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Mathematical Modeling in Systems Biology

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Entropy and Biology".

Deadline for manuscript submissions: closed (31 May 2023) | Viewed by 15608

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A printed edition of this Special Issue is available here.

Special Issue Editor

Academy of Integrated Science, Division of Systems Biology, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
Interests: systems biology; mathematical modeling; cell cycle; gene regulatory networks; cell death mechanism

Special Issue Information

Dear Colleagues,

Systems biology is a rapidly growing field, yet many biological systems remain qualitatively undescribed. We use mathematical modeling and develop computational methods and tools to study how the parts of biological systems interact to produce complex dynamic behavior. Without these computational techniques and mathematical modeling, overall behavior of the complex biological systems cannot be intuitively understood.

This Special Issue is dedicated to publishing the research that takes computational and mathematical approaches to study biological systems. We encourage all contributions, including: mathematical, theoretical, analytical and computational studies of system-level properties of biological systems; dynamic models of biological networks; control theory; systems-oriented approaches; big data analysis methods and tools; studies using deterministic and stochastic approaches. Computational studies of gene–protein interaction networks, metabolic networks, cell communication, population dynamics, infectious disease evolution and ecological systems are all supportively invited for this Special Issue.

Dr. Pavel Kraikivski
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematical modeling
  • biological systems
  • computational methods
  • network dynamics
  • systems-oriented approaches
  • data analysis tools
  • regulatory networks
  • complex biological mechanisms
  • mechanistic approaches

Published Papers (12 papers)

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15 pages, 775 KiB  
Review
Self-Organization and Genomic Causality in Models of Morphogenesis
by Ute Deichmann
Entropy 2023, 25(6), 873; https://doi.org/10.3390/e25060873 - 30 May 2023
Cited by 2 | Viewed by 1067
Abstract
The debate about what causes the generation of form and structure in embryological development goes back to antiquity. Most recently, it has focused on the divergent views as to whether the generation of patterns and form in development is a largely self-organized process [...] Read more.
The debate about what causes the generation of form and structure in embryological development goes back to antiquity. Most recently, it has focused on the divergent views as to whether the generation of patterns and form in development is a largely self-organized process or is mainly determined by the genome, in particular, complex developmental gene regulatory processes. This paper presents and analyzes pertinent models of pattern formation and form generation in a developing organism in the past and the present, with a special emphasis on Alan Turing’s 1952 reaction–diffusion model. I first draw attention to the fact that Turing’s paper remained, at first, without a noticeable impact on the community of biologists because purely physical–chemical models were unable to explain embryological development and often also simple repetitive patterns. I then show that from the year 2000 and onwards, Turing’s 1952 paper was increasingly cited also by biologists. The model was updated to include gene products and now seemed able to account for the generation of biological patterns, though discrepancies between models and biological reality remained. I then point out Eric Davidson’s successful theory of early embryogenesis based on gene-regulatory network analysis and its mathematical modeling that not only was able to provide a mechanistic and causal explanation for gene regulatory events controlling developmental cell fate specification but, unlike reaction–diffusion models, also addressed the effects of evolution and organisms’ longstanding developmental and species stability. The paper concludes with an outlook on further developments of the gene regulatory network model. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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16 pages, 2220 KiB  
Article
A Well-Posed Fractional Order Cholera Model with Saturated Incidence Rate
by Isa Abdullahi Baba, Usa Wannasingha Humphries and Fathalla A. Rihan
Entropy 2023, 25(2), 360; https://doi.org/10.3390/e25020360 - 15 Feb 2023
Cited by 2 | Viewed by 1061
Abstract
A fractional-order cholera model in the Caputo sense is constructed. The model is an extension of the Susceptible–Infected–Recovered (SIR) epidemic model. The transmission dynamics of the disease are studied by incorporating the saturated incidence rate into the model. This is particularly important since [...] Read more.
A fractional-order cholera model in the Caputo sense is constructed. The model is an extension of the Susceptible–Infected–Recovered (SIR) epidemic model. The transmission dynamics of the disease are studied by incorporating the saturated incidence rate into the model. This is particularly important since assuming that the increase in incidence for a large number of infected individualsis equivalent to a small number of infected individualsdoes not make much sense. The positivity, boundedness, existence, and uniqueness of the solution of the model are also studied. Equilibrium solutions are computed, and their stability analyses are shown to depend on a threshold quantity, the basic reproduction ratio (R0). It is clearly shown that if R0<1, the disease-free equilibrium is locally asymptotically stable, whereas if R0>1, the endemic equilibrium exists and is locally asymptotically stable. Numerical simulations are carried out to support the analytic results and to show the significance of the fractional order from the biological point of view. Furthermore, the significance of awareness is studied in the numerical section. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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36 pages, 1236 KiB  
Article
Mechanistic Modelling of Biomass Growth, Glucose Consumption and Ethanol Production by Kluyveromyces marxianus in Batch Fermentation
by Yolocuauhtli Salazar, Paul A. Valle, Emmanuel Rodríguez, Nicolás O. Soto-Cruz, Jesús B. Páez-Lerma and Francisco J. Reyes-Sánchez
Entropy 2023, 25(3), 497; https://doi.org/10.3390/e25030497 - 14 Mar 2023
Cited by 2 | Viewed by 2371
Abstract
This paper presents results concerning mechanistic modeling to describe the dynamics and interactions between biomass growth, glucose consumption and ethanol production in batch culture fermentation by Kluyveromyces marxianus (K. marxianus). The mathematical model was formulated based on the biological assumptions underlying [...] Read more.
This paper presents results concerning mechanistic modeling to describe the dynamics and interactions between biomass growth, glucose consumption and ethanol production in batch culture fermentation by Kluyveromyces marxianus (K. marxianus). The mathematical model was formulated based on the biological assumptions underlying each variable and is given by a set of three coupled nonlinear first-order Ordinary Differential Equations. The model has ten parameters, and their values were fitted from the experimental data of 17 K. marxianus strains by means of a computational algorithm design in Matlab. The latter allowed us to determine that seven of these parameters share the same value among all the strains, while three parameters concerning biomass maximum growth rate, and ethanol production due to biomass and glucose had specific values for each strain. These values are presented with their corresponding standard error and 95% confidence interval. The goodness of fit of our system was evaluated both qualitatively by in silico experimentation and quantitative by means of the coefficient of determination and the Akaike Information Criterion. Results regarding the fitting capabilities were compared with the classic model given by the logistic, Pirt, and Luedeking–Piret Equations. Further, nonlinear theories were applied to investigate local and global dynamics of the system, the Localization of Compact Invariant Sets Method was applied to determine the so-called localizing domain, i.e., lower and upper bounds for each variable; whilst Lyapunov’s stability theories allowed to establish sufficient conditions to ensure asymptotic stability in the nonnegative octant, i.e., R+,03. Finally, the predictive ability of our mechanistic model was explored through several numerical simulations with expected results according to microbiology literature on batch fermentation. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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22 pages, 443 KiB  
Article
Quantifying Parameter Interdependence in Stochastic Discrete Models of Biochemical Systems
by Samaneh Gholami and Silvana Ilie
Entropy 2023, 25(8), 1168; https://doi.org/10.3390/e25081168 - 05 Aug 2023
Cited by 1 | Viewed by 783
Abstract
Stochastic modeling of biochemical processes at the cellular level has been the subject of intense research in recent years. The Chemical Master Equation is a broadly utilized stochastic discrete model of such processes. Numerous important biochemical systems consist of many species subject to [...] Read more.
Stochastic modeling of biochemical processes at the cellular level has been the subject of intense research in recent years. The Chemical Master Equation is a broadly utilized stochastic discrete model of such processes. Numerous important biochemical systems consist of many species subject to many reactions. As a result, their mathematical models depend on many parameters. In applications, some of the model parameters may be unknown, so their values need to be estimated from the experimental data. However, the problem of parameter value inference can be quite challenging, especially in the stochastic setting. To estimate accurately the values of a subset of parameters, the system should be sensitive with respect to variations in each of these parameters and they should not be correlated. In this paper, we propose a technique for detecting collinearity among models’ parameters and we apply this method for selecting subsets of parameters that can be estimated from the available data. The analysis relies on finite-difference sensitivity estimations and the singular value decomposition of the sensitivity matrix. We illustrated the advantages of the proposed method by successfully testing it on several models of biochemical systems of practical interest. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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3 pages, 180 KiB  
Editorial
Mathematical Modeling in Systems Biology
by Pavel Kraikivski
Entropy 2023, 25(10), 1380; https://doi.org/10.3390/e25101380 - 25 Sep 2023
Viewed by 789
Abstract
Mathematical modeling is a key tool used in the field of systems biology to determine the mechanisms with which the elements of biological systems interact to produce complex dynamic behavior [...] Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
10 pages, 458 KiB  
Article
Precise Traits from Sloppy Components: Perception and the Origin of Phenotypic Response
by Steven A. Frank
Entropy 2023, 25(8), 1162; https://doi.org/10.3390/e25081162 - 03 Aug 2023
Cited by 2 | Viewed by 609
Abstract
Organisms perceive their environment and respond. The origin of perception–response traits presents a puzzle. Perception provides no value without response. Response requires perception. Recent advances in machine learning may provide a solution. A randomly connected network creates a reservoir of perceptive information about [...] Read more.
Organisms perceive their environment and respond. The origin of perception–response traits presents a puzzle. Perception provides no value without response. Response requires perception. Recent advances in machine learning may provide a solution. A randomly connected network creates a reservoir of perceptive information about the recent history of environmental states. In each time step, a relatively small number of inputs drives the dynamics of the relatively large network. Over time, the internal network states retain a memory of past inputs. To achieve a functional response to past states or to predict future states, a system must learn only how to match states of the reservoir to the target response. In the same way, a random biochemical or neural network of an organism can provide an initial perceptive basis. With a solution for one side of the two-step perception–response challenge, evolving an adaptive response may not be so difficult. Two broader themes emerge. First, organisms may often achieve precise traits from sloppy components. Second, evolutionary puzzles often follow the same outlines as the challenges of machine learning. In each case, the basic problem is how to learn, either by artificial computational methods or by natural selection. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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30 pages, 5583 KiB  
Article
VeVaPy, a Python Platform for Efficient Verification and Validation of Systems Biology Models with Demonstrations Using Hypothalamic-Pituitary-Adrenal Axis Models
by Christopher Parker, Erik Nelson and Tongli Zhang
Entropy 2022, 24(12), 1747; https://doi.org/10.3390/e24121747 - 29 Nov 2022
Cited by 1 | Viewed by 1666
Abstract
In order for mathematical models to make credible contributions, it is essential for them to be verified and validated. Currently, verification and validation (V&V) of these models does not meet the expectations of the system biology and systems pharmacology communities. Partially as a [...] Read more.
In order for mathematical models to make credible contributions, it is essential for them to be verified and validated. Currently, verification and validation (V&V) of these models does not meet the expectations of the system biology and systems pharmacology communities. Partially as a result of this shortfall, systemic V&V of existing models currently requires a lot of time and effort. In order to facilitate systemic V&V of chosen hypothalamic-pituitary-adrenal (HPA) axis models, we have developed a computational framework named VeVaPy—taking care to follow the recommended best practices regarding the development of mathematical models. VeVaPy includes four functional modules coded in Python, and the source code is publicly available. We demonstrate that VeVaPy can help us efficiently verify and validate the five HPA axis models we have chosen. Supplied with new and independent data, VeVaPy outputs objective V&V benchmarks for each model. We believe that VeVaPy will help future researchers with basic modeling and programming experience to efficiently verify and validate mathematical models from the fields of systems biology and systems pharmacology. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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15 pages, 416 KiB  
Article
On the Emergence of the Deviation from a Poisson Law in Stochastic Mathematical Models for Radiation-Induced DNA Damage: A System Size Expansion
by Francesco Giuseppe Cordoni
Entropy 2023, 25(9), 1322; https://doi.org/10.3390/e25091322 - 11 Sep 2023
Cited by 2 | Viewed by 509
Abstract
In this paper, we study the system size expansion of a stochastic model for radiation-induced DNA damage kinetics and repair. In particular, we characterize both the macroscopic deterministic limit and the fluctuation around it. We further show that such fluctuations are Gaussian-distributed. In [...] Read more.
In this paper, we study the system size expansion of a stochastic model for radiation-induced DNA damage kinetics and repair. In particular, we characterize both the macroscopic deterministic limit and the fluctuation around it. We further show that such fluctuations are Gaussian-distributed. In deriving such results, we provide further insights into the relationship between stochastic and deterministic mathematical models for radiation-induced DNA damage repair. Specifically, we demonstrate how the governing deterministic equations commonly employed in the field arise naturally within the stochastic framework as a macroscopic limit. Additionally, by examining the fluctuations around this macroscopic limit, we uncover deviations from a Poissonian behavior driven by interactions and clustering among DNA damages. Although such behaviors have been empirically observed, our derived results represent the first rigorous derivation that incorporates these deviations from a Poissonian distribution within a mathematical model, eliminating the need for specific ad hoc corrections. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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12 pages, 313 KiB  
Article
DNA Code from Cyclic and Skew Cyclic Codes over F4[v]/v3
by Om Prakash, Ashutosh Singh, Ram Krishna Verma, Patrick Solé and Wei Cheng
Entropy 2023, 25(2), 239; https://doi.org/10.3390/e25020239 - 28 Jan 2023
Cited by 2 | Viewed by 1261
Abstract
The main motivation of this work is to study and obtain some reversible and DNA codes of length n with better parameters. Here, we first investigate the structure of cyclic and skew cyclic codes over the chain ring [...] Read more.
The main motivation of this work is to study and obtain some reversible and DNA codes of length n with better parameters. Here, we first investigate the structure of cyclic and skew cyclic codes over the chain ring R:=F4[v]/v3. We show an association between the codons and the elements of R using a Gray map. Under this Gray map, we study reversible and DNA codes of length n. Finally, several new DNA codes are obtained that have improved parameters than previously known codes. We also determine the Hamming and the Edit distances of these codes. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
22 pages, 1845 KiB  
Review
Mathematical Models of Death Signaling Networks
by Madhumita Srinivasan, Robert Clarke and Pavel Kraikivski
Entropy 2022, 24(10), 1402; https://doi.org/10.3390/e24101402 - 01 Oct 2022
Cited by 2 | Viewed by 1671
Abstract
This review provides an overview of the progress made by computational and systems biologists in characterizing different cell death regulatory mechanisms that constitute the cell death network. We define the cell death network as a comprehensive decision-making mechanism that controls multiple death execution [...] Read more.
This review provides an overview of the progress made by computational and systems biologists in characterizing different cell death regulatory mechanisms that constitute the cell death network. We define the cell death network as a comprehensive decision-making mechanism that controls multiple death execution molecular circuits. This network involves multiple feedback and feed-forward loops and crosstalk among different cell death-regulating pathways. While substantial progress has been made in characterizing individual cell death execution pathways, the cell death decision network is poorly defined and understood. Certainly, understanding the dynamic behavior of such complex regulatory mechanisms can be only achieved by applying mathematical modeling and system-oriented approaches. Here, we provide an overview of mathematical models that have been developed to characterize different cell death mechanisms and intend to identify future research directions in this field. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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23 pages, 22784 KiB  
Article
Entropy as a Geometrical Source of Information in Biological Organizations
by Juan Lopez-Sauceda, Philipp von Bülow, Carlos Ortega-Laurel, Francisco Perez-Martinez, Kalina Miranda-Perkins and José Gerardo Carrillo-González
Entropy 2022, 24(10), 1390; https://doi.org/10.3390/e24101390 - 29 Sep 2022
Cited by 2 | Viewed by 1283
Abstract
Considering both biological and non-biological polygonal shape organizations, in this paper we introduce a quantitative method which is able to determine informational entropy as spatial differences between heterogeneity of internal areas from simulation and experimental samples. According to these data (i.e., heterogeneity), we [...] Read more.
Considering both biological and non-biological polygonal shape organizations, in this paper we introduce a quantitative method which is able to determine informational entropy as spatial differences between heterogeneity of internal areas from simulation and experimental samples. According to these data (i.e., heterogeneity), we are able to establish levels of informational entropy using statistical insights of spatial orders using discrete and continuous values. Given a particular state of entropy, we establish levels of information as a novel approach which can unveil general principles of biological organization. Thirty-five geometric aggregates are tested (biological, non-biological, and polygonal simulations) in order to obtain the theoretical and experimental results of their spatial heterogeneity. Geometrical aggregates (meshes) include a spectrum of organizations ranging from cell meshes to ecological patterns. Experimental results for discrete entropy using a bin width of 0.5 show that a particular range of informational entropy (0.08 to 0.27 bits) is intrinsically associated with low rates of heterogeneity, which indicates a high degree of uncertainty in finding non-homogeneous configurations. In contrast, differential entropy (continuous) results reflect negative entropy within a particular range (−0.4 to −0.9) for all bin widths. We conclude that the differential entropy of geometrical organizations is an important source of neglected information in biological systems. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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19 pages, 5225 KiB  
Article
A Numerical Study of the Dynamics of Vector-Born Viral Plant Disorders Using a Hybrid Artificial Neural Network Approach
by Hosam Alhakami, Muhammad Umar, Muhammad Sulaiman, Wajdi Alhakami and Abdullah Baz
Entropy 2022, 24(11), 1511; https://doi.org/10.3390/e24111511 - 22 Oct 2022
Cited by 5 | Viewed by 1207
Abstract
Most plant viral infections are vector-borne. There is a latent period of disease inside the vector after obtaining the virus from the infected plant. Thus, after interacting with an infected vector, the plant demonstrates an incubation time before becoming diseased. This paper analyzes [...] Read more.
Most plant viral infections are vector-borne. There is a latent period of disease inside the vector after obtaining the virus from the infected plant. Thus, after interacting with an infected vector, the plant demonstrates an incubation time before becoming diseased. This paper analyzes a mathematical model for persistent vector-borne viral plant disease dynamics. The backpropagated neural network based on the Levenberg—Marquardt algorithm (NN-BLMA) is used to study approximate solutions for fluctuations in natural plant mortality and vector mortality rates. A state-of-the-art numerical technique is utilized to generate reference data for obtaining surrogate solutions for multiple cases through NN-BLMA. Curve fitting, regression analysis, error histograms, and convergence analysis are used to assess accuracy of the calculated solutions. It is evident from our simulations that NN-BLMA is accurate and reliable. Full article
(This article belongs to the Special Issue Mathematical Modeling in Systems Biology)
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