A Bloch equation approach to intensity dependent optical spectra of light harvesting complex II

On the basis of the recent progress in the resolution of the structure of the antenna light harvesting complex II (LHC II) of the photosystem II, we propose a microscopically motivated theory to predict excitation intensity-dependent spectra. We show that optical Bloch equations provide the means to include all 2( N ) excited states of an oligomer complex of N coupled two-level systems and analyze the effects of Pauli Blocking and exciton-exciton annihilation on pump-probe spectra. We use LHC Bloch equations for 14 Coulomb coupled two-level systems, which describe the S (0) and S (1) level of every chlorophyll molecule. All parameter introduced into the Hamiltonian are based on microscopic structure and a quantum chemical model. The derived Bloch equations describe not only linear absorption but also the intensity dependence of optical spectra in a regime where the interplay of Pauli Blocking effects as well as exciton-exciton annihilation effects are important. As an example, pump-probe spectra are discussed. The observed saturation of the spectra for high intensities can be viewed as a relaxation channel blockade on short time scales due to Pauli blocking. The theoretical investigation is useful for the interpretation of the experimental data, if the experimental conditions exceed the low intensity pump limit and effects like strong Pauli Blocking and exciton-exciton annihilation need to be considered. These effects become important when multiple excitations are generated by the pump pulse in the complex.     

Optimal HIV treatment by maximising immune response

We present an optimal control model of drug treatment of the human immunodeficiency virus (HIV). Our model is based upon ordinary differential equations that describe the interaction between HIV and the specific immune response as measured by levels of natural killer cells. We establish stability results for the model. We approach the treatment problem by posing it as an optimal control problem in which we maximise the benefit based on levels of healthy CD4+ T cells and immune response cells, less the systemic cost of chemotherapy. We completely characterise the optimal control and compute a numerical solution of the optimality system via analytic continuation.Research supported by the Natural Science and Engineering Research Council (NSERC) and the Mathematics of Information Technology and Complex Systems (MITACS) of Canada     

Improving the time efficiency of the Fourier synthesis method for slice selection in magnetic resonance imaging

The design of slice selective pulses for magnetic resonance imaging can be cast as an optimal control problem. The Fourier synthesis method is an existing approach to solve these optimal control problems. In this method the gradient field as well as the excitation field are switched rapidly and their amplitudes are calculated based on a Fourier series expansion. Here, we provide a novel insight into the Fourier synthesis method via representing the Bloch equation in spherical coordinates. Based on the spherical Bloch equation, we propose an alternative sequence of pulses that can be used for slice selection which is more time efficient compared to the original method. Simulation results demonstrate that while the performance of both methods is approximately the same, the required time for the proposed sequence of pulses is half of the original sequence of pulses. Furthermore, the slice selectivity of both sequences of pulses changes with radio frequency field inhomogeneities in a similar way. We also introduce a measure, referred to as gradient complexity, to compare the performance of both sequences of pulses. This measure indicates that for a desired level of uniformity in the excited slice, the gradient complexity for the proposed sequence of pulses is less than the original sequence.     

Free Terminal Time Optimal Control Problem of an HIV Model Based on a Conjugate Gradient Method

The minimum duration of treatment periods and the optimal multidrug therapy for human immunodeficiency virus (HIV) type 1 infection are considered. We formulate an optimal tracking problem, attempting to drive the states of the model to a “healthy” steady state in which the viral load is low and the immune response is strong. We study an optimal time frame as well as HIV therapeutic strategies by analyzing the free terminal time optimal tracking control problem. The minimum duration of treatment periods and the optimal multidrug therapy are found by solving the corresponding optimality systems with the additional transversality condition for the terminal time. We demonstrate by numerical simulations that the optimal dynamic multidrug therapy can lead to the long-term control of HIV by the strong immune response after discontinuation of therapy.     

Theory of photoselection by intense light pulses. Influence of reorientational dynamics and chemical kinetics on absorbance measurements.

The theory of absorbance measurements on a system (e.g., chromophore(s) in a protein) that undergoes a sequence of reactions initiated by a linearly polarized light pulse is developed for excitation pulses of arbitrary intensity. This formalism is based on a set of master equations describing the time evolution of the orientational distribution function of the various species resulting from excitation, reorientational dynamics, and chemical kinetics. For intense but short excitation pulses, the changes in absorbance (for arbitrary polarization directions of the excitation and probe pulses) and the absorption anisotropy are expressed in terms of reorientational correlation functions. The influence of the internal motions of the chromophore as well as the overall motions of the molecules is considered. When the duration of the excitation pulse is long compared to the time-scale of internal motions but comparable to the overall correlation time of the molecule that is reorienting isotropically, the problem of calculating the changes in absorbance is reduced to the solution of a set of first-order coupled differential equations. Emphasis is placed on obtaining explicit results for quantities that are measured in photolysis and fluorescence experiments so as to facilitate the analysis of experimental data.     

密度制约竞争二种群Volterra方程解的有界性及参数估计

本文给出密度制约且相互竞争二种群Volterra方程解的一种有界性及存在唯一性的证明。基于此,参考单种群Logistic方程反问题的方法”,给出了该Volterra方程参数的一种估计。     

Interpreting Frequency Responses to Dose-Conserved Pulsatile Input Signals in Simple Cell Signaling Motifs

Many hormones are released in pulsatile patterns. This pattern can be modified, for instance by changing pulse frequency, to encode relevant physiological information. Often other properties of the pulse pattern will also change with frequency. How do signaling pathways of cells targeted by these hormones respond to different input patterns? In this study, we examine how a given dose of hormone can induce different outputs from the target system, depending on how this dose is distributed in time. We use simple mathematical models of feedforward signaling motifs to understand how the properties of the target system give rise to preferences in input pulse pattern. We frame these problems in terms of frequency responses to pulsatile inputs, where the amplitude or duration of the pulses is varied along with frequency to conserve input dose. We find that the form of the nonlinearity in the steady state input-output function of the system predicts the optimal input pattern. It does so by selecting an optimal input signal amplitude. Our results predict the behavior of common signaling motifs such as receptor binding with dimerization, and protein phosphorylation. The findings have implications for experiments aimed at studying the frequency response to pulsatile inputs, as well as for understanding how pulsatile patterns drive biological responses via feedforward signaling pathways.     

Wavelet Domain Radiofrequency Pulse Design Applied to Magnetic Resonance Imaging

A new method for designing radiofrequency (RF) pulses with numerical optimization in the wavelet domain is presented. Numerical optimization may yield solutions that might otherwise have not been discovered with analytic techniques alone. Further, processing in the wavelet domain reduces the number of unknowns through compression properties inherent in wavelet transforms, providing a more tractable optimization problem. This algorithm is demonstrated with simultaneous multi-slice (SMS) spin echo refocusing pulses because reduced peak RF power is necessary for SMS diffusion imaging with high acceleration factors. An iterative, nonlinear, constrained numerical minimization algorithm was developed to generate an optimized RF pulse waveform. Wavelet domain coefficients were modulated while iteratively running a Bloch equation simulator to generate the intermediate slice profile of the net magnetization. The algorithm minimizes the L2-norm of the slice profile with additional terms to penalize rejection band ripple and maximize the net transverse magnetization across each slice. Simulations and human brain imaging were used to demonstrate a new RF pulse design that yields an optimized slice profile and reduced peak energy deposition when applied to a multiband single-shot echo planar diffusion acquisition. This method may be used to optimize factors such as magnitude and phase spectral profiles and peak RF pulse power for multiband simultaneous multi-slice (SMS) acquisitions. Wavelet-based RF pulse optimization provides a useful design method to achieve a pulse waveform with beneficial amplitude reduction while preserving appropriate magnetization response for magnetic resonance imaging.     

Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind

In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.     

可描述横向电场中外周神经兴奋的电缆方程

传统的电缆方程只能用于描述纵向电场中外周神经的兴奋,无法描述外周神经在横向电场作用下的兴奋,其于两阶段过程模型,提出一种改进的电缆方程,可以描述外周神经在横向电场中的兴奋,其结果和Struijk的离体实验数据相吻合。此改进的电缆方程可用于描述任意电场中外周神经的兴奋。     

Subthreshold oscillatory responses of the Hodgkin-Huxley cable model for the squid giant axon

The classical cable equation, in which membrane conductance is considered constant, is modified by including the linearized effect of membrane potential on sodium and potassium ionic currents, as formulated in the Hodgkin-Huxley equations for the squid giant axon. The resulting partial differential equation is solved by numerical inversion of the Laplace transform of the voltage response to current and voltage inputs. The voltage response is computed for voltage step, current step, and current pulse inputs, and the effect of temperature on the response to a current step input is also calculated.The validity of the linearized approximation is examined by comparing the linearized response to a current step input with the solution of the nonlinear partial differential cable equation for various subthreshold current step inputs.All the computed responses for the squid giant axon show oscillatory behavior and depart significantly from what is predicted on the basis of the classical cable equation. The linearization procedure, coupled with numerical inversion of the Laplace transform, proves to be a convenient approach which predicts at least qualitatively the subthreshold behavior of the nonlinear system.     

Modeling epithelial cell homeostasis: Assessing recovery and control mechanisms

Critical to epithelial cell viability is prompt and direct recovery, following a perturbation of cellular conditions. Although a number of transporters are known to be activated by changes in cell volume, cell pH, or cell membrane potential, their importance to cellular homeostasis has been difficult to establish. Moreover, the coordination among such regulated transporters to enhance recovery has received no attention in mathematical models of cellular function. In this paper, a previously developed model of proximal tubule (Weinstein, 1992, Am. J. Physiol. 263, F784–F798), has been approximated by its linearization about a reference condition. This yields a system of differential equations and auxiliary linear equations, which estimate cell volume and composition and transcellular fluxes in response to changes in bath conditions or membrane transport coefficients. Using the singular value decomposition, this system is reduced to a linear dynamical system, which is stable and reproduces the full model behavior in a useful neighborhood of the reference. Cost functions on trajectories formulated in the model variables (e.g., time for cell volume recovery) are translated into cost functions for the dynamical system. When the model is extended by the inclusion of linear dependence of membrane transport coefficients on model variables, the impact of each such controller on the recovery cost can be estimated with the solution of a Lyapunov matrix equation. Alternatively, solution of an algebraic Riccati equation provides the ensemble of controllers that constitute optimal state feedback for the dynamical system. When translated back into the physiological variables, the optimal controller contains some expected components, as well as unanticipated controllers of uncertain significance. This approach provides a means of relating cellular homeostasis to optimization of a dynamical system.     

Comparison of experimentally determined protein structures by solution of Bloch equations

Nuclear Overhauser enhancement (NOESY) spectra were theoretically generated by solving the generalized Bloch equations with the appropriate initial conditions. The input to the equations were the coordinates of the protons of two similar crystal structures of basic pancreatic trypsin inhibitor. The two NOESY spectra obtained were compared to published experimental spectra of the protein in solution. It was found that the two crystal structures of basic pancreatic trypsin inhibitor give different theoretical spectra. The solution of the Bloch equations is very sensitive to small variations in the distance between protons (approx. 0.2 A), and to differences in the surrounding configurations. The method allows a detailed comparison of the crystal and solution structures of proteins. The structure of the trypsin inhibitor in solution was found to be similar to either one or the other crystal forms in different regions of the molecule.     

一类生物种群投资的最优采收控制

在原有的Gauss白噪声刻画环境噪声项的基础上,考虑环境不可预知的跳跃性变化,运用Lévy白噪声建立了有界环境中的随机生物种群模型.并且,引入随机奇异控制来描述投资者的最优采收策略.进一步地,构造一族有着不同起点的控制问题,利用动态规划的思想,给出了最优采收控制问题解的充分条件,进而,将随机控制问题的求解转化为确定型偏微分方程的求解.     

Modelling inert gas exchange in tissue and mixed-venous blood return to the lungs

Inert gas exchange in tissue has been almost exclusively modelled by using an ordinary differential equation. The mathematical model that is used to derive this ordinary differential equation assumes that the partial pressure of an inert gas (which is proportional to the content of that gas) is a function only of time. This mathematical model does not allow for spatial variations in inert gas partial pressure. This model is also dependent only on the ratio of blood flow to tissue volume, and so does not take account of the shape of the body compartment or of the density of the capillaries that supply blood to this tissue. The partial pressure of a given inert gas in mixed-venous blood flowing back to the lungs is calculated from this ordinary differential equation. In this study, we write down the partial differential equations that allow for spatial as well as temporal variations in inert gas partial pressure in tissue. We then solve these partial differential equations and compare them to the solution of the ordinary differential equations described above. It is found that the solution of the ordinary differential equation is very different from the solution of the partial differential equation, and so the ordinary differential equation should not be used if an accurate calculation of inert gas transport to tissue is required. Further, the solution of the PDE is dependent on the shape of the body compartment and on the density of the capillaries that supply blood to this tissue. As a result, techniques that are based on the ordinary differential equation to calculate the mixed-venous blood partial pressure may be in error.     

Molecular clock fork phylogenies: closed form analytic maximum likelihood solutions

Maximum likelihood (ML) is increasingly used as an optimality criterion for selecting evolutionary trees, but finding the global optimum is a hard computational task. Because no general analytic solution is known, numeric techniques such as hill climbing or expectation maximization (EM) are used in order to find optimal parameters for a given tree. So far, analytic solutions were derived only for the simplest model-three-taxa, two-state characters, under a molecular clock. Quoting Ziheng Yang, who initiated the analytic approach,"this seems to be the simplest case, but has many of the conceptual and statistical complexities involved in phylogenetic estimation."In this work, we give general analytic solutions for a family of trees with four-taxa, two-state characters, under a molecular clock. The change from three to four taxa incurs a major increase in the complexity of the underlying algebraic system, and requires novel techniques and approaches. We start by presenting the general maximum likelihood problem on phylogenetic trees as a constrained optimization problem, and the resulting system of polynomial equations. In full generality, it is infeasible to solve this system, therefore specialized tools for the molecular clock case are developed. Four-taxa rooted trees have two topologies-the fork (two subtrees with two leaves each) and the comb (one subtree with three leaves, the other with a single leaf). We combine the ultrametric properties of molecular clock fork trees with the Hadamard conjugation to derive a number of topology dependent identities. Employing these identities, we substantially simplify the system of polynomial equations for the fork. We finally employ symbolic algebra software to obtain closed formanalytic solutions (expressed parametrically in the input data). In general, four-taxa trees can have multiple ML points. In contrast, we can now prove that each fork topology has a unique(local and global) ML point.     

A tractable revenue management model for capacity allocation and overbooking over an airline network

In this paper, we develop a revenue management model to jointly make the capacity allocation and overbooking decisions over an airline network. The crucial observation behind our model is that if the penalty cost of denying boarding to the reservations were given by a separable function, then the optimality equation for the joint capacity allocation and overbooking problem would decompose by the itineraries. We exploit this observation by building an approximation to the penalty cost that is separable by the numbers of reservations for different itineraries. In this case, we can obtain an approximate solution to the optimality equation by plugging the separable approximation into the boundary condition of the optimality equation. Our computational experiments compare our approach with a standard deterministic linear programming formulation, as well as a recent joint capacity allocation and overbooking model. When compared with the standard deterministic linear programming formulation, our approach can provide significant profit improvements. On the other hand, when compared with the recent joint capacity allocation and overbooking model, our approach can provide similar profit performance with substantially shorter runtimes.     

Nonlinear extraordinary wave in dense plasma

Conditions for the propagation of a slow extraordinary wave in dense magnetized plasma are found. A solution to the set of relativistic hydrodynamic equations and Maxwell’s equations under the plasma resonance conditions, when the phase velocity of the nonlinear wave is equal to the speed of light, is obtained. The deviation of the wave frequency from the resonance frequency is accompanied by nonlinear longitudinal-transverse oscillations. It is shown that, in this case, the solution to the set of self-consistent equations obtained by averaging the initial equations over the period of high-frequency oscillations has the form of an envelope soliton. The possibility of excitation of a nonlinear wave in plasma by an external electromagnetic pulse is confirmed by numerical simulations.