Mathematical Models in Biology

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Formation of diverse Glc6P isotopomers in a haemolysate, obtained by solving the adapted model by Berthon et al. Gibbs and Kandler described it in and [ 8 , 9 ], when they observed the atypical and asymmetrical incorporation of radioactive 14 CO 2 in hexoses. An example of label incorporation by the CBB cycle intermediates is presented schematically in Figure 5.

Finally, our package provides a solid foundation for additional extensions to the framework architecture, its classes and modelling routines. We would like to thank the students working on their Bachelor and Masters projects in our lab, who applied and tested this software while investigating their scientific problems.

Bringing Mathematics to Life

Metabolic Engineering , 38— Oliphant, T E Python for scientific computing. Computing in Science and Engineering , 9 3 : 10— Biochem , 3 : — Eur J Biochem , 2 : — IEE Proc. Biol , 5 : — Bioinformatics , 21 4 : — Biochemical Society Transactions. Plant physiology , 31 5 : Mathematics and Computers in Simulation , 26— Computing In Science and Engineering , 9 3 : 90— BMC Systems Biology , 7 1 : 1.

Applications of Mathematical Modelling to Biological Pattern Formation

Philosophical Transactions B , Journal of Experimental Botany , 51 : — Biochem J , — Biosystems , 2 : — BMC Systems Biology , 6 1 : In: Schmidt, B and Loizides, F eds. IOS Press. Building Mathematical Models of Biological Systems with modelbase. Journal of Open Research Software , 6 1 , p. Journal of Open Research Software. Journal of Open Research Software , 6 1 , Journal of Open Research Software 6 1 : Journal of Open Research Software 6, no.

Journal of Open Research Software , vol. Start Submission Become a Reviewer. Nima P. Abstract The modelbase package is a free expandable Python package for building and analysing dynamic mathematical models of biological systems. Published on 16 Nov Peer Reviewed. CC BY 4. Motivation In the course of our photosynthetic research, we identified several shortcomings that are not adequately met by available free and open source research software.

Implementation and architecture modelbase is a console-based application written in Python. Model construction The user has the possibility to build two types of models, using one of the classes defined in the module model : Model , for differential-equation based systems, or LabelModel , for isotope-labelled models. Class Model Every model object is defined by: model parameters, model variables, rate equations, stoichiometries.

Model d After instantiation, the keys of the parameter dictionary d become accessible as attributes of an object of the internal class modelbase. Box 1: Basic model use We use modelbase to simulate a simple chain of reactions, in which the two state variables X and Y describe the concentrations of the intermediates. Working with algebraic modules A particularly useful function of the class Model has been developed to facilitate the incorporation of algebraic expressions, by which dependent variables can be computed from independent ones.

Various analysis methods With import modelbase. Class LabelModel for isotope-labelled models The modelbase package includes a class to construct isotope-labelled versions of developed models.

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LabelModel d , where d is again a dictionary holding parameters. Box 2: Isotope-labelled model A minimal example of an isotope-label specific model simulates equilibration of isotope distribution in a system consisting of the two reactions of triose-phosphate isomerase and fructose-bisphosphate aldolase: 2. Integration methods and simulation subpackages Methods for the numeric integration of models are provided by the two subclasses Simulate and LabelSimulate , where the latter inherits many methods from the first.

Simulator m returns an instance of either Simulate or LabelSimulate , depending on the class of model m , providing all methods to numerically simulate the differential equation system and to analyse the results.


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When needed, almost every integrator option can be overridden by the user by simply accessing s. Quality control modelbase has been continuously developed and used within our lab since Programming language modelbase is written in the Python programming language, a general-purpose interpreted, interactive, object-oriented, and high-level programming language. Dependencies Dependencies are provided in the setup. Language modelbase was entirely developed in English.

Table 1 Mathematical models originally published without the source-code, reconstructed in our lab using the modelbase package. Process Original publication GitHub. Calvin-Benson-Bassham Cycle [ 16 ]. Pentose Phosphate Pathway [ 4 , 17 ]. Modelling the PETC to study photoprotective mechanisms Part of our research focuses on understanding the dynamics of various photoprotective mechanisms present in photosynthetic organisms [ 18 , 15 , 19 ].

PPP and isotope labelling experiments We envisage that especially our LabelModel extension will find a wide application in metabolic network analysis.

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Explore and analyze the solutions of mathematical models from diverse disciplines As biology increasingly depends on data, algorithms, and models, it has become necessary to use a computing language, such as the user-friendly MapleTM, to focus more on building and analyzing models as opposed to configuring tedious calculations.

Explorations of Mathematical Models in Biology with Maple provides an introduction to model creation using Maple, followed by the translation, analysis, interpretation, and observation of the models. With an integrated and interdisciplinary approach that embeds mathematical modeling into biological applications, the book illustrates numerous applications of mathematical techniques within biology, ecology, and environmental sciences.

Dynamics of Mathematical Models in Biology | SpringerLink

Featuring a quantitative, computational, and mathematical approach, the book includes:. He has extensive background and experience in designing interdisciplinary instructional materials that integrate mathematics and other disciplines, such as biology, ecology, and finance. Request permission to reuse content from this site.

Despite the accumulation of an overwhelming amount of data on cell movement patterns in developing tissues of different geometry Beloussov , little is known about the mechanisms governing the migration and rearrangement of cells in embryonic tissue. One of the commonly discussed mechanisms is chemotaxis when the migration of a cell is driven by the gradient of a substance, which is, in this case, called a chemotactic agent Dormann et al.

Other known mechanisms such as apical constriction Odell et al. No mathematical formalism has been developed so far to describe tissue dynamics due to apical constriction or cellular intercalation. Studies of the developmental cycle in D. Here, we will give an overview of models and modelling results from these studies. The classical illustration of how a morphogen can provide positional information is given by the French Flag model suggested by Lewis Wolpert Wolpert This model demonstrates how a simple, linear concentration profile of a morphogen Fig.

The linear concentration profiles can form naturally in various settings. The simplest case is when the production and degradation of the morphogen take place outside the tissue on its opposing sides and the morphogen passively diffuses along the tissue, from the side where it is produced to the side where it is degraded. Mathematically, the stationary concentration profile of the morphogen in this system should obey the so-called Laplace's equation with Dirichlet boundary conditions, which for a tissue represented by a one-dimensional domain of length L , is given by the following mathematical formulation: Figure 1 Download Figure Download figure as PowerPoint slide Modelling morphogen gradient.

A Linear gradient in the French Flag model forms due to passive diffusion of the morphogen equation 1. B Exponential gradient forms when the diffusion is combined with decay equation 2. C A periodic Turing pattern forms due to non-linear interactions between the involved morphogens equation 3. A and B T 1 and T 2 are the threshold concentrations of the morphogen, defining the borders of the cell determination domains. C The sharp profile of the activator makes the locations of cell determination domains insensitive to the exact value of the threshold concentrations.

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The advantage of a linear profile, the solution of equation 1 , is that it scales with the size of the tissue. This means that if, for example, we double the size of the tissue, then the sizes of all domains of cellular determination as defined by the threshold concentration values T 1 and T 2 , see Fig. However, the experimental observations do not always confirm the linear shape of a morphogen gradient. Formation of the exponential profile can be shown mathematically under the assumption that the morphogen not only diffuses but also degrades inside the domain.

The concentration of morphogen can be buffered fixed on the boundaries of the tissue similar to the above case expressed in equation 1.

Math can help uncover cancer's secrets - Irina Kareva

Alternatively, we can assume that the tissue is isolated no flow on the boundaries and the production of the morphogen takes place in a restricted area inside the domain Fig. These assumptions are perfectly reasonable for many studied cases.


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A stationary concentration profile of the morphogen in this system will satisfy the following equation:. The main problem with the exponential or nearly exponential as in Fig. For example, if, in the case shown in Fig. This property of the solution of equation 2 contradicts the observation that the exponential gradient of Bicoid in D.

The models represented by equations 1 and 2 fall into the class of linear models that is, the variable u and its derivative appear only in their first power. Since the introduction of a classical predator—prey model Lotka , many non-linear models have been developed and used for studies in mathematical biology.

Contrary to linear models, which have a limited range of possible solutions, non-linear models can be used to reproduce virtually any kind of known dynamics in concentration fields of morphogens. This is especially true if more than one morphogen is considered and the model is represented by a set of reaction—diffusion equations. The dynamics of two interacting morphogens can be described by the following reaction—diffusion equations:.

These equations state that the rates of change of morphogen concentrations are defined by two processes, diffusion first term on the right-hand side and reaction second term with the latter accounting for the production and decay of morphogens. The FHN model was originally designed as a generic model for signal propagation along a nerve fibre. The morphogen, with concentration u , is called the activator as it promotes its own production and the production of the second morphogen, the inhibitor.

The inhibitor promotes the degradation of the activator. According to this mechanism, a spatially periodic pattern can spontaneously arise under certain conditions due to the intrinsic noise in the system. Figure 1 C shows a periodic stationary pattern that has emerged in system 3 from nearly homogeneous noisy initial conditions. This pattern, which could represent a morphogen gradient, only occurs when the inhibitor diffuses quickly and has slow kinetics i.

Recent experimental studies of pigmentation patterns of fish Yamaguchi et al. However, the Turing patterns fail the scaling test: the distance between spikes Fig. This means that the number of spikes should increase with an increase in the size of the medium. In models represented by equations 1 , 2 and 3 , formation of the gradient is conditioned by diffusion of the morphogen. A number of mathematical studies have addressed the formation of gradients in the absence of morphogen diffusion, for example due to proliferation Ibanes et al. Non-diffusive patterning mechanisms can be provided by direct contacts between cells, of which the classical example is Delta-Notch signalling associated with the binding of non-diffusive Delta to the Notch receptor of neighbouring cells Collier et al.

The shape of a developing biological structure often demonstrates remarkable robustness with respect to various changes in its developmental conditions. Scaling is a particular case of robustness and there are many instances recorded including classical experiments of Hans Driesch Sander where perfectly capable organisms emerge from embryos of different sizes. Possible mechanisms ensuring the robustness of dorso-ventral patterning to changes in production rates of involved proteins BMP and Sog in the D. A modification of this model Ben-Zvi et al.

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