PeerJ Preprints: Mathematical Biologyhttps://peerj.com/preprints/index.atom?journal=peerj&subject=1900Mathematical Biology articles published in PeerJ PreprintsEllipse packing in 2D cell tessellation: A theoretical explanation for Lewis’s law and Aboav-Weaire’s lawhttps://peerj.com/preprints/274212018-12-092018-12-09Kai Xu
Background: To date, the theoretical bases of Lewis’s law and Aboav-Weaire’s law are still unclear.
Methods: Software R with package Conicfit was used to fit ellipses based on geometric parameters of polygonal cells of red alga Pyropia haitanensis.
Results: The average form deviation of vertexes from the fitted ellipse was 0±3.1 % (8,291 vertices in 1375 cells were examined). Area of the polygonal cell was 0.9±0.1 times of area of the ellipse’s maximal inscribed polygon (EMIP). These results indicated that the polygonal cells can be considered as ellipse’s inscribed polygons (EIPs) and tended to form EMIPs. This phenomenon was named as ellipse packing. Then, an improved relation of Lewis’s law for a n-edged cell was derived
\[cell\ area=0.5nab\sin(\frac{2\pi}{n})(1-\frac{3}{n^2})\]
where, a and b are the semi-major axis and the semi-minor axis of fitted ellipse, respectively. This study also improved the relation of Aboav-Weaire’s law
\[number\ of\ neighboring\ cells=6+\frac{6-n}{n}(\frac{a}{b}+\frac{3}{n^2})\]
Conclusions: Ellipse packing is a short-range order which places restrictions on the direction of cell division and the turning angles of cell edges. The ellipse packing requires allometric growth of cell edges. Lewis’s law describes the effect of deformation from EMIP to EIP on area. Aboav-Weaire’s law mainly reflects the effect of deformation from circle to ellipse on number of neighboring cells, and the deformation from EMIP to EIP has only a minor effect. The results of this study could help to simulate the dynamics of cell topology during growth.
Background: To date, the theoretical bases of Lewis’s law and Aboav-Weaire’s law are still unclear.Methods: Software R with package Conicfit was used to fit ellipses based on geometric parameters of polygonal cells of red alga Pyropia haitanensis.Results: The average form deviation of vertexes from the fitted ellipse was 0±3.1 % (8,291 vertices in 1375 cells were examined). Area of the polygonal cell was 0.9±0.1 times of area of the ellipse’s maximal inscribed polygon (EMIP). These results indicated that the polygonal cells can be considered as ellipse’s inscribed polygons (EIPs) and tended to form EMIPs. This phenomenon was named as ellipse packing. Then, an improved relation of Lewis’s law for a n-edged cell was derived\[cell\ area=0.5nab\sin(\frac{2\pi}{n})(1-\frac{3}{n^2})\]where, a and b are the semi-major axis and the semi-minor axis of fitted ellipse, respectively. This study also improved the relation of Aboav-Weaire’s law\[number\ of\ neighboring\ cells=6+\frac{6-n}{n}(\frac{a}{b}+\frac{3}{n^2})\]Conclusions: Ellipse packing is a short-range order which places restrictions on the direction of cell division and the turning angles of cell edges. The ellipse packing requires allometric growth of cell edges. Lewis’s law describes the effect of deformation from EMIP to EIP on area. Aboav-Weaire’s law mainly reflects the effect of deformation from circle to ellipse on number of neighboring cells, and the deformation from EMIP to EIP has only a minor effect. The results of this study could help to simulate the dynamics of cell topology during growth.Discrete stochastic marine metapopulation disease modelhttps://peerj.com/preprints/264542018-12-042018-12-04Gorka BidegainTal Ben-Horin
Some marine microparasitic pathogens can survive several months outside the host in the water column to make contact with hosts or to be absorbed or filtered by hosts. Once inside, pathogens invade the host if they find suitable conditions for reproduction within the host. This transmission from the environment occurs via pathogens released from infected animals and dead infected animals. Some recent modeling studies concentrated on the disease dynamic imposed by this complex interaction between population and water column at the host-pathogen level in single populations. However, only when a marine disease can be understood at the metapopulation scale effective approaches to management will become routinely achievable. In this paper we explore the disease dynamics at the metapopulation applying a stochastic version. The discrete-time disease model in this paper investigates both spatial and temporal dynamics of hosts and waterborne pathogens in a three patch system. This metapopulation with a patch providing infective particles and susceptible and infected individuals by dispersal tries to imitate the effect of current forces in the ocean on the passive dispersal of organisms. The model detects system behaviors that are not present in single population continuous-time and deterministic models.
Some marine microparasitic pathogens can survive several months outside the host in the water column to make contact with hosts or to be absorbed or filtered by hosts. Once inside, pathogens invade the host if they find suitable conditions for reproduction within the host. This transmission from the environment occurs via pathogens released from infected animals and dead infected animals. Some recent modeling studies concentrated on the disease dynamic imposed by this complex interaction between population and water column at the host-pathogen level in single populations. However, only when a marine disease can be understood at the metapopulation scale effective approaches to management will become routinely achievable. In this paper we explore the disease dynamics at the metapopulation applying a stochastic version. The discrete-time disease model in this paper investigates both spatial and temporal dynamics of hosts and waterborne pathogens in a three patch system. This metapopulation with a patch providing infective particles and susceptible and infected individuals by dispersal tries to imitate the effect of current forces in the ocean on the passive dispersal of organisms. The model detects system behaviors that are not present in single population continuous-time and deterministic models.Comparison of genome sequences via projection extractor upon virtual mixerhttps://peerj.com/preprints/273332018-11-082018-11-08Hongjie YuYuan-Ting ZhangWei Fang
To compare multiple genome sequences, we transform each primary genome sequence into corresponding k-mer-based vectors. According to the principle of independent component analysis (ICA), the operation can be regarded as mixing multiple source genomic signals via several sensors, through which we can obtain the mixed vectors with equal-length from the corresponding genome sequences with different length. However, this mixing operation is performed by counting all the k-mer-based frequencies, instead of using real hardware of sensors. Thus, we name this preprocessing operation as virtual mixer (VM). Using ICA-based transformation, we projected all the vectors upon their independent components to capture the coefficients-based feature vector through the projection extractor (PE), which has been proved to have a property of distance preserving. Then, we used the proposed VMPE model upon three representative real datasets of genome sequence to test the efficiency for the model. The contrastive analysis results indicate that the proposed VMPE model performs well in similarity analysis.
To compare multiple genome sequences, we transform each primary genome sequence into corresponding k-mer-based vectors. According to the principle of independent component analysis (ICA), the operation can be regarded as mixing multiple source genomic signals via several sensors, through which we can obtain the mixed vectors with equal-length from the corresponding genome sequences with different length. However, this mixing operation is performed by counting all the k-mer-based frequencies, instead of using real hardware of sensors. Thus, we name this preprocessing operation as virtual mixer (VM).Using ICA-based transformation, we projected all the vectors upon their independent components to capture the coefficients-based feature vector through the projection extractor (PE), which has been proved to have a property of distance preserving. Then, we used the proposed VMPE model upon three representative real datasets of genome sequence to test the efficiency for the model. The contrastive analysis results indicate that the proposed VMPE model performs well in similarity analysis.Approach for numerically describing and classifying benthic animals’ polymorphic cells morphologyhttps://peerj.com/preprints/272712018-10-112018-10-11Yura A KaretinEduardas Cicinskas
Describing cell morphology is a tricky task, prone to misinterpretation due to subjective nature of the human observer and his vocabulary limitations. Consequently, these limitations actuate prevalence of non-formalized, statistically unverifiable language use. This determines the reason for overlooking cell shape as a viable parameter for describing cell's functional state intricacies. In this study we demonstrate the use of mathematical parameters set for describing two-dimensional fractals, such as: convex hull, density, roundness and asymmetry, for comparative in vitro morphological analysis of sprawled starfishes' Aphelasterias japonica and Patiria pectinifera (Echinodermata: Asteroidea) coelomocytes, and bivalve's Callista brevisiphonata (Mollusca: Bivalvia) hemocytes. We found that these parameters allow us to describe visually distinguishable but verbally indescribable "chaotic" sprawled cell shapes. Furthermore, resulting numerical cell descriptions differs significantly, enabling for their species-specific grouping and classification. We argue that presented morphometric methodology can be used for describing and classifying cells of any arbitrary morphology, as well as compiling "cell shape - cell functional state" match library for later use in in vitro analysis, potentially for cells of any animal.
Describing cell morphology is a tricky task, prone to misinterpretation due to subjective nature of the human observer and his vocabulary limitations. Consequently, these limitations actuate prevalence of non-formalized, statistically unverifiable language use. This determines the reason for overlooking cell shape as a viable parameter for describing cell's functional state intricacies. In this study we demonstrate the use of mathematical parameters set for describing two-dimensional fractals, such as: convex hull, density, roundness and asymmetry, for comparative in vitro morphological analysis of sprawled starfishes' Aphelasterias japonica and Patiria pectinifera (Echinodermata: Asteroidea) coelomocytes, and bivalve's Callista brevisiphonata (Mollusca: Bivalvia) hemocytes. We found that these parameters allow us to describe visually distinguishable but verbally indescribable "chaotic" sprawled cell shapes. Furthermore, resulting numerical cell descriptions differs significantly, enabling for their species-specific grouping and classification. We argue that presented morphometric methodology can be used for describing and classifying cells of any arbitrary morphology, as well as compiling "cell shape - cell functional state" match library for later use in in vitro analysis, potentially for cells of any animal.Alpha shapes: Determining 3D shape complexity across morphologically diverse structureshttps://peerj.com/preprints/272702018-10-102018-10-10James D GardinerJulia BehnsenCharlotte A Brassey
Background. Following recent advances in bioimaging, high-resolution 3D models of biological structures are now generated rapidly and at low-cost. To utilise this data to address evolutionary and ecological questions, an array of tools has been developed to conduct 3D shape analysis and quantify topographic complexity. Here we focus particularly on shape techniques applied to irregular-shaped objects lacking clear homologous landmarks, and propose the new ‘alpha-shapes’ method for quantifying 3D shape complexity.
Methods. We apply alpha-shapes to quantify shape complexity in the mammalian baculum as an example of a morphologically disparate structure. Micro- computed-tomography (μCT) scans of bacula were conducted. Bacula were binarised and converted into point clouds. Following application of a scaling factor to account for absolute differences in size, a suite of alpha-shapes was fitted to each specimen. An alpha shape is a formed from a subcomplex of the Delaunay triangulation of a given set of points, and ranges in refinement from a very coarse mesh (approximating convex hulls) to a very fine fit. ‘Optimal’ alpha was defined as the degree of refinement necessary in order for alpha-shape volume to equal CT voxel volume, and was taken as a metric of overall shape ‘complexity’.
Results Our results show that alpha-shapes can be used to quantify interspecific variation in shape ‘complexity’ within biological structures of disparate geometry. The ‘stepped’ nature of alpha curves is informative with regards to the contribution of specific morphological features to overall shape ‘complexity’. Alpha-shapes agrees with other measures of topographic complexity (dissection index, Dirichlet normal energy) in identifying ursid bacula as having low shape complexity. However, alpha-shapes estimates mustelid bacula as possessing the highest topographic complexity, contrasting with other shape metrics. 3D fractal dimension is found to be an inappropriate metric of complexity when applied to bacula.
Conclusions. The alpha-shapes methodology can be used to calculate ‘optimal’ alpha refinement as a proxy for shape ‘complexity’ without identifying landmarks. The implementation of alpha-shapes is straightforward, and is automated to process large datasets quickly. Beyond genital shape, we consider the alpha-shapes technique to hold considerable promise for new applications across evolutionary, ecological and palaeoecological disciplines.
Background. Following recent advances in bioimaging, high-resolution 3D models of biological structures are now generated rapidly and at low-cost. To utilise this data to address evolutionary and ecological questions, an array of tools has been developed to conduct 3D shape analysis and quantify topographic complexity. Here we focus particularly on shape techniques applied to irregular-shaped objects lacking clear homologous landmarks, and propose the new ‘alpha-shapes’ method for quantifying 3D shape complexity.Methods. We apply alpha-shapes to quantify shape complexity in the mammalian baculum as an example of a morphologically disparate structure. Micro- computed-tomography (μCT) scans of bacula were conducted. Bacula were binarised and converted into point clouds. Following application of a scaling factor to account for absolute differences in size, a suite of alpha-shapes was fitted to each specimen. An alpha shape is a formed from a subcomplex of the Delaunay triangulation of a given set of points, and ranges in refinement from a very coarse mesh (approximating convex hulls) to a very fine fit. ‘Optimal’ alpha was defined as the degree of refinement necessary in order for alpha-shape volume to equal CT voxel volume, and was taken as a metric of overall shape ‘complexity’.Results Our results show that alpha-shapes can be used to quantify interspecific variation in shape ‘complexity’ within biological structures of disparate geometry. The ‘stepped’ nature of alpha curves is informative with regards to the contribution of specific morphological features to overall shape ‘complexity’. Alpha-shapes agrees with other measures of topographic complexity (dissection index, Dirichlet normal energy) in identifying ursid bacula as having low shape complexity. However, alpha-shapes estimates mustelid bacula as possessing the highest topographic complexity, contrasting with other shape metrics. 3D fractal dimension is found to be an inappropriate metric of complexity when applied to bacula.Conclusions. The alpha-shapes methodology can be used to calculate ‘optimal’ alpha refinement as a proxy for shape ‘complexity’ without identifying landmarks. The implementation of alpha-shapes is straightforward, and is automated to process large datasets quickly. Beyond genital shape, we consider the alpha-shapes technique to hold considerable promise for new applications across evolutionary, ecological and palaeoecological disciplines.Seasonality in ecology: Progress and prospects in theoryhttps://peerj.com/preprints/272352018-09-212018-09-21Easton R WhiteAlan Hastings
Seasonality is an important feature of essentially all natural systems but the consequences of seasonality have been vastly underappreciated. Early work emphasized the role of seasonality in driving cyclic population dynamics, but the consequences of seasonality for ecological processes are far broader. In ecological systems, seasonality may include variations in temperature, precipitation, or other processes. Seasonality is typically not explicitly included in either empirical or theoretical studies. However, many aspects of ecological dynamics can only be understood when seasonality is included, ranging from the oscillations in the incidence of childhood diseases to the coexistence of species. Further, studies of phenology and global climate change only make sense in the context of seasonal dynamics. Our goal is to outline what is now known about seasonality and to set the stage for future efforts. We review the effects of seasonality on ecological systems in both laboratory and field settings. We then discuss approaches for incorporating seasonality in mathematical models, including Floquet theory. We argue, however, that these tools are still limited in scope and more approaches need to be developed. We demonstrate the range of impacts of seasonality on ecological systems and show the necessity of incorporating seasonality to understand ecological dynamics.
Seasonality is an important feature of essentially all natural systems but the consequences of seasonality have been vastly underappreciated. Early work emphasized the role of seasonality in driving cyclic population dynamics, but the consequences of seasonality for ecological processes are far broader. In ecological systems, seasonality may include variations in temperature, precipitation, or other processes. Seasonality is typically not explicitly included in either empirical or theoretical studies. However, many aspects of ecological dynamics can only be understood when seasonality is included, ranging from the oscillations in the incidence of childhood diseases to the coexistence of species. Further, studies of phenology and global climate change only make sense in the context of seasonal dynamics. Our goal is to outline what is now known about seasonality and to set the stage for future efforts. We review the effects of seasonality on ecological systems in both laboratory and field settings. We then discuss approaches for incorporating seasonality in mathematical models, including Floquet theory. We argue, however, that these tools are still limited in scope and more approaches need to be developed. We demonstrate the range of impacts of seasonality on ecological systems and show the necessity of incorporating seasonality to understand ecological dynamics.The mathematics of extinction across scales: from populations to the biospherehttps://peerj.com/preprints/33672018-09-112018-09-11Colin CarlsonKevin BurgioTad DallasWayne Getz
The sixth mass extinction poses an unparalleled quantitative challenge to conservation biologists. Mathematicians and ecologists alike face the problem of developing models that can scale predictions of extinction rates from populations to the level of a species, or even to an entire ecosystem. We review some of the most basic stochastic and analytical methods of calculating extinction risk at different scales, including population viability analysis, stochastic metapopulation occupancy models, and the species area relationship. We also consider two major extensions of theory: the possibility of evolutionary rescue from extinction in a changing environment, and the posthumous assignment of an extinction date from sighting records. In the case of the latter, we provide an example using data on Spix's macaw (Cyanopsitta spixii), the "rarest bird in the world," to demonstrate the challenges associated with extinction date research.
The sixth mass extinction poses an unparalleled quantitative challenge to conservation biologists. Mathematicians and ecologists alike face the problem of developing models that can scale predictions of extinction rates from populations to the level of a species, or even to an entire ecosystem. We review some of the most basic stochastic and analytical methods of calculating extinction risk at different scales, including population viability analysis, stochastic metapopulation occupancy models, and the species area relationship. We also consider two major extensions of theory: the possibility of evolutionary rescue from extinction in a changing environment, and the posthumous assignment of an extinction date from sighting records. In the case of the latter, we provide an example using data on Spix's macaw (Cyanopsitta spixii), the "rarest bird in the world," to demonstrate the challenges associated with extinction date research.Synchronization, oscillator death, and frequency modulation in a class of biologically inspired coupled oscillatorshttps://peerj.com/preprints/264472018-09-112018-09-11Alessio FranciMarco A Herrera-ValdezMiguel Lara-AparicioPablo Padilla-Longoria
The general purpose of this paper is to build up on our understanding of the basic mathematical principles that underlie the emergence of synchronous biological rhythms, in particular, the circadian clock. To do so, we study the role that the coupling strength, coupling type, and noise play in the synchronization of a system of coupled, nonlinear oscillators. First, we study a deterministic model based on Van der Pol coupled oscillators, modeling a population of diffusively coupled cells, to find regions in the parameter space for which synchronous oscillations emerge and to provide conditions under which diffusive coupling kills the synchronous oscillation. Second, we study how noise and coupling interact and lead to synchronous oscillations in linearly coupled oscillators, modeling the interaction between various pacemaker populations, each having an endogenous circadian clock. To do so, we use the Fokker-Planck equation associated to the system. We show how coupling can tune the frequency of the emergent synchronous oscillation, which provides a general mechanism to make fast (ultradian) pacemakers slow (circadian) and synchronous via coupling. The basic mechanisms behind the generation of oscillations and the emergence of synchrony that we describe here can be used to guide further studies of coupled
oscillations in biophysical nonlinear models.
The general purpose of this paper is to build up on our understanding of the basic mathematical principles that underlie the emergence of synchronous biological rhythms, in particular, the circadian clock. To do so, we study the role that the coupling strength, coupling type, and noise play in the synchronization of a system of coupled, nonlinear oscillators. First, we study a deterministic model based on Van der Pol coupled oscillators, modeling a population of diffusively coupled cells, to find regions in the parameter space for which synchronous oscillations emerge and to provide conditions under which diffusive coupling kills the synchronous oscillation. Second, we study how noise and coupling interact and lead to synchronous oscillations in linearly coupled oscillators, modeling the interaction between various pacemaker populations, each having an endogenous circadian clock. To do so, we use the Fokker-Planck equation associated to the system. We show how coupling can tune the frequency of the emergent synchronous oscillation, which provides a general mechanism to make fast (ultradian) pacemakers slow (circadian) and synchronous via coupling. The basic mechanisms behind the generation of oscillations and the emergence of synchrony that we describe here can be used to guide further studies of coupledoscillations in biophysical nonlinear models.The time distribution of biological phenomena – illustrated with the London marathonhttps://peerj.com/preprints/271752018-09-062018-09-06Miguel Franco
Background. The time distribution of biological phenomena (phenology) is a subject of wide interest, but a general statistical distribution to describe and quantify its essential properties is lacking. Existing distributions are limiting, if not entirely inappropriate, because their parameters do not in general correlate with biologically relevant attributes of the organism and the conditions under which they find themselves. Methods. A distribution function that allows quantification of three essential properties of a biological dynamic process occurring over a continuous timescale was derived from first principles. The distribution turned out to have three parameters with clear meanings and units: (i) a scaled rate of completion (dimensionless), (ii) a measure of temporal concentration of the process (units: time-1), and (iii) an overall measure of temporal delay (units: time). Its performance as an accurate description of the process was tested with completion data for the London Marathon employing non-linear regression. Results. The parameters of the distribution correlated with biological attributes of the runners (gender and age) and with the maximum temperature on the day of the race. These relationships mirrored known differences in morphology and physiology of participants and the deterioration of these biological attributes with age (senescence), as well as the known effects of hypo- and hyperthermia. Discussion. By relating the variation in parameter values to possible biological and environmental variables, the marathon example demonstrates the ability of the distribution to help identify possible triggers and drivers of the duration, shape and temporal shift of its temporal distribution. This more detailed account of the effect of biological and environmental variables would provide a deeper insight into the drivers of a wide variety of phenological phenomena of high current interest, such as the shifting patterns of leafing, flowering, growth, migration, etc. of many organisms worldwide.
Background. The time distribution of biological phenomena (phenology) is a subject of wide interest, but a general statistical distribution to describe and quantify its essential properties is lacking. Existing distributions are limiting, if not entirely inappropriate, because their parameters do not in general correlate with biologically relevant attributes of the organism and the conditions under which they find themselves. Methods. A distribution function that allows quantification of three essential properties of a biological dynamic process occurring over a continuous timescale was derived from first principles. The distribution turned out to have three parameters with clear meanings and units: (i) a scaled rate of completion (dimensionless), (ii) a measure of temporal concentration of the process (units: time-1), and (iii) an overall measure of temporal delay (units: time). Its performance as an accurate description of the process was tested with completion data for the London Marathon employing non-linear regression. Results. The parameters of the distribution correlated with biological attributes of the runners (gender and age) and with the maximum temperature on the day of the race. These relationships mirrored known differences in morphology and physiology of participants and the deterioration of these biological attributes with age (senescence), as well as the known effects of hypo- and hyperthermia. Discussion. By relating the variation in parameter values to possible biological and environmental variables, the marathon example demonstrates the ability of the distribution to help identify possible triggers and drivers of the duration, shape and temporal shift of its temporal distribution. This more detailed account of the effect of biological and environmental variables would provide a deeper insight into the drivers of a wide variety of phenological phenomena of high current interest, such as the shifting patterns of leafing, flowering, growth, migration, etc. of many organisms worldwide.Optimal exponent-pairs for the Bertalanffy-Pütter growth modelhttps://peerj.com/preprints/271522018-08-282018-08-28Katharina 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 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). For data fitting using general exponents, five model parameters need to be optimized, the pair a<b of non-negative exponents, the non-negative constants p and q, and a positive initial value m0 for the differential equation. For the case b=1 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 p, q, m0 were optimized). Thereby, data fitting used the method of least squares, minimizing the sum of squared errors (SSE). It was conjectured that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and thereby reduce SSE. This conjecture was tested for a data set for the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. Compared to the Bertalanffy exponent-pair the optimal exponent-pair achieved a reduction of SSE by 10%. However, when the optimization of additional parameters was penalized, using the Akaike information criterion (AIC), then the optimal exponent-pair model had a higher (worse) AIC, when compared to the Bertalanffy exponent-pair. Thereby SSE and AIC are different ways to compare models. SSE is used, when predictive power is needed alone, and AIC is used, when simplicity of the model and explanatory power are needed.
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 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). For data fitting using general exponents, five model parameters need to be optimized, the pair a<b of non-negative exponents, the non-negative constants p and q, and a positive initial value m0 for the differential equation. For the case b=1 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 p, q, m0 were optimized). Thereby, data fitting used the method of least squares, minimizing the sum of squared errors (SSE). It was conjectured that the optimization of both exponents would result in a significantly better fit of the optimal growth function to the data and thereby reduce SSE. This conjecture was tested for a data set for the mass-growth of Walleye (Sander vitreus), a fish from Lake Erie, USA. Compared to the Bertalanffy exponent-pair the optimal exponent-pair achieved a reduction of SSE by 10%. However, when the optimization of additional parameters was penalized, using the Akaike information criterion (AIC), then the optimal exponent-pair model had a higher (worse) AIC, when compared to the Bertalanffy exponent-pair. Thereby SSE and AIC are different ways to compare models. SSE is used, when predictive power is needed alone, and AIC is used, when simplicity of the model and explanatory power are needed.