A review on the FvCB biochemical model of photosynthesis and the measurement of A-Ci curves

The biochemical model of photosynthesis proposed by Farquhar, von Caemmerer and Berry is a CO₂ response model based on photosynthetic processes. It hypothesizes that leaf CO₂ assimilation rate (A) of C₃ plants is decided by the minimum of three biochemical processes: the carboxylation rate supported by ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco), the ribulose-1,5-bisphosphate (RuBP) regeneration rate supported by electron transport and the triose-phosphate (TP) use rate. Fitting leaf CO₂ assimilation rate versus intercellular CO₂ concentration (A-C_i) curves with the modified FvCB model could provide several important biochemical parameters, including maximum Rubisco carboxylation rate, maximum rate of electron transport, TP use rate, day respiration rate and mesophyll conductance. The FvCB model has greatly improved our understanding and prediction of plant photosynthetic physiology and its response to environmental changes. In this review, we firstly described the FvCB model, and analysed the characteristics of this model: segmentation and overparameterization. We reviewed the estimation of biochemical parameters which by fitting A-C_i curves with the FvCB model. The biochemical parameters were estimated previously by segmenting subjectively and fitting each limitation state separately, whereas now by segmenting objectively and fitting all limitation simultaneously. In comparison to the previously conventional ordinary least squares (OLS), terativgorithms (eg. Genetic Algorithm, Simulated Annealing Algorithm) based on the modern computer technology are now in common use. However, to further improve the reliability and the precision of the parameters estimation, more studies about Rubisco kinetics parameters and their temperature dependence are needed. In the end, to obtain efficient photosynthetic data for biochemical parameters estimation, we integrated and modified methods concerning the measurement of A-C_i curves according to current knowledge about FvCB model fitting. We expect this review would advance our understanding and application of the FvCB model and A-C_i curves.     

Leaf photosynthesis and respiration of three bioenergy crops in relation to temperature and leaf nitrogen: how conserved are biochemical model parameters among crop species?

Given the need for parallel increases in food and energy production from crops in the context of global change, crop simulation models and data sets to feed these models with photosynthesis and respiration parameters are increasingly important. This study provides information on photosynthesis and respiration for three energy crops (sunflower, kenaf, and cynara), reviews relevant information for five other crops (wheat, barley, cotton, tobacco, and grape), and assesses how conserved photosynthesis parameters are among crops. Using large data sets and optimization techniques, the C(3) leaf photosynthesis model of Farquhar, von Caemmerer, and Berry (FvCB) and an empirical night respiration model for tested energy crops accounting for effects of temperature and leaf nitrogen were parameterized. Instead of the common approach of using information on net photosynthesis response to CO(2) at the stomatal cavity (A(n)-C(i)), the model was parameterized by analysing the photosynthesis response to incident light intensity (A(n)-I(inc)). Convincing evidence is provided that the maximum Rubisco carboxylation rate or the maximum electron transport rate was very similar whether derived from A(n)-C(i) or from A(n)-I(inc) data sets. Parameters characterizing Rubisco limitation, electron transport limitation, the degree to which light inhibits leaf respiration, night respiration, and the minimum leaf nitrogen required for photosynthesis were then determined. Model predictions were validated against independent sets. Only a few FvCB parameters were conserved among crop species, thus species-specific FvCB model parameters are needed for crop modelling. Therefore, information from readily available but underexplored A(n)-I(inc) data should be re-analysed, thereby expanding the potential of combining classical photosynthetic data and the biochemical model.     

Optimal designs for a multiresponse Emax model and efficient parameter estimation

The aim of dose finding studies is sometimes to estimate parameters in a fitted model. The precision of the parameter estimates should be as high as possible. This can be obtained by increasing the number of subjects in the study, N, choosing a good and efficient estimation approach, and by designing the dose finding study in an optimal way. Increasing the number of subjects is not always feasible because of increasing cost, time limitations, etc. In this paper, we assume fixed N and consider estimation approaches and study designs for multiresponse dose finding studies. We work with diabetes dose–response data and compare a system estimation approach that fits a multiresponse Emax model to the data to equation‐by‐equation estimation that fits uniresponse Emax models to the data. We then derive some optimal designs for estimating the parameters in the multi‐ and uniresponse Emax model and study the efficiency of these designs.     

Distinct cellular states determine calcium signaling response

The heterogeneity in mammalian cells signaling response is largely a result of pre‐existing cell‐to‐cell variability. It is unknown whether cell‐to‐cell variability rises from biochemical stochastic fluctuations or distinct cellular states. Here, we utilize calcium response to adenosine trisphosphate as a model for investigating the structure of heterogeneity within a population of cells and analyze whether distinct cellular response states coexist. We use a functional definition of cellular state that is based on a mechanistic dynamical systems model of calcium signaling. Using Bayesian parameter inference, we obtain high confidence parameter value distributions for several hundred cells, each fitted individually. Clustering the inferred parameter distributions revealed three major distinct cellular states within the population. The existence of distinct cellular states raises the possibility that the observed variability in response is a result of structured heterogeneity between cells. The inferred parameter distribution predicts, and experiments confirm that variability in IP3R response explains the majority of calcium heterogeneity. Our work shows how mechanistic models and single‐cell parameter fitting can uncover hidden population structure and demonstrate the need for parameter inference at the single‐cell level.     

Properties of cell death models calibrated and compared using Bayesian approaches

Using models to simulate and analyze biological networks requires principled approaches to parameter estimation and model discrimination. We use Bayesian and Monte Carlo methods to recover the full probability distributions of free parameters (initial protein concentrations and rate constants) for mass‐action models of receptor‐mediated cell death. The width of the individual parameter distributions is largely determined by non‐identifiability but covariation among parameters, even those that are poorly determined, encodes essential information. Knowledge of joint parameter distributions makes it possible to compute the uncertainty of model‐based predictions whereas ignoring it (e.g., by treating parameters as a simple list of values and variances) yields nonsensical predictions. Computing the Bayes factor from joint distributions yields the odds ratio (～20‐fold) for competing ‘direct’ and ‘indirect’ apoptosis models having different numbers of parameters. Our results illustrate how Bayesian approaches to model calibration and discrimination combined with single‐cell data represent a generally useful and rigorous approach to discriminate between competing hypotheses in the face of parametric and topological uncertainty.     

Modeling of an aerobic biofilm reactor with double-limiting substrate kinetics: bifurcational and dynamical analysis

A mathematical model of an aerobic biofilm reactor is presented to investigate the bifurcational patterns and the dynamical behavior of the reactor as a function of different key operating parameters. Suspended cells and biofilm are assumed to grow according to double limiting kinetics with phenol inhibition (carbon source) and oxygen limitation. The model presented by Russo et al. is extended to embody key features of the phenomenology of the granular‐supported biofilm: biofilm growth and detachment, gas–liquid oxygen transport, phenol, and oxygen uptake by both suspended and immobilized cells, and substrate diffusion into the biofilm. Steady‐state conditions and stability, and local dynamic behavior have been characterized. The multiplicity of steady states and their stability depend on key operating parameter values (dilution rate, gas–liquid mass transfer coefficient, biofilm detachment rate, and inlet substrate concentration). Small changes in the operating conditions may be coupled with a drastic change of the steady‐state scenario with transcritical and saddle‐node bifurcations. The relevance of concentration profiles establishing within the biofilm is also addressed. When the oxygen level in the liquid phase is <10% of the saturation level, the biofilm undergoes oxygen starvation and the active biofilm fraction becomes independent of the dilution rate. © 2011 American Institute of Chemical Engineers Biotechnol. Prog., 2011     

Development of a dynamic continuous-discrete-continuous model describing the lag phase of individual bacterial cells

AIMS: A previous model for adaptation and growth of individual bacterial cells was not dynamic in the lag phase, and could not be used to perform simulations of growth under non-isothermal conditions. The aim of the present study was to advance this model by adding a continuous adaptation step, prior to the discrete step, to form a continuous-discrete-continuous (CDC) model. METHODS AND RESULTS: The revised model uses four parameters: N(0), initial population; N(max), maximum population; p0, mean initial individual cell physiological state; SD(p0), standard deviation of the distribution of individual physiological states. A truncated normal distribution was used to generate tables of distributions to allow fitting of the CDC model to viable count data for Listeria monocytogenes grown at 5 degrees C to 35 degrees C. The p0 values increased with increasing SD(p0) and were, on average, greater than the corresponding population physiological states (h0); p0 and h0 were equivalent for individual cells. CONCLUSION: The CDC model has improved the ability to simulate the behaviour of individual bacterial cells by using a physiological state parameter and a distribution function to handle inter-cell variability. The stages of development of this model indicate the importance of physiological state parameters over the population lag concept, and provide a potential approach for making growth models more mechanistic by incorporating actual physiological events. SIGNIFICANCE AND IMPACT OF THE STUDY: Individual cell behaviour is important in modelling bacterial growth in foods. The CDC model provides a means of improving existing growth models, and increases the value of mathematical modelling to the food industry.     

The MIAmaxent R package: Variable transformation and model selection for species distribution models

The widely used “Maxent” software for modeling species distributions from presence‐only data (Phillips et al., Ecological Modelling, 190, 2006, 231) tends to produce models with high‐predictive performance but low‐ecological interpretability, and implications of Maxent's statistical approach to variable transformation, model fitting, and model selection remain underappreciated. In particular, Maxent's approach to model selection through lasso regularization has been shown to give less parsimonious distribution models—that is, models which are more complex but not necessarily predictively better—than subset selection. In this paper, we introduce the MIAmaxent R package, which provides a statistical approach to modeling species distributions similar to Maxent's, but with subset selection instead of lasso regularization. The simpler models typically produced by subset selection are ecologically more interpretable, and making distribution models more grounded in ecological theory is a fundamental motivation for using MIAmaxent. To that end, the package executes variable transformation based on expected occurrence–environment relationships and contains tools for exploring data and interrogating models in light of knowledge of the modeled system. Additionally, MIAmaxent implements two different kinds of model fitting: maximum entropy fitting for presence‐only data and logistic regression (GLM) for presence–absence data. Unlike Maxent, MIAmaxent decouples variable transformation, model fitting, and model selection, which facilitates methodological comparisons and gives the modeler greater flexibility when choosing a statistical approach to a given distribution modeling problem.     

Use of the covariance matrix in directly fitting kinetic parameters: application to GABAA receptors

A new method of analysis is described that begins to explore the relationship between the phases of ion channel desensitization and the underlying states of the channel. The method, referred to as covariance fitting (CVF), couples Q-matrix calculations with a maximum likelihood algorithm to fit macroscopic desensitization data directly to kinetic models. Unlike conventional sum-of-squares minimization, CVF fits both the magnitude of the recorded current and the strength of the correlations between different time points. When applied to simulated data generated using various kinetic models with up to 11 free parameters, CVF leads to reasonable parameter estimates. Coupled with the likelihood ratio test, it accurately discriminates between models with different numbers of states, discriminates between most models with the same number but a different arrangement of states, and extracts meaningful information on the relationship between the desensitized states and the phases of macroscopic desensitization. When applied to GABA(A) receptor traces (outside out patches, alpha 1 beta 2 gamma 2S, 1 mM GABA, >2.5 s), a model with two open states and three desensitized states is favored. When applied to simulated data generated using a consensus model, CVF leads to reasonable parameter estimates and accurately discriminates between this and other models.     

Applications of hidden hybrid Markov/semi‐Markov models: from stopover duration to breeding success dynamics

Usually in capture–recapture, a model parameter is time or time since first capture dependent. However, the case where the probability of staying in one state depends on the time spent in that particular state is not rare. Hidden Markov models are not appropriate to manage these situations. A more convenient approach would be to consider models that incorporate semi‐Markovian states which explicitly define the waiting time distribution and have been used in previous biologic studies as a convenient framework for modeling the time spent in a given physiological state. Here, we propose hidden Markovian models that combine several nonhomogeneous Markovian states with one semi‐Markovian state and which (i) are well adapted to imperfect and variable detection and (ii) allow us to consider time, time since first capture, and time spent in one state effects. Implementation details depending on the number of semi‐Markovian states are discussed. From a user's perspective, the present approach enhances the toolbox for analyzing capture–recapture data. We then show the potential of this framework by means of two ecological examples: (i) stopover duration and (ii) breeding success dynamics.     

Do all leaf photosynthesis parameters of rice acclimate to elevated CO2, elevated temperature,and their combination,in FACE environments?

Leaf photosynthesis of crops acclimates to elevated CO₂ and temperature, but studies quantifying responses of leaf photosynthetic parameters to combined CO₂ and temperature increases under field conditions are scarce. We measured leaf photosynthesis of rice cultivars Changyou 5 and Nanjing 9108 grown in two free‐air CO₂ enrichment (FACE) systems, respectively, installed in paddy fields. Each FACE system had four combinations of two levels of CO₂ (ambient and enriched) and two levels of canopy temperature (no warming and warmed by 1.0–2.0°C). Parameters of the C₃ photosynthesis model of Farquhar, von Caemmerer and Berry (the FvCB model), and of a stomatal conductance (g_s) model were estimated for the four conditions. Most photosynthetic parameters acclimated to elevated CO₂, elevated temperature, and their combination. The combination of elevated CO₂ and temperature changed the functional relationships between biochemical parameters and leaf nitrogen content for Changyou 5. The g_s model significantly underestimated g_s under the combination of elevated CO₂ and temperature by 19% for Changyou 5 and by 10% for Nanjing 9108 if no acclimation was assumed. However, our further analysis applying the coupled g_s–FvCB model to an independent, previously published FACE experiment showed that including such an acclimation response of g_s hardly improved prediction of leaf photosynthesis under the four combinations of CO₂ and temperature. Therefore, the typical procedure that crop models using the FvCB and g_s models are parameterized from plants grown under current ambient conditions may not result in critical errors in projecting productivity of paddy rice under future global change.     

Making use of multiple surveys: Estimating breeding probability using a multievent‐robust design capture–recapture model

Increased environmental stochasticity due to climate change will intensify temporal variance in the life‐history traits, and especially breeding probabilities, of long‐lived iteroparous species. These changes may decrease individual fitness and population viability and is therefore important to monitor. In wild animal populations with imperfect individual detection, breeding probabilities are best estimated using capture–recapture methods. However, in many vertebrate species (e.g., amphibians, turtles, seabirds), nonbreeders are unobservable because they are not tied to a territory or breeding location. Although unobservable states can be used to model temporary emigration of nonbreeders, there are disadvantages to having unobservable states in capture–recapture models. The best solution to deal with unobservable life‐history states is therefore to eliminate them altogether. Here, we achieve this objective by fitting novel multievent‐robust design models which utilize information obtained from multiple surveys conducted throughout the year. We use this approach to estimate annual breeding probabilities of capital breeding female elephant seals (Mirounga leonina). Conceptually, our approach parallels a multistate version of the Barker/robust design in that it combines robust design capture data collected during discrete breeding seasons with observations made at other times of the year. A substantial advantage of our approach is that the nonbreeder state became “observable” when multiple data sources were analyzed together. This allowed us to test for the existence of state‐dependent survival (with some support found for lower survival in breeders compared to nonbreeders), and to estimate annual breeding transitions to and from the nonbreeder state with greater precision (where current breeders tended to have higher future breeding probabilities than nonbreeders). We used program E‐SURGE (2.1.2) to fit the multievent‐robust design models, with uncertainty in breeding state assignment (breeder, nonbreeder) being incorporated via a hidden Markov process. This flexible modeling approach can easily be adapted to suit sampling designs from numerous species which may be encountered during and outside of discrete breeding seasons.     

Evaluating the performance of neutrality tests of a local community using a niche‐structured simulation model

Understanding the processes that underlie species diversity and abundance in a community is a fundamental issue in community ecology. While the species abundance distributions (SADs) of various natural communities may be well explained by Hubbell's neutral model, it has been repeatedly pointed out that Hubbell's SAD‐fitting approach lacks the ability to detect the effects of non‐neutral factors such as niche differentiation; however, our understanding of its quantitative effect is limited. Herein, we conducted extensive simulations to quantitatively evaluate the performance of the SAD‐fitting method and other recently developed tests. For simulations, we developed a niche model that incorporates the random stochastic demography of individuals and the nonrandom replacements of those individuals, i.e. niche differentiation. It therefore allows us to explore situations with various degrees of niche differentiation. We found that niche differentiation has strong effects on SADs and the number of species in the community under this model. We then examined the performance of these neutrality tests, including Hubbell's SAD‐fitting method, using extensive simulations. It was demonstrated that all these tests have relatively poor performance except for the cases with very strong niche structure, which is in accordance with previous studies. This is likely because two important parameters in Hubbell's model are usually unknown and are commonly estimated from the data to be tested. To demonstrate this point, we showed that the precise estimation of the two parameters substantially improved the performance of these neutrality tests, indicating that poor performance can be owed to overfitting Hubbell's neutral model with unrealistic parameters. Our results therefore emphasize the importance of accurate parameter estimation, which should be obtained from data independent of the local community to be tested.     

Model‐based identifiable parameter determination applied to a simultaneous saccharification and fermentation process model for bio‐ethanol production

In this work, a methodology for the model‐based identifiable parameter determination (MBIPD) is presented. This systematic approach is proposed to be used for structure and parameter identification of nonlinear models of biological reaction networks. Usually, this kind of problems are over‐parameterized with large correlations between parameters. Hence, the related inverse problems for parameter determination and analysis are mathematically ill‐posed and numerically difficult to solve. The proposed MBIPD methodology comprises several tasks: (i) model selection, (ii) tracking of an adequate initial guess, and (iii) an iterative parameter estimation step which includes an identifiable parameter subset selection (SsS) algorithm and accuracy analysis of the estimated parameters. The SsS algorithm is based on the analysis of the sensitivity matrix by rank revealing factorization methods. Using this, a reduction of the parameter search space to a reasonable subset, which can be reliably and efficiently estimated from available measurements, is achieved. The simultaneous saccharification and fermentation (SSF) process for bio‐ethanol production from cellulosic material is used as case study for testing the methodology. The successful application of MBIPD to the SSF process demonstrates a relatively large reduction in the identified parameter space. It is shown by a cross‐validation that using the identified parameters (even though the reduction of the search space), the model is still able to predict the experimental data properly. Moreover, it is shown that the model is easily and efficiently adapted to new process conditions by solving reduced and well conditioned problems. © 2013 American Institute of Chemical Engineers Biotechnol. Prog., 29:1064–1082, 2013     

Universally sloppy parameter sensitivities in systems biology models

Quantitative computational models play an increasingly important role in modern biology. Such models typically involve many free parameters, and assigning their values is often a substantial obstacle to model development. Directly measuring in vivo biochemical parameters is difficult, and collectively fitting them to other experimental data often yields large parameter uncertainties. Nevertheless, in earlier work we showed in a growth-factor-signaling model that collective fitting could yield well-constrained predictions, even when it left individual parameters very poorly constrained. We also showed that the model had a “sloppy” spectrum of parameter sensitivities, with eigenvalues roughly evenly distributed over many decades. Here we use a collection of models from the literature to test whether such sloppy spectra are common in systems biology. Strikingly, we find that every model we examine has a sloppy spectrum of sensitivities. We also test several consequences of this sloppiness for building predictive models. In particular, sloppiness suggests that collective fits to even large amounts of ideal time-series data will often leave many parameters poorly constrained. Tests over our model collection are consistent with this suggestion. This difficulty with collective fits may seem to argue for direct parameter measurements, but sloppiness also implies that such measurements must be formidably precise and complete to usefully constrain many model predictions. We confirm this implication in our growth-factor-signaling model. Our results suggest that sloppy sensitivity spectra are universal in systems biology models. The prevalence of sloppiness highlights the power of collective fits and suggests that modelers should focus on predictions rather than on parameters.     

The continuous smooth hockey stick: a newly proposed spawner‐recruitment model

Spawner‐recruit relationships are important components of fisheries management. The two most widely used models have been criticized for unsatisfactory fits and biologically unreasonable extrapolations. A simple hockey stick model has been shown to provide more robust predictions, however, this model is not widely used, possibly because the abrupt change from density‐dependence to density‐independence is unrealistic and the piecewise model is difficult to fit. Here I present a continuous two‐parameter model that resembles a smoothed hockey stick and provides parameter estimates similar to the piecewise hockey stick. The new model is easily parameterized with regular curve‐fitting routines.     

Extension of a biochemical model for the generalized stoichiometry of electron transport limited C3 photosynthesis

The widely used steady‐state model of Farquhar et al. (Planta 149: 78–90, 1980) for C₃ photosynthesis was developed on the basis of linear whole‐chain (non‐cyclic) electron transport. In this model, calculation of the RuBP‐regeneration limited CO₂‐assimilation rate depends on whether it is insufficient ATP or NADPH that causes electron transport limitation. A new, generalized equation that allows co‐limitation of NADPH and ATP on electron transport is presented herein. The model is based on the assumption that other thylakoid pathways (the Q‐cycle, cyclic photophosphorylation, and pseudocyclic electron transport) interplay with the linear chain to co‐contribute to a balanced production of NADPH and ATP as required by stromal metabolism. The original model assuming linear electron transport limited either by NADPH or by ATP, predicts quantum yields for CO₂ uptake that represent the highest and the lowest values, respectively, of the range given by the new equation. The applicability of the new equation is illustrated for a number of C₃ crop species, by curve fitting to gas exchange data in the literature. In comparison with the original model, the new model enables analysis of photosynthetic regulation via the electron transport pathways in response to environmental stresses.     

Plantecophys - An R Package for Analysing and Modelling Leaf Gas Exchange Data

Here I present the R package ''plantecophys'', a toolkit to analyse and model leaf gas exchange data. Measurements of leaf photosynthesis and transpiration are routinely collected with portable gas exchange instruments, and analysed with a few key models. These models include the Farquhar-von Caemmerer-Berry (FvCB) model of leaf photosynthesis, the Ball-Berry models of stomatal conductance, and the coupled leaf gas exchange model which combines the supply and demand functions for CO₂ in the leaf. The ''plantecophys'' R package includes functions for fitting these models to measurements, as well as simulating from the fitted models to aid in interpreting experimental data. Here I describe the functionality and implementation of the new package, and give some examples of its use. I briefly describe functions for fitting the FvCB model of photosynthesis to measurements of photosynthesis-CO₂ response curves (''A-C_i curves''), fitting Ball-Berry type models, modelling C3 photosynthesis with the coupled photosynthesis-stomatal conductance model, modelling C4 photosynthesis, numerical solution of optimal stomatal behaviour, and energy balance calculations using the Penman-Monteith equation. This open-source package makes technically challenging calculations easily accessible for many users and is freely available on CRAN.     

Molecular variability elicits a tunable switch with discrete neuromodulatory response phenotypes

Conductance-based compartment modeling requires tuning of many parameters to fit the neuron model to target electrophysiological data. Automated parameter optimization via evolutionary algorithms (EAs) is a common approach to accomplish this task, using error functions to quantify differences between model and target. We present a three-stage EA optimization protocol for tuning ion channel conductances and kinetics in a generic neuron model with minimal manual intervention. We use the technique of Latin hypercube sampling in a new way, to choose weights for error functions automatically so that each function influences the parameter search to a similar degree. This protocol requires no specialized physiological data collection and is applicable to commonly-collected current clamp data and either single- or multi-objective optimization. We applied the protocol to two representative pyramidal neurons from layer 3 of the prefrontal cortex of rhesus monkeys, in which action potential firing rates are significantly higher in aged compared to young animals. Using an idealized dendritic topology and models with either 4 or 8 ion channels (10 or 23 free parameters respectively), we produced populations of parameter combinations fitting the target datasets in less than 80 hours of optimization each. Passive parameter differences between young and aged models were consistent with our prior results using simpler models and hand tuning. We analyzed parameter values among fits to a single neuron to facilitate refinement of the underlying model, and across fits to multiple neurons to show how our protocol will lead to predictions of parameter differences with aging in these neurons.