Optimal nitrogen distribution within a leaf canopy under direct and diffuse light

Nitrogen distribution within a leaf canopy is an important determinant of canopy carbon gain. Previous theoretical studies have predicted that canopy photosynthesis is maximized when the amount of photosynthetic nitrogen is proportionally allocated to the absorbed light. However, most of such studies used a simple Beer's law for light extinction to calculate optimal distribution, and it is not known whether this holds true when direct and diffuse light are considered together. Here, using an analytical solution and model simulations, optimal nitrogen distribution is shown to be very different between models using Beer's law and direct–diffuse light. The presented results demonstrate that optimal nitrogen distribution under direct–diffuse light is steeper than that under diffuse light only. The whole‐canopy carbon gain is considerably increased by optimizing nitrogen distribution compared with that in actual canopies in which nitrogen distribution is not optimized. This suggests that optimization of nitrogen distribution can be an effective target trait for improving plant productivity.     

Determining residual reduction algorithm kinematic tracking weights for a sidestep cut via numerical optimization

Musculoskeletal modeling allows for the determination of various parameters during dynamic maneuvers by using in vivo kinematic and ground reaction force (GRF) data as inputs. Differences between experimental and model marker data and inconsistencies in the GRFs applied to these musculoskeletal models may not produce accurate simulations. Therefore, residual forces and moments are applied to these models in order to reduce these differences. Numerical optimization techniques can be used to determine optimal tracking weights of each degree of freedom of a musculoskeletal model in order to reduce differences between the experimental and model marker data as well as residual forces and moments. In this study, the particle swarm optimization (PSO) and simplex simulated annealing (SIMPSA) algorithms were used to determine optimal tracking weights for the simulation of a sidestep cut. The PSO and SIMPSA algorithms were able to produce model kinematics that were within 1.4° of experimental kinematics with residual forces and moments of less than 10 N and 18 Nm, respectively. The PSO algorithm was able to replicate the experimental kinematic data more closely and produce more dynamically consistent kinematic data for a sidestep cut compared to the SIMPSA algorithm. Future studies should use external optimization routines to determine dynamically consistent kinematic data and report the differences between experimental and model data for these musculoskeletal simulations.     

Estimation of muscle response using three-dimensional musculoskeletal models before impact situation: a simulation study

When car crash experiments are performed using cadavers or dummies, the active muscles' reaction on crash situations cannot be observed. The aim of this study is to estimate muscles' response of the major muscle groups using three-dimensional musculoskeletal model by dynamic simulations of low-speed sled-impact. The three-dimensional musculoskeletal models of eight subjects were developed, including 241 degrees of freedom and 86 muscles. The muscle parameters considering limb lengths and the force-generating properties of the muscles were redefined by optimization to fit for each subject. Kinematic data and external forces measured by motion tracking system and dynamometer were then input as boundary conditions. Through a least-squares optimization algorithm, active muscles' responses were calculated during inverse dynamic analysis tracking the motion of each subject. Electromyography for major muscles at elbow, knee, and ankle joints was measured to validate each model. For low-speed sled-impact crash, experiment and simulation with optimized and unoptimized muscle parameters were performed at 9.4 m/h and 10 m/h and muscle activities were compared among them. The muscle activities with optimized parameters were closer to experimental measurements than the results without optimization. In addition, the extensor muscle activities at knee, ankle, and elbow joint were found considerably at impact time, unlike previous studies using cadaver or dummies. This study demonstrated the need to optimize the muscle parameters to predict impact situation correctly in computational studies using musculoskeletal models. And to improve accuracy of analysis for car crash injury using humanlike dummies, muscle reflex function, major extensor muscles' response at elbow, knee, and ankle joints, should be considered.     

Designing connected nature reserve networks using a graph theory approach

Habitat fragmentation has been cited as one of the critical reasons for biodiversity loss. Establishing connected nature reserve networks is an effective way to reduce habit fragmentation. However, the resources devoted to nature reserves have always been scarce. Therefore it is important to allocate our scarce resources in an optimal way. The optimal design of a reserve network which is effective both ecologically and economically has become an important research topic in the reserve design literature. The problem of optimal selection of a subset from a larger group of potential habitat sites is solved using either heuristic or formal optimization methods. The heuristic methods, although flexible and computationally fast, can not guarantee the solution is optimal therefore may lead to scarce resources being used in an ineffective way. The formal optimization methods, on the other hand, guarantees the solution is optimal, but it has been argued that it would be difficult to model site selection process using optimization models, especially when spatial attributes of the reserve have to be taken into account. This paper presents a linear integer programming model for the design of a minimal connected reserve network using a graph theory approach. A connected tree is determined corresponding to a connected reserve. Computational performance of the model is tested using datasets randomly generated by the software GAMS. Results show that the model can solve a connected reserve design problem which includes 100 potential sites and 30 species in a reasonable period of time. As an empirical application, the model is applied to the protection of endangered and threatened bird species in the Cache River basin area in Illinois, US. Two connected reserve networks are determined for 13 bird species.     

Designing connected nature reserve networks using a graph theory approach

Habitat fragmentation has been cited as one of the critical reasons for biodiversity loss. Establishing connected nature reserve networks is an effective way to reduce habit fragmentation. However, the resources devoted to nature reserves have always been scarce. Therefore it is important to allocate our scarce resources in an optimal way. The optimal design of a reserve network which is effective both ecologically and economically has become an important research topic in the reserve design literature. The problem of optimal selection of a subset from a larger group of potential habitat sites is solved using either heuristic or formal optimization methods. The heuristic methods, although flexible and computationally fast, can not guarantee the solution is optimal therefore may lead to scarce resources being used in an ineffective way. The formal optimization methods, on the other hand, guarantees the solution is optimal, but it has been argued that it would be difficult to model site selection process using optimization models, especially when spatial attributes of the reserve have to be taken into account. This paper presents a linear integer programming model for the design of a minimal connected reserve network using a graph theory approach. A connected tree is determined corresponding to a connected reserve. Computational performance of the model is tested using datasets randomly generated by the software GAMS. Results show that the model can solve a connected reserve design problem which includes 100 potential sites and 30 species in a reasonable period of time. As an empirical application, the model is applied to the protection of endangered and threatened bird species in the Cache River basin area in Illinois, US. Two connected reserve networks are determined for 13 bird species.     

A neuromusculoskeletal tracking method for estimating individual muscle forces in human movement

A neuromusculoskeletal tracking (NMT) method was developed to estimate muscle forces from observed motion data. The NMT method combines skeletal motion tracking and optimal neuromuscular tracking to produce forward simulations of human movement quickly and accurately. The skeletal motion tracker calculates the joint torques needed to actuate a skeletal model and track observed segment angles and ground forces in a forward simulation of the motor task. The optimal neuromuscular tracker resolves the muscle redundancy problem dynamically and finds the muscle excitations (and muscle forces) needed to produce the joint torques calculated by the skeletal motion tracker. To evaluate the accuracy of the NMT method, kinematics and ground forces obtained from an optimal control (parameter optimization) solution for maximum-height jumping were contaminated with both random and systematic noise. These data served as input observations to the NMT method as well as an inverse dynamics analysis. The NMT solution was compared to the input observations, the original optimal solution, and a simulation driven by the inverse dynamics torques. The results show that, in contrast to inverse dynamics, the NMT method is able to produce an accurate forward simulation consistent with the optimal control solution. The NMT method also requires 3 orders-of-magnitude less CPU time than parameter optimization. The speed and accuracy of the NMT method make it a promising new tool for estimating muscle forces using experimentally obtained kinematics and ground force data.     

Interrogating the effect of enzyme kinetics on metabolism using differentiable constraint-based models

Metabolic models are typically characterized by a large number of parameters. Traditionally, metabolic control analysis is applied to differential equation-based models to investigate the sensitivity of predictions to parameters. A corresponding theory for constraint-based models is lacking, due to their formulation as optimization problems. Here, we show that optimal solutions of optimization problems can be efficiently differentiated using constrained optimization duality and implicit differentiation. We use this to calculate the sensitivities of predicted reaction fluxes and enzyme concentrations to turnover numbers in an enzyme-constrained metabolic model of Escherichia coli. The sensitivities quantitatively identify rate limiting enzymes and are mathematically precise, unlike current finite difference based approaches used for sensitivity analysis. Further, efficient differentiation of constraint-based models unlocks the ability to use gradient information for parameter estimation. We demonstrate this by improving, genome-wide, the state-of-the-art turnover number estimates for E. coli. Finally, we show that this technique can be generalized to arbitrarily complex models. By differentiating the optimal solution of a model incorporating both thermodynamic and kinetic rate equations, the effect of metabolite concentrations on biomass growth can be elucidated. We benchmark these metabolite sensitivities against a large experimental gene knockdown study, and find good alignment between the predicted sensitivities and in vivo metabolome changes. In sum, we demonstrate several applications of differentiating optimal solutions of constraint-based metabolic models, and show how it connects to classic metabolic control analysis.     

Evolutionarily stable strategies in food selection models with fitness sets

Most current models for optimal food selection apply to ecological and behavioural optimization. In this paper optimal food selection theory is extended to apply to evolutionary optimization. A general evolutionary model for optimal food selection must incorporate the concept of fitness sets--or that variables, changing as a result of natural selection in evolutionary time, cannot, in general, vary independently of each other. A "Charnov type" optimal food selection model with a fitness set is investigated, and evolutionarily stable strategy (ESS) solutions of the evolutionary variables (i.e., the efficiencies of using available food types) are found. From this analysis it follows that the relative frequency of various food types in the environment may, under specified conditions, influence the evolutionarily optimal diet. Secondly, the analysis demonstrates that a food type not in the optimal diet may, in evolutionary time, be added to this by becoming more abundant. Thirdly, it follows from the analysis that the ecological result of MacArthur and Pianka, that food types are worth eating even if there is competition for them, is not generally applicable when referring to an evolutionary time scale. Finally, it is pointed out that for the diet to be an ESS, it is necessary that the consumer's density is stable and that the consumer's population dynamics are subjected to some density-dependent factor.     

Dipole characterization of single neurons from their extracellular action potentials

The spatial variation of the extracellular action potentials (EAP) of a single neuron contains information about the size and location of the dominant current source of its action potential generator, which is typically in the vicinity of the soma. Using this dependence in reverse in a three-component realistic probe + brain + source model, we solved the inverse problem of characterizing the equivalent current source of an isolated neuron from the EAP data sampled by an extracellular probe at multiple independent recording locations. We used a dipole for the model source because there is extensive evidence it accurately captures the spatial roll-off of the EAP amplitude, and because, as we show, dipole localization, beyond a minimum cell-probe distance, is a more accurate alternative to approaches based on monopole source models. Dipole characterization is separable into a linear dipole moment optimization where the dipole location is fixed, and a second, nonlinear, global optimization of the source location. We solved the linear optimization on a discrete grid via the lead fields of the probe, which can be calculated for any realistic probe + brain model by the finite element method. The global source location was optimized by means of Tikhonov regularization that jointly minimizes model error and dipole size. The particular strategy chosen reflects the fact that the dipole model is used in the near field, in contrast to the typical prior applications of dipole models to EKG and EEG source analysis. We applied dipole localization to data collected with stepped tetrodes whose detailed geometry was measured via scanning electron microscopy. The optimal dipole could account for 96% of the power in the spatial variation of the EAP amplitude. Among various model error contributions to the residual, we address especially the error in probe geometry, and the extent to which it biases estimates of dipole parameters. This dipole characterization method can be applied to any recording technique that has the capabilities of taking multiple independent measurements of the same single units.     

Optimal posture control of a musculo-skeletal arm model

 In this paper maximal performance posture control of the human arm is investigated by means of model simulations. Recent experiments (F.C.T. van der Helm, submitted, 2000) have shown that the reflexive feedback during postural control varies with the bandwidth of the applied force disturbances. This paper focusses on the influence of the frequency content of force disturbances on the reflexive feedback gains by means of optimization. The arm is modelled by a non-linear musculo-skeletal model with two degrees of freedom and six muscles. To facilitate the optimization of the model parameters, the arm model is linearized. A performance criterion is minimized for stochastic force disturbances in a two-step procedure: (1) optimization of static muscle activations using an additional energy criterion to obtain a unique and energy-efficient solution; and (2) optimization of reflex gains using an additional control effort criterion to obtain a unique solution. The optimization reveals that for the given task and posture, the shoulder muscles have the largest contribution, whereas the bi-articular muscles have a relatively small contribution to the behaviour. The dynamics at the endpoint level are estimated so that a comparison can be made with the experiments. Compared to the experiments, the intrinsic damping of the model is relatively large (about 150%), whereas the intrinsic stiffness is relatively small (about 60%). These differences can be attributed to unmodelled mechanical effects of cross-bridges in Hill-type muscle models. The optimized reflex gains show remarkable similarities with the values found in the experiments, implying that humans can adjust their reflexive feedback gains in an optimal way, weighting the performance and energy. The approach in this paper could be useful in the study of various posture tasks, for example in the prediction of the relation between the control parameters of various musculo-skeletal models and different experimental variables. Received: 24 January 2000 / Accepted in revised form: 7 July 2000     

Design and verification of a simple 3D dynamic model of speed skating which mimics observed forces and motions

Advice about the optimal coordination pattern for an individual speed skater, could be addressed by simulation and optimization of a biomechanical speed skating model. But before getting to this optimization approach one needs a model that can reasonably match observed behaviour. Therefore, the objective of this study is to present a verified three dimensional inverse skater model with minimal complexity, which models the speed skating motion on the straights. The model simulates the upper body transverse translation of the skater together with the forces exerted by the skates on the ice. The input of the model is the changing distance between the upper body and the skate, referred to as the leg extension (Euclidean distance in 3 D space). Verification shows that the model mimics the observed forces and motions well. The model is most accurate for the position and velocity estimation (respectively 1.2% and 2.9% maximum residuals) and least accurate for the force estimations (underestimation of 4.5–10%). The model can be used to further investigate variables in the skating motion. For this, the input of the model, the leg extension, can be optimized to obtain a maximal forward velocity of the upper body.     

Mathematical shape optimization of hip prosthesis design

The long-term success of artificial-joint replacement depends partly on the chances for acrylic cement failure and interface disruption. These chances can be diminished by an optimal load-transfer mechanism, whereby stress concentrations are avoided. The present paper introduces a method for numerical shape optimization, whereby the finite element method is used iteratively to determine optimal prosthetic designs, which minimize interface stresses. The method is first applied in a simplified one-dimensional model of a cemented femoral stem fixation, using acrylic cement. The results show that 30-70% cement and interface stress reductions can be obtained in principle with an optimized design. Although the actual optimal shape is susceptible to the characteristics of the joint load, the stem length, stem modulus, cement modulus and bone properties, its general geometrical characteristics are consistent, featuring proximal and distal tapers, and a belly-shaped middle region. These general characteristics are confirmed in a more realistic two-dimensional FEM model. It is concluded that this method of shape optimization can provide a meaningful basis for prosthetic design and analysis activities in general.     

单纯形加速法拟合生态学中的非线性模型

本文以Logistic模型,Taylor幂法则模型,Holling功能反应模型,以及种群内禀增长力Rm等模型的拟合和参数估计为例,探讨单纯形加速法在生态模型优化拟合和参数估计中的应用.结果表明,单纯形加速法拟合生态学中的非线性模型不仅适用广泛,而且拟合过程是直接求原来非线性模型的最优拟合,因而优于生态学中通常使用的将原模型“线性化后再拟合”的方法,而与其它一些最优化方法,如:麦夸方法、枚举选优法等比较,由于单纯形法不需计算目标函数的偏导数,因而计算不受目标函数及其偏导函数复杂程度的限制,而且对于各种模型其求优计算过程十分相似,可以编制统一的计算程序.本研究所编制的计算机程序对于本文未提到的其它一些模型也是完全适用的,在应用时仅需修改定义目标函数的自定义函数语句即可.研究也发现,在求优过程中,只要搜索系数选择适当和实际数据合理,是可以保证寻优成功的.     

Hybrid neuromusculoskeletal modeling to best track joint moments using a balance between muscle excitations derived from electromyograms and optimization

Current electromyography (EMG)-driven musculoskeletal models are used to estimate joint moments measured from an individual?s extremities during dynamic movement with varying levels of accuracy. The main benefit is the underlying musculoskeletal dynamics is simulated as a function of realistic, subject-specific, neural-excitation patterns provided by the EMG data. The main disadvantage is surface EMG cannot provide information on deeply located muscles. Furthermore, EMG data may be affected by cross-talk, recording and post-processing artifacts that could adversely influence the EMG?s information content. This limits the EMG-driven model?s ability to calculate the multi-muscle dynamics and the resulting joint moments about multiple degrees of freedom. We present a hybrid neuromusculoskeletal model that combines calibration, subject-specificity, EMG-driven and static optimization methods together. In this, the joint moment tracking errors are minimized by balancing the information content extracted from the experimental EMG data and from that generated by a static optimization method. Using movement data from five healthy male subjects during walking and running we explored the hybrid model?s best configuration to minimally adjust recorded EMGs and predict missing EMGs while attaining the best tracking of joint moments. Minimally adjusted and predicted excitations substantially improved the experimental joint moment tracking accuracy than current EMG-driven models. The ability of the hybrid model to predict missing muscle EMGs was also examined. The proposed hybrid model enables muscle-driven simulations of human movement while enforcing physiological constraints on muscle excitation patterns. This might have important implications for studying pathological movement for which EMG recordings are limited.     

Metabolic design based on a coupled gene expression-metabolic network model of tryptophan production in Escherichia coli

The presumably high potential of a holistic design approach for complex biochemical reaction networks is exemplified here for the network of tryptophan biosynthesis from glucose, a system whose components have been investigated thoroughly before. A dynamic model that combines the behavior of the trp operon gene expression with the metabolic network of central carbon metabolism and tryptophan biosynthesis is investigated. This model is analyzed in terms of metabolic fluxes, metabolic control, and nonlinear optimization. We compare two models for a wild-type strain and another model for a tryptophan producer. An integrated optimization of the whole network leads to a significant increase in tryptophan production rate for all systems under study. This enhancement is well above the increase that can be achieved by an optimization of subsystems. A constant ratio of control coefficients on tryptophan synthesis rate has been identified for the models regarding or disregarding trp operon expression. Although we found some examples where flux control coefficients even contradict the trends of enzyme activity changes in an optimized profile, flux control can be used as an indication for enzymes that have to be taken into account in optimization.     

Performance modeling and simulation of biochemical process sequences with interacting unit operations

Many biochemical processes consist of a sequence of operations for which optimal operating conditions (setpoints) have to be determined. If such optimization is performed for each operation separately with respect to objectives defined for each operation individually, overall process performance is likely to be suboptimal. Interactions between unit operations have to be considered, and a unique objective has to be defined for the whole process. This paper shows how a suitable optimization problem can be formulated and solved to obtain the best overall set of operating conditions for a process. A typical enzyme production process has been chosen as an example. In order to arrive at a demonstrative model for the entire sequence of unit operations, it is shown how interaction effects may be accommodated in the models. Optimal operating conditions are then determined subject to a global process objective and are shown to be different from those resulting from optimization of each separate operation. As this strategy may result in an economic benefit, it merits further research into interaction modeling and performance optimization.     

Dynamic optimization of hybridoma growth in a fed-batch bioreactor

This study addressed the problem of maximizing cell mass and monoclonal antibody production from a fed-batch hybridoma cell culture. We hypothesized that inaccuracies in the process model limited the mathematical optimization. On the basis of shaker flask data, we established a simple phenomenological model with cell mass and lactate production as the controlled variables. We then formulated an optimal control algorithm, which calculated the process-model mismatch at each sampling time, updated the model parameters, and re-optimized the substrate concentrations dynamically throughout the time course of the batch. Manipulated variables were feed rates of glucose and glutamine. Dynamic parameter adjustment was done using a fuzzy logic technique, while a heuristic random optimizer (HRO) optimized the feed rates. The parameters selected for updating were specific growth rate and the yield coefficient of lactate from glucose. These were chosen by a sensitivity analysis. The cell mass produced using dynamic optimization was compared to the cell mass produced for an unoptimized case, and for a one-time optimization at the beginning of the batch. Substantial improvements in reactor productivity resulted from dynamic re-optimization and parameter adjustment. We demonstrated first that a single offline optimization of substrate concentration at the start of the batch significantly increased the yield of cell mass by 27% over an unoptimized fermentation. Periodic optimization online increased yield of cell mass per batch by 44% over the single offline optimization. Concomitantly, the yield of monoclonal antibody increased by 31% over the off-line optimization case. For batch and fed-batch processes, this appears to be a suitable arrangement to account for inaccuracies in process models. This suggests that implementation of advanced yet inexpensive techniques can improve performance of fed-batch reactors employed in hybridoma cell culture.     

Cue integration in categorical tasks: insights from audio-visual speech perception

Previous cue integration studies have examined continuous perceptual dimensions (e.g., size) and have shown that human cue integration is well described by a normative model in which cues are weighted in proportion to their sensory reliability, as estimated from single-cue performance. However, this normative model may not be applicable to categorical perceptual dimensions (e.g., phonemes). In tasks defined over categorical perceptual dimensions, optimal cue weights should depend not only on the sensory variance affecting the perception of each cue but also on the environmental variance inherent in each task-relevant category. Here, we present a computational and experimental investigation of cue integration in a categorical audio-visual (articulatory) speech perception task. Our results show that human performance during audio-visual phonemic labeling is qualitatively consistent with the behavior of a Bayes-optimal observer. Specifically, we show that the participants in our task are sensitive, on a trial-by-trial basis, to the sensory uncertainty associated with the auditory and visual cues, during phonemic categorization. In addition, we show that while sensory uncertainty is a significant factor in determining cue weights, it is not the only one and participants' performance is consistent with an optimal model in which environmental, within category variability also plays a role in determining cue weights. Furthermore, we show that in our task, the sensory variability affecting the visual modality during cue-combination is not well estimated from single-cue performance, but can be estimated from multi-cue performance. The findings and computational principles described here represent a principled first step towards characterizing the mechanisms underlying human cue integration in categorical tasks.     

The simplest formal argument for fitness optimization

The Formal Darwinism Project aims to provide a formal argument linking population genetics to fitness optimization, which of necessity includes defining fitness. This bridges the gulf between those biologists who assume that natural selection leads to something close to fitness optimization and those biologists who believe on theoretical grounds that there is no sense of fitness that can usefully be said to be optimized. The current paper’s main objective is to provide a careful mathematical introduction to the project, and it also reflects on the project’s scope and limitations. The central argument is the proof of close ties between the mathematics of motion, as embodied in the Price equation, and the mathematics of optimization, as represented by optimization programmes. To make these links, a general and abstract model linking genotype, phenotype and number of successful gametes is assumed. The project has begun with simple dynamic models and simple linking models, and its progress will involve more realistic versions of them. The versions given here are fully mathematically rigorous, but elementary enough to serve as an introduction.     

Modelling whole-plant allocation in relation to carbon and nitrogen supply: Coordination versus optimization: Opinion

One of the few integrating theories related to allocation is the hypothesis of optimization. While optimization theory has great heuristic appeal and has been used to describe a range of physiological and ecological phenomena, it has major limitations. Optimization is necessarily based on a definite time integral and an optimal control strategy must be specific to the same patterns exhibited by the driving variables over this same period of time. Optimization tends to employ the use of oversimplifications in order to facilitate analytical solutions to the optimal control strategy, i.e. the mechanism governing the response of plants, which is the critical issue of interest. It is difficult to define objective criteria that can account for the natural variability in plants and testing the quantitative predictions of optimality models is also difficult. Thus, we suggest that optimization theory is too limited for practical use in modelling whole plant allocation. In this paper, we introduce the use of coordination theory as a practical alternative. We develop a simple plant growth allocation model using both coordination and optimization approaches and show that coordination theory is easily applied, produces results that are quantitatively similar to optimization, and overcomes the inherent limitations of optimization theory.