PeerJ:Mathematical Biologyhttps://peerj.com/articles/index.atom?journal=peerj&subject=1900Mathematical Biology articles published in PeerJEstimating probabilistic context-free grammars for proteins using contact map constraintshttps://peerj.com/articles/65592019-03-182019-03-18Witold DyrkaMateusz PyzikFrançois CosteHugo Talibart
Interactions between amino acids that are close in the spatial structure, but not necessarily in the sequence, play important structural and functional roles in proteins. These non-local interactions ought to be taken into account when modeling collections of proteins. Yet the most popular representations of sets of related protein sequences remain the profile Hidden Markov Models. By modeling independently the distributions of the conserved columns from an underlying multiple sequence alignment of the proteins, these models are unable to capture dependencies between the protein residues. Non-local interactions can be represented by using more expressive grammatical models. However, learning such grammars is difficult. In this work, we propose to use information on protein contacts to facilitate the training of probabilistic context-free grammars representing families of protein sequences. We develop the theory behind the introduction of contact constraints in maximum-likelihood and contrastive estimation schemes and implement it in a machine learning framework for protein grammars. The proposed framework is tested on samples of protein motifs in comparison with learning without contact constraints. The evaluation shows high fidelity of grammatical descriptors to protein structures and improved precision in recognizing sequences. Finally, we present an example of using our method in a practical setting and demonstrate its potential beyond the current state of the art by creating a grammatical model of a meta-family of protein motifs. We conclude that the current piece of research is a significant step towards more flexible and accurate modeling of collections of protein sequences. The software package is made available to the community.
Interactions between amino acids that are close in the spatial structure, but not necessarily in the sequence, play important structural and functional roles in proteins. These non-local interactions ought to be taken into account when modeling collections of proteins. Yet the most popular representations of sets of related protein sequences remain the profile Hidden Markov Models. By modeling independently the distributions of the conserved columns from an underlying multiple sequence alignment of the proteins, these models are unable to capture dependencies between the protein residues. Non-local interactions can be represented by using more expressive grammatical models. However, learning such grammars is difficult. In this work, we propose to use information on protein contacts to facilitate the training of probabilistic context-free grammars representing families of protein sequences. We develop the theory behind the introduction of contact constraints in maximum-likelihood and contrastive estimation schemes and implement it in a machine learning framework for protein grammars. The proposed framework is tested on samples of protein motifs in comparison with learning without contact constraints. The evaluation shows high fidelity of grammatical descriptors to protein structures and improved precision in recognizing sequences. Finally, we present an example of using our method in a practical setting and demonstrate its potential beyond the current state of the art by creating a grammatical model of a meta-family of protein motifs. We conclude that the current piece of research is a significant step towards more flexible and accurate modeling of collections of protein sequences. The software package is made available to the community.From data compilation to model validation: a comprehensive analysis of a full deep-sea ecosystem model of the Chatham Risehttps://peerj.com/articles/65172019-02-222019-02-22Vidette L. McGregorPeter L. HornElizabeth A. FultonMatthew R. Dunn
The Chatham Rise is a highly productive deep-sea ecosystem that supports numerous substantial commercial fisheries, and is a likely candidate for an ecosystem based approach to fisheries management in New Zealand. We present the first end-to-end ecosystem model of the Chatham Rise, which is also to the best of our knowledge, the first end-to-end ecosystem model of any deep-sea ecosystem. We describe the process of data compilation through to model validation and analyse the importance of knowledge gaps with respect to model dynamics and results. The model produces very similar results to fisheries stock assessment models for key fisheries species, and the population dynamics and system interactions are realistic. Confidence intervals based on bootstrapping oceanographic variables are produced. The model components that have knowledge gaps and are most likely to influence model results were oceanographic variables, and the aggregate species groups ‘seabird’ and ‘cetacean other’. We recommend applications of the model, such as forecasting biomasses under various fishing regimes, include alternatives that vary these components.
The Chatham Rise is a highly productive deep-sea ecosystem that supports numerous substantial commercial fisheries, and is a likely candidate for an ecosystem based approach to fisheries management in New Zealand. We present the first end-to-end ecosystem model of the Chatham Rise, which is also to the best of our knowledge, the first end-to-end ecosystem model of any deep-sea ecosystem. We describe the process of data compilation through to model validation and analyse the importance of knowledge gaps with respect to model dynamics and results. The model produces very similar results to fisheries stock assessment models for key fisheries species, and the population dynamics and system interactions are realistic. Confidence intervals based on bootstrapping oceanographic variables are produced. The model components that have knowledge gaps and are most likely to influence model results were oceanographic variables, and the aggregate species groups ‘seabird’ and ‘cetacean other’. We recommend applications of the model, such as forecasting biomasses under various fishing regimes, include alternatives that vary these components.Geometric morphometric analysis in female freshwater crabs of Sarawak (Borneo) permits addressing taxonomy-related problemshttps://peerj.com/articles/62052019-02-142019-02-14Jongkar GrinangIndraneil DasPeter K.L. Ng
The taxonomy of freshwater crabs requires a paradigm change in methodological approaches, particularly in investigations that use morphological techniques. The traditional morphometric approach (two-dimensional measurements) tends to be inappropriate for the identification of freshwater crabs due to their variable external morphology and lack of gonopods (conventionally used for the identification of male crabs) in females. In this study, we explore the potential use of the geometric morphometric technique for identification of female freshwater crabs, and identify taxonomic key characteristics of species. The shape of the carapace could be a good characteristic for the identification of female crabs, especially when the geometric morphometric technique is used. It was observed that the shape of the carapace has an advantage over the shape of the pleon and chela because its relatively flat orientation allows more consistent and easier data preparation for geometric morphometric analysis. The geometric morphometric technique is inexpensive, relatively less time consuming to employ, and accurate. This technique is convenient when dissection to examine the gonopods is not possible, which can damage the specimen in the case of endangered or rare species. Since the technique was used herein for only two species, more compelling and extensive evidence is needed before the reliability of the method can be proven.
The taxonomy of freshwater crabs requires a paradigm change in methodological approaches, particularly in investigations that use morphological techniques. The traditional morphometric approach (two-dimensional measurements) tends to be inappropriate for the identification of freshwater crabs due to their variable external morphology and lack of gonopods (conventionally used for the identification of male crabs) in females. In this study, we explore the potential use of the geometric morphometric technique for identification of female freshwater crabs, and identify taxonomic key characteristics of species. The shape of the carapace could be a good characteristic for the identification of female crabs, especially when the geometric morphometric technique is used. It was observed that the shape of the carapace has an advantage over the shape of the pleon and chela because its relatively flat orientation allows more consistent and easier data preparation for geometric morphometric analysis. The geometric morphometric technique is inexpensive, relatively less time consuming to employ, and accurate. This technique is convenient when dissection to examine the gonopods is not possible, which can damage the specimen in the case of endangered or rare species. Since the technique was used herein for only two species, more compelling and extensive evidence is needed before the reliability of the method can be proven.GRID-independent molecular descriptor analysis and molecular docking studies to mimic the binding hypothesis of γ-aminobutyric acid transporter 1 (GAT1) inhibitorshttps://peerj.com/articles/62832019-01-312019-01-31Sadia ZafarIshrat Jabeen
Background
The γ-aminobutyric acid (GABA) transporter GAT1 is involved in GABA transport across the biological membrane in and out of the synaptic cleft. The efficiency of this Na+ coupled GABA transport is regulated by an electrochemical gradient, which is directed inward under normal conditions. However, in certain pathophysiological situations, including strong depolarization or an imbalance in ion homeostasis, the GABA influx into the cytoplasm is increased by re-uptake transport mechanism. This mechanism may lead to extra removal of extracellular GABA which results in numerous neurological disorders such as epilepsy. Thus, small molecule inhibitors of GABA re-uptake may enhance GABA activity at the synaptic clefts.
Methods
In the present study, various GRID-independent molecular descriptor (GRIND) models have been developed to shed light on the 3D structural features of human GAT1 (hGAT1) inhibitors using nipecotic acid and N-diarylalkenyl piperidine analogs. Further, a binding hypothesis has been developed for the selected GAT1 antagonists by molecular docking inside the binding cavity of hGAT1 homology model.
Results
Our results indicate that two hydrogen bond acceptors, one hydrogen bond donor and one hydrophobic region at certain distances from each other play an important role in achieving high inhibitory potency against hGAT1. Our docking results elucidate the importance of the COOH group in hGAT1 antagonists by considering substitution of the COOH group with an isoxazol ring in compound 37, which subsequently leads to a three order of magnitude decrease in biological activity of 37 (IC50 = 38 µM) as compared to compound 1 (IC50 = 0.040 µM).
Discussion
Our docking results are strengthened by the structure activity relationship of the data series as well as by GRIND models, thus providing a significant structural basis for understanding the binding of antagonists, which may be useful for guiding the design of hGAT1 inhibitors.
Background
The γ-aminobutyric acid (GABA) transporter GAT1 is involved in GABA transport across the biological membrane in and out of the synaptic cleft. The efficiency of this Na+ coupled GABA transport is regulated by an electrochemical gradient, which is directed inward under normal conditions. However, in certain pathophysiological situations, including strong depolarization or an imbalance in ion homeostasis, the GABA influx into the cytoplasm is increased by re-uptake transport mechanism. This mechanism may lead to extra removal of extracellular GABA which results in numerous neurological disorders such as epilepsy. Thus, small molecule inhibitors of GABA re-uptake may enhance GABA activity at the synaptic clefts.
Methods
In the present study, various GRID-independent molecular descriptor (GRIND) models have been developed to shed light on the 3D structural features of human GAT1 (hGAT1) inhibitors using nipecotic acid and N-diarylalkenyl piperidine analogs. Further, a binding hypothesis has been developed for the selected GAT1 antagonists by molecular docking inside the binding cavity of hGAT1 homology model.
Results
Our results indicate that two hydrogen bond acceptors, one hydrogen bond donor and one hydrophobic region at certain distances from each other play an important role in achieving high inhibitory potency against hGAT1. Our docking results elucidate the importance of the COOH group in hGAT1 antagonists by considering substitution of the COOH group with an isoxazol ring in compound 37, which subsequently leads to a three order of magnitude decrease in biological activity of 37 (IC50 = 38 µM) as compared to compound 1 (IC50 = 0.040 µM).
Discussion
Our docking results are strengthened by the structure activity relationship of the data series as well as by GRIND models, thus providing a significant structural basis for understanding the binding of antagonists, which may be useful for guiding the design of hGAT1 inhibitors.Seasonality and trend prediction of scarlet fever incidence in mainland China from 2004 to 2018 using a hybrid SARIMA-NARX modelhttps://peerj.com/articles/61652019-01-172019-01-17Yongbin WangChunjie XuZhende WangJuxiang Yuan
Background
Scarlet fever is recognized as being a major public health issue owing to its increase in notifications in mainland China, and an advanced response based on forecasting techniques is being adopted to tackle this. Here, we construct a new hybrid method incorporating seasonal autoregressive integrated moving average (SARIMA) with a nonlinear autoregressive with external input(NARX) to analyze its seasonality and trend in order to efficiently prevent and control this re-emerging disease.
Methods
Four statistical models, including a basic SARIMA, basic nonlinear autoregressive (NAR) method, traditional SARIMA-NAR and new SARIMA-NARX hybrid approaches, were developed based on scarlet fever incidence data between January 2004 and July 2018 to evaluate its temporal patterns, and their mimic and predictive capacities were compared to discover the optimal using the mean absolute percentage error, root mean square error, mean error rate, and root mean square percentage error.
Results
The four preferred models identified were comprised of the SARIMA(0,1,0)(0,1,1)12, NAR with 14 hidden neurons and five delays, SARIMA-NAR with 33 hidden neurons and five delays, and SARIMA-NARX with 16 hidden neurons and 4 delays. Among which presenting the lowest values of the aforementioned indices in both simulation and prediction horizons is the SARIMA-NARX method. Analyses from the data suggested that scarlet fever was a seasonal disease with predominant peaks of summer and winter and a substantial rising trend in the scarlet fever notifications was observed with an acceleration of 9.641% annually, particularly since 2011 with 12.869%, and moreover such a trend will be projected to continue in the coming year.
Conclusions
The SARIMA-NARX technique has the promising ability to better consider both linearity and non-linearity behind scarlet fever data than the others, which significantly facilitates its prevention and intervention of scarlet fever. Besides, under current trend of ongoing resurgence, specific strategies and countermeasures should be formulated to target scarlet fever.
Background
Scarlet fever is recognized as being a major public health issue owing to its increase in notifications in mainland China, and an advanced response based on forecasting techniques is being adopted to tackle this. Here, we construct a new hybrid method incorporating seasonal autoregressive integrated moving average (SARIMA) with a nonlinear autoregressive with external input(NARX) to analyze its seasonality and trend in order to efficiently prevent and control this re-emerging disease.
Methods
Four statistical models, including a basic SARIMA, basic nonlinear autoregressive (NAR) method, traditional SARIMA-NAR and new SARIMA-NARX hybrid approaches, were developed based on scarlet fever incidence data between January 2004 and July 2018 to evaluate its temporal patterns, and their mimic and predictive capacities were compared to discover the optimal using the mean absolute percentage error, root mean square error, mean error rate, and root mean square percentage error.
Results
The four preferred models identified were comprised of the SARIMA(0,1,0)(0,1,1)12, NAR with 14 hidden neurons and five delays, SARIMA-NAR with 33 hidden neurons and five delays, and SARIMA-NARX with 16 hidden neurons and 4 delays. Among which presenting the lowest values of the aforementioned indices in both simulation and prediction horizons is the SARIMA-NARX method. Analyses from the data suggested that scarlet fever was a seasonal disease with predominant peaks of summer and winter and a substantial rising trend in the scarlet fever notifications was observed with an acceleration of 9.641% annually, particularly since 2011 with 12.869%, and moreover such a trend will be projected to continue in the coming year.
Conclusions
The SARIMA-NARX technique has the promising ability to better consider both linearity and non-linearity behind scarlet fever data than the others, which significantly facilitates its prevention and intervention of scarlet fever. Besides, under current trend of ongoing resurgence, specific strategies and countermeasures should be formulated to target scarlet fever.Comparing enzyme activity modifier equations through the development of global data fitting templates in Excelhttps://peerj.com/articles/60822018-12-142018-12-14Ryan Walsh
The classical way of defining enzyme inhibition has obscured the distinction between inhibitory effect and the inhibitor binding constant. This article examines the relationship between the simple binding curve used to define biomolecular interactions and the standard inhibitory term (1 + ([I]∕Ki)). By understanding how this term relates to binding curves which are ubiquitously used to describe biological processes, a modifier equation which distinguishes between inhibitor binding and the inhibitory effect, is examined. This modifier equation which can describe both activation and inhibition is compared to standard inhibitory equations with the development of global data fitting templates in Excel and via the global fitting of these equations to simulated and previously published datasets. In both cases, this modifier equation was able to match or outperform the other equations by providing superior fits to the datasets. The ability of this single equation to outperform the other equations suggests an over-complication of the field. This equation and the template developed in this article should prove to be useful tools in the study of enzyme inhibition and activation.
The classical way of defining enzyme inhibition has obscured the distinction between inhibitory effect and the inhibitor binding constant. This article examines the relationship between the simple binding curve used to define biomolecular interactions and the standard inhibitory term (1 + ([I]∕Ki)). By understanding how this term relates to binding curves which are ubiquitously used to describe biological processes, a modifier equation which distinguishes between inhibitor binding and the inhibitory effect, is examined. This modifier equation which can describe both activation and inhibition is compared to standard inhibitory equations with the development of global data fitting templates in Excel and via the global fitting of these equations to simulated and previously published datasets. In both cases, this modifier equation was able to match or outperform the other equations by providing superior fits to the datasets. The ability of this single equation to outperform the other equations suggests an over-complication of the field. This equation and the template developed in this article should prove to be useful tools in the study of enzyme inhibition and activation.Modelling of the SDF-1/CXCR4 regulated in vivo homing of therapeutic mesenchymal stem/stromal cells in micehttps://peerj.com/articles/60722018-12-062018-12-06Wang JinXiaowen LiangAnastasia BrooksKathryn FutregaXin LiuMichael R. DoranMatthew J. SimpsonMichael S. RobertsHaolu Wang
Background
Mesenchymal stem/stromal cells (MSCs) are a promising tool for cell-based therapies in the treatment of tissue injury. The stromal cell-derived factor-1 (SDF-1)/CXC chemokine receptor 4 (CXCR4) axis plays a significant role in directing MSC homing to sites of injury. However in vivo MSC distribution following intravenous transplantation remains poorly understood, potentially hampering the precise prediction and evaluation of therapeutic efficacy.
Methods
A murine model of partial ischemia/reperfusion (I/R) is used to induce liver injury, increase the hepatic levels of SDF-1, and study in vivo MSC distribution. Hypoxia-preconditioning increases the expression of CXCR4 in human bone marrow-derived MSCs. Quantitative assays for human DNA using droplet digital PCR (ddPCR) allow us to examine the in vivo kinetics of intravenously infused human MSCs in mouse blood and liver. A mathematical model-based system is developed to characterize in vivo homing of human MSCs in mouse models with SDF-1 levels in liver and CXCR4 expression on the transfused MSCs. The model is calibrated to experimental data to provide novel estimates of relevant parameter values.
Results
Images of immunohistochemistry for SDF-1 in the mouse liver with I/R injury show a significantly higher SDF-1 level in the I/R injured liver than that in the control. Correspondingly, the ddPCR results illustrate a higher MSC concentration in the I/R injured liver than the normal liver. CXCR4 is overexpressed in hypoxia-preconditioned MSCs. An increased number of hypoxia-preconditioned MSCs in the I/R injured liver is observed from the ddPCR results. The model simulations align with the experimental data of control and hypoxia-preconditioned human MSC distribution in normal and injured mouse livers, and accurately predict the experimental outcomes with different MSC doses.
Discussion
The modelling results suggest that SDF-1 in organs is an effective in vivo attractant for MSCs through the SDF-1/CXCR4 axis and reveal the significance of the SDF-1/CXCR4 chemotaxis on in vivo homing of MSCs. This in vivo modelling approach allows qualitative characterization and prediction of the MSC homing to normal and injured organs on the basis of clinically accessible variables, such as the MSC dose and SDF-1 concentration in blood. This model could also be adapted to abnormal conditions and/or other types of circulating cells to predict in vivo homing patterns.
Background
Mesenchymal stem/stromal cells (MSCs) are a promising tool for cell-based therapies in the treatment of tissue injury. The stromal cell-derived factor-1 (SDF-1)/CXC chemokine receptor 4 (CXCR4) axis plays a significant role in directing MSC homing to sites of injury. However in vivo MSC distribution following intravenous transplantation remains poorly understood, potentially hampering the precise prediction and evaluation of therapeutic efficacy.
Methods
A murine model of partial ischemia/reperfusion (I/R) is used to induce liver injury, increase the hepatic levels of SDF-1, and study in vivo MSC distribution. Hypoxia-preconditioning increases the expression of CXCR4 in human bone marrow-derived MSCs. Quantitative assays for human DNA using droplet digital PCR (ddPCR) allow us to examine the in vivo kinetics of intravenously infused human MSCs in mouse blood and liver. A mathematical model-based system is developed to characterize in vivo homing of human MSCs in mouse models with SDF-1 levels in liver and CXCR4 expression on the transfused MSCs. The model is calibrated to experimental data to provide novel estimates of relevant parameter values.
Results
Images of immunohistochemistry for SDF-1 in the mouse liver with I/R injury show a significantly higher SDF-1 level in the I/R injured liver than that in the control. Correspondingly, the ddPCR results illustrate a higher MSC concentration in the I/R injured liver than the normal liver. CXCR4 is overexpressed in hypoxia-preconditioned MSCs. An increased number of hypoxia-preconditioned MSCs in the I/R injured liver is observed from the ddPCR results. The model simulations align with the experimental data of control and hypoxia-preconditioned human MSC distribution in normal and injured mouse livers, and accurately predict the experimental outcomes with different MSC doses.
Discussion
The modelling results suggest that SDF-1 in organs is an effective in vivo attractant for MSCs through the SDF-1/CXCR4 axis and reveal the significance of the SDF-1/CXCR4 chemotaxis on in vivo homing of MSCs. This in vivo modelling approach allows qualitative characterization and prediction of the MSC homing to normal and injured organs on the basis of clinically accessible variables, such as the MSC dose and SDF-1 concentration in blood. This model could also be adapted to abnormal conditions and/or other types of circulating cells to predict in vivo homing patterns.Shielding effects of myelin sheath on axolemma depolarization under transverse electric field stimulationhttps://peerj.com/articles/60202018-12-032018-12-03Hui YeJeffrey Ng
Axonal stimulation with electric currents is an effective method for controlling neural activity. An electric field parallel to the axon is widely accepted as the predominant component in the activation of an axon. However, recent studies indicate that the transverse component to the axolemma is also effective in depolarizing the axon. To quantitatively investigate the amount of axolemma polarization induced by a transverse electric field, we computed the transmembrane potential (Vm) for a conductive body that represents an unmyelinated axon (or the bare axon between the myelin sheath in a myelinated axon). We also computed the transmembrane potential of the sheath-covered axonal segment in a myelinated axon. We then systematically analyzed the biophysical factors that affect axonal polarization under transverse electric stimulation for both the bare and sheath-covered axons. Geometrical patterns of polarization of both axon types were dependent on field properties (magnitude and field orientation to the axon). Polarization of both axons was also dependent on their axolemma radii and electrical conductivities. The myelin provided a significant “shielding effect” against the transverse electric fields, preventing excessive axolemma depolarization. Demyelination could allow for prominent axolemma depolarization in the transverse electric field, via a significant increase in myelin conductivity. This shifts the voltage drop of the myelin sheath to the axolemma. Pathological changes at a cellular level should be considered when electric fields are used for the treatment of demyelination diseases. The calculated term for membrane polarization (Vm) could be used to modify the current cable equation that describes axon excitation by an external electric field to account for the activating effects of both parallel and transverse fields surrounding the target axon.
Axonal stimulation with electric currents is an effective method for controlling neural activity. An electric field parallel to the axon is widely accepted as the predominant component in the activation of an axon. However, recent studies indicate that the transverse component to the axolemma is also effective in depolarizing the axon. To quantitatively investigate the amount of axolemma polarization induced by a transverse electric field, we computed the transmembrane potential (Vm) for a conductive body that represents an unmyelinated axon (or the bare axon between the myelin sheath in a myelinated axon). We also computed the transmembrane potential of the sheath-covered axonal segment in a myelinated axon. We then systematically analyzed the biophysical factors that affect axonal polarization under transverse electric stimulation for both the bare and sheath-covered axons. Geometrical patterns of polarization of both axon types were dependent on field properties (magnitude and field orientation to the axon). Polarization of both axons was also dependent on their axolemma radii and electrical conductivities. The myelin provided a significant “shielding effect” against the transverse electric fields, preventing excessive axolemma depolarization. Demyelination could allow for prominent axolemma depolarization in the transverse electric field, via a significant increase in myelin conductivity. This shifts the voltage drop of the myelin sheath to the axolemma. Pathological changes at a cellular level should be considered when electric fields are used for the treatment of demyelination diseases. The calculated term for membrane polarization (Vm) could be used to modify the current cable equation that describes axon excitation by an external electric field to account for the activating effects of both parallel and transverse fields surrounding the target axon.Cochlear impulse responses resolved into sets of gammatones: the case for beating of closely spaced local resonanceshttps://peerj.com/articles/60162018-11-272018-11-27Andrew BellHero P. Wit
Gammatones have had a long history in auditory studies, and recent theoretical work suggests they may play an important role in cochlear mechanics as well. Following this lead, the present paper takes five examples of basilar membrane impulse responses and uses a curve-fitting algorithm to decompose them into a number of discrete gammatones. The limits of this ‘sum of gammatones’ (SOG) method to accurately represent the impulse response waveforms were tested and it was found that at least two and up to six gammatones could be isolated from each example. Their frequencies were stable and largely independent of stimulus parameters. The gammatones typically formed a regular series in which the frequency ratio between successive members was about 1.1. Adding together the first few gammatones in a set produced beating-like waveforms which mimicked waxing and waning, and the instantaneous frequencies of the waveforms were also well reproduced, providing an explanation for frequency glides. Consideration was also given to the impulse response of a pair of elastically coupled masses—the basis of two-degree-of-freedom models comprised of coupled basilar and tectorial membranes—and the resulting waveform was similar to a pair of beating gammatones, perhaps explaining why the SOG method seems to work well in describing cochlear impulse responses. A major limitation of the SOG method is that it cannot distinguish a waveform resulting from an actual physical resonance from one derived from overfitting, but taken together the method points to the presence of a series of closely spaced local resonances in the cochlea.
Gammatones have had a long history in auditory studies, and recent theoretical work suggests they may play an important role in cochlear mechanics as well. Following this lead, the present paper takes five examples of basilar membrane impulse responses and uses a curve-fitting algorithm to decompose them into a number of discrete gammatones. The limits of this ‘sum of gammatones’ (SOG) method to accurately represent the impulse response waveforms were tested and it was found that at least two and up to six gammatones could be isolated from each example. Their frequencies were stable and largely independent of stimulus parameters. The gammatones typically formed a regular series in which the frequency ratio between successive members was about 1.1. Adding together the first few gammatones in a set produced beating-like waveforms which mimicked waxing and waning, and the instantaneous frequencies of the waveforms were also well reproduced, providing an explanation for frequency glides. Consideration was also given to the impulse response of a pair of elastically coupled masses—the basis of two-degree-of-freedom models comprised of coupled basilar and tectorial membranes—and the resulting waveform was similar to a pair of beating gammatones, perhaps explaining why the SOG method seems to work well in describing cochlear impulse responses. A major limitation of the SOG method is that it cannot distinguish a waveform resulting from an actual physical resonance from one derived from overfitting, but taken together the method points to the presence of a series of closely spaced local resonances in the cochlea.Optimal and near-optimal exponent-pairs for the Bertalanffy-Pütter growth modelhttps://peerj.com/articles/59732018-11-232018-11-23Katharina Renner-MartinNorbert BrunnerManfred KühleitnerWerner-Georg NowakKlaus Scheicher
The Bertalanffy–Pütter growth model describes mass m at age t by means of the differential equation dm/dt = p * ma − q * mb. The special case using the von Bertalanffy exponent-pair a = 2/3 and b = 1 is most common (it corresponds to the von Bertalanffy growth function VBGF for length in fishery literature). Fitting VBGF to size-at-age data requires the optimization of three model parameters (the constants p, q, and an initial value for the differential equation). For the general Bertalanffy–Pütter model, two more model parameters are optimized (the pair a < b of non-negative exponents). While this reduces bias in growth estimates, it increases model complexity and more advanced optimization methods are needed, such as the Nelder–Mead amoeba method, interior point methods, or simulated annealing. Is the improved performance worth these efforts? For the case, where the exponent b = 1 remains fixed, it is known that for most fish data any exponent a < 1 could be used to model growth without affecting the fit to the data significantly (when the other parameters were optimized). We hypothesized that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and we tested this conjecture for a data set (20,166 fish) about the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. To this end, we assessed the fit on a grid of 14,281 exponent-pairs (a, b) and identified the best fitting model curve on the boundary a = b of the grid (a = b = 0.686); it corresponds to the generalized Gompertz equation dm/dt = p * ma − q * ln(m) * ma. Using the Akaike information criterion for model selection, the answer to the conjecture was no: The von Bertalanffy exponent-pair model (but not the logistic model) remained parsimonious. However, the bias reduction attained by the optimal exponent-pair may be worth the tradeoff with complexity in some situations where predictive power is solely preferred. Therefore, we recommend the use of the Bertalanffy–Pütter model (and of its limit case, the generalized Gompertz model) in natural resources management (such as in fishery stock assessments), as it relies on careful quantitative assessments to recommend policies for sustainable resource usage.
The Bertalanffy–Pütter growth model describes mass m at age t by means of the differential equation dm/dt = p * ma − q * mb. The special case using the von Bertalanffy exponent-pair a = 2/3 and b = 1 is most common (it corresponds to the von Bertalanffy growth function VBGF for length in fishery literature). Fitting VBGF to size-at-age data requires the optimization of three model parameters (the constants p, q, and an initial value for the differential equation). For the general Bertalanffy–Pütter model, two more model parameters are optimized (the pair a < b of non-negative exponents). While this reduces bias in growth estimates, it increases model complexity and more advanced optimization methods are needed, such as the Nelder–Mead amoeba method, interior point methods, or simulated annealing. Is the improved performance worth these efforts? For the case, where the exponent b = 1 remains fixed, it is known that for most fish data any exponent a < 1 could be used to model growth without affecting the fit to the data significantly (when the other parameters were optimized). We hypothesized that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and we tested this conjecture for a data set (20,166 fish) about the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. To this end, we assessed the fit on a grid of 14,281 exponent-pairs (a, b) and identified the best fitting model curve on the boundary a = b of the grid (a = b = 0.686); it corresponds to the generalized Gompertz equation dm/dt = p * ma − q * ln(m) * ma. Using the Akaike information criterion for model selection, the answer to the conjecture was no: The von Bertalanffy exponent-pair model (but not the logistic model) remained parsimonious. However, the bias reduction attained by the optimal exponent-pair may be worth the tradeoff with complexity in some situations where predictive power is solely preferred. Therefore, we recommend the use of the Bertalanffy–Pütter model (and of its limit case, the generalized Gompertz model) in natural resources management (such as in fishery stock assessments), as it relies on careful quantitative assessments to recommend policies for sustainable resource usage.