through the numerical solution of its coupled equations of motion. Fluid equations are solved by a finite difference method on a fixed, regular computational lattice. Fiber points move freely through this lattice without being constrained to lie at the lattice intersections. Communication between fibers and fluid involves interpolation of the fluid velocity to the fiber points and the spreading of the fiber forces to the computational lattice of the fluid. Both of these operations make use of a smoothed approximation to the Dirac delta function. The entire method is suitable for implementation on vector, parallel, or parallel-vector hardware. Applications include the investigation of normal cardiac function, the simulation of disease processes affecting the mechanical function of the heart or its valves, and the computer-assisted design of prosthetic cardiac valves.

The derivation and verification of a non-stationary, optimal smoothing filter for nuclear medicine image data.

A non-stationary optimal smoothing filter for digital nuclear medicine image data, degraded by Poisson noise, has been derived and applied to temporal simulated and clinical gated blood pool study (GBPS) data. The derived filter is automatically calculated from a large group (library) of similar GBPS which are representative of all studies acquired according to the same protocol in a defined patient population (the ensemble). The filter is designed to minimize the mean-square difference between the filtered data and the true image values; it provides an optimal trade-off between noise reduction and signal degradation for members of the ensemble. The filter is evaluated using a computer simulated ensemble of GBPS. Libraries of Poisson-degraded and non-degraded studies were generated. Libraries of up to 400 Poisson-degraded simulated studies were used to estimate optimal temporal filters that, when applied to Poisson-degraded members of the ensemble not included in the libraries, reduced the mean-square error in the raw data by 65%. When the non-degraded studies were used instead to compute the optimal filter values, the corresponding reduction in the error was 83%. Libraries of previously acquired clinical GBPS were then used to estimate optimal temporal filters for an ensemble of similarly acquired studies. These filters were subsequently applied to studies of 13 patients (not in the original libraries) who received multiple sequential repeat studies. Comparisons of both the filtered and raw data to averages of the repeat studies demonstrated that optimal filters calculated from 400 and 800 clinical studies reduced the mean-square error in the clinical data by 56% and 63% respectively.

A C version of Fourier-derived affinity spectrum analysis (FASA) to resolve binding heterogeneity.

The computer program described in this paper facilitates resolution of binding affinity heterogeneity by transforming binding curve data (bound versus free) into affinity spectra (density versus affinity). The original program, written in FORTRAN, is extended and presented here in the language C. New applications include an ability to transform competition curves into affinity spectra and to evaluate the effects of sampling and experimental error on spectrum analysis. We propose that this program be incorporated in the routine evaluation of binding systems.

Hemodynamics in transposition of the great arteries with comparison to ventricular septal defect.

This paper uses a mathematical model of the circulations to study the hemodynamics of transposition of the great arteries (TGA) with comparison to ventricular septal defect (VSD). Computer experiments are conducted to determine the influence of the defect conductance and the pulmonary vascular conductance on the pulsatile pressures, flows, and oxygen concentrations of the circulation. In particular, the model is used to determine the waveform of the (possibly bidirectional) shunt through the ventricular and atrial septal defects. The results of the computer experiments consist of two parts. The first set of experiments is devoted to the comparison of VSD and TGA with a ventricular septal defect. The results are theoretical in the sense that most parameters have been fixed at the same levels. In each case TGA is represented by changing the connection of the chambers and reversing the compliance of the two ventricles. In the second set of experiments we attempt to simulate conditions clinically observed in a variety of cases of TGA. In each case we use clinical observations to infer parameters as the input to the model. We find that the model (with appropriate choice of parameters) generally exhibits blood pressure, blood flows and oxygen concentrations similar to the clinical observations. As a byproduct of these computer experiments we predict the effects of changing the pulmonary conductance. The comparison between TGA and VSD shows that as the defect conductance increases, the systemic oxygen concentrations decrease in VSD and increase in TGA. Even at large defect conductance, the two conditions remain distinct, however, since the mixing of the right and left ventricular blood pools is incomplete. This phenomenon of incomplete mixing sets quantitative limits on the benefits that can be achieved by surgical enlargement of the defect. A result of this study that may be useful in the management of TGA patients with a ventricular septal defect is the finding that there is a value of the pulmonary conductance that maximizes the effective flow and hence the systemic oxygen concentrations. The optimal pulmonary conductance is approximately equal to the systemic conductance when the defect is large.

A "give" in tension and sarcomere dynamics in cardiac muscle relaxation.

Isometric relaxation in cardiac sarcomeres is characterised by an early, very slow phase of tension fall which is terminated by a 'give' in tension. A 'give' which occurs during relaxation in cardiac muscle can not be attributed to decrease in myofilament overlap. After the 'give' asynchronous motion occurs between sarcomeres, but the duration and extent of their displacement is limited. Intriguingly, the effect of isotonic displacements on the early fall in the velocity of sarcomere shortening indicates that an internal resistance increases near the peak of contraction. The complex shape of the sarcomere's complete force-velocity relation, with lengthening motions in particular, was consistent with an idealized model of cross-bridge cycling. The sarcomere's resistance to stretch is high at low velocity, but it diminishes to reveal yielding at larger velocities. Relative to tension, the resistance to yielding does not decrease during relaxation, and it may actually increase. The decay of isometric tension after a controlled stretch also slows during relaxation. Consequently, cycling slows in those cross-bridges which form (or persist but produce less force) later in contraction. Changes in cross-bridge properties may restrict sarcomere shortening, prolong activation, but promote a disequilibrium which favors rapid relaxation in cardiac muscle.

Light adaptation in the turtle retina: embedding a parametric family of linear models in a single nonlinear model.

A method for constructing nonlinear models for light adaptation in the retina is introduced. The components of the models are linear filters and static (instantaneous) nonlinear elements configured in a feedback arrangement. The signals in the models are combined through algebraic addition or multiplication. We apply the method to model light adaptation measured in turtle horizontal cells. Given a particular wiring diagram for the components, the functional forms of the static nonlinearities and frequency responses of the linear filters are determined by constraining the model to give temporal frequency responses (linear regime behavior) consistent with a family of linear feedback models which has been shown to provide a good description of adaptation in these cells. Two particular models, quite different in structure, are presented. Each model responds to perturbations around a mean light level as a feedback circuit in which the gain (strength) of feedback is adjusted to be proportional to the mean light level, but neither model has a separate pathway for measuring the mean light level. Thus, each of these nonlinear models embeds an entire family of linear models parametric in mean light level. Harmonic distortion in the responses of these models to sinusoidal input is found to be qualitatively consistent with physiological data. An alternative class of nonlinear models in which feedback gain is set by a separate slow pathway which tracks the mean light level is ruled out on the basis of its incorrect steady-state input-output behavior. The methods presented can be used to develop specific physical models for light adaptation based on the chemical kinetics of phototransduction or on nonlinear neural feedback. The relevance of the nonlinear models and construction techniques to modeling phototransduction is discussed.

Resolution of steroid binding heterogeneity by Fourier-derived affinity spectrum analysis (FASA).

A new mathematical method of analyzing radioreceptor assay data is presented. When there are many binding classes with different affinities, the probability-density function B(p) is described by the equation B(p) = (integral negative infinity to infinity) q(k)f(p-k)dk, where q(k) is the affinity spectrum (density of a particular binding class as a function of affinity) and f(p-k) is a probability function (probability that dissociation constants will fall between k and p-k, where p is the free ligand concentration). This equation is solved for q(k) and evaluated explicitly by Fourier transformation, namely, q(w) = b(w)/f(w), where w is frequency. Since division by f(w) can amplify and high frequency noise present in the experimental data, a Gaussian smoothing function is introduced thus: qs(w) = q(w)e(-w/W0)2, where W0 is a constant. This produces an affinity spectrum defined as a plot of the number of binding sites, qs(k), versus their respective dissociation constants, k. Using a FORTRAN computer program, we verify this algorithm using simulated data. We also apply the procedure to resolve heterogeneous populations of estrogen binders in human endometrium using [3H]estradiol as ligand. Two estrogen binder classes are revealed with dissociation constants approximately 2.5 natural logarithmic units apart. We identify one high-affinity (Kd = 0.18 nM)-low density (70 pM [or 72 fmol/mg protein]) subpopulation and one low affinity (Kd = 2.5 nM)-high density (101 pM [or 102 fmol/mg protein]) subpopulation of estradiol binders. The management of experimental error, sampling limitations, and nonspecific binding are discussed. This method directly transforms experimental data into an easily interpretable representation without mathematical modeling or statistical procedures.

Hemodynamics in congenital heart disease.

This paper introduces a very general and flexible model for the study of hemodynamic changes in congenital heart disease. The generality of the model makes it possible to use the same computer program (which is included in an Appendix) to study both the fetal circulation and the adult circulation, as well as such diverse disease states as patent ductus arteriosus, ventricular septal defect, atrial septal defect, tetralogy of Fallot and transposition of the great arteries. In this paper, only patent ductus and ventricular or atrial septal defect are studied, with special emphasis on the influence of increasing pulmonary vascular resistance on the shunt flow. In the case of patent ductus and ventricular septal defect, the computed shunt flow is very time-dependent and the left-to-right shunt becomes first bidirectional and then right-to-left as the pulmonary resistance increases. By contrast, the computed shunt flow of atrial septal defect is nearly time-independent and is also somewhat less sensitive to the pulmonary vascular resistance.

Effects of timing of atrial systole on LV filling and mitral valve closure: computer and dog studies.

Atrioventricular (AV) delay that results in maximum ventricular filling and physiological mechanisms that govern dependence of filling on timing of atrial systole were studied by combining computer experiments with experiments in the anesthetized dog instrumented to measure phasic mitral flow. Ventricular filling volume is maximized at AV delay of 100 ms in the computer study and 80 ms in the dog study. At any time in diastole atrial contraction accelerates mitral flow, opening the mitral valve widely; atrial relaxation then decelerates mitral flow, moving the valve leaflets toward closure. The time the valve remains closed following atrial systole varies inversely with AV delay. When AV delay is optimal, the mitral valve is moving rapidly toward closure but is not yet closed at onset of ventricular systole. The decline in filling volume as AV delay decreases below its optimum value is primarily the result of premature termination of atrial ejection by ventricular systole. As AV delay increases above its optimal value, filling volume progressively decreases because of premature mitral valve closure that limits effective diastolic filling period. There is no significant retrograde mitral flow at any point in diastole for any AV delay.