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Biological homochirality and stoichiometric network analysis: Variations on Frank’s model
Homoquiralidad biológica y el análisis de redes estequiométricas: el modelo de Frank y sus variantes
Homoquiralidade biológica e análise de redes estequiométricas: O modelo de Frank e as suas variantes
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https://doi.org/10.15446/rev.colomb.quim.v50n3.96921Palabras clave:
Homochirality, Stochiometric Network Analysis, Chemical Reaction Mechanisms, Feedback, Autocatalysis (en)homoquiralidade, análise de redes estequiométricas, mecanismos de reação, retroalimentação, autocatálise (pt)
homoquiralidad, análisis de redes estequiométricas, mecanismos de reacción, retroalimentación, autocatálisis (es)
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Biological homochirality is modelled using chemical reaction mechanisms that include autocatalytic and inhibition reactions as well as input and output flows. From the mathematical point of view, the differential equations associated with those mechanisms have to exhibit bistability. The search for those bifurcations can be carried out using stoichiometric network analysis. This algorithm simplifies the mathematical analysis and can be implemented in a computer programme, which can help us to analyse chemical networks. However, regardless of the reduction to linear polynomials, which is made possible by this algorithm, in some cases, the complexity and length of the polynomials involved make the analysis unfeasible. This problem has been partially solved by extending the stoichiometric matrix with rows that code the duality relations between the different reactions occurring in the network given as input. All these facts allow us to analyse 28 different network models, highlighting the basic requirements needed by a chemical mechanism to have spontaneous mirror symmetry breaking.
El origen de la homoquiralidad biológica se ha modelado usando mecanismos de reacción con pasos autocatalíticos, de inhibición y flujos de entrada y salida. Desde el punto de vista de las matemáticas, las ecuaciones diferenciales asociadas a tales mecanismos deben exhibir biestabilidad. La búsqueda de tales bifurcaciones se puede hacer usando el análisis de redes estequiométricas. Tal algoritmo facilita el trabajo matemático y se puede implementar en un programa de computadora, con lo que se simplifica el análisis y ayuda a entender y mejorar los mecanismos de reacción. No obstante, y a pesar de la reducción en la complejidad que es alcanzada usando el análisis de redes estequiométricas, la dificultad y la longitud de los polinomios involucrados hacen que, en los casos más difíciles y de mayor envergadura, la solución de estos no sea posible. En este trabajo se ha superado parcialmente el problema, adicionando a la matriz estequiométrica un conjunto de filas que codifican la relacion de dualidad entre las diferentes reacciones presentes en la red química dada como entrada al programa. Así, hemos logrado analizar 28 modelos diferentes de homoquiralidad biológica, extrayendo de ellos el conjunto de requisitos necesarios para tener un modelo cinética y termodinámicamente consistente.
A origem da homoquiralidade biológica foi modelada usando mecanismos de reação com etapas autocatalíticas, de inibição e fluxos de entrada e saída. Do ponto de vista da matemática, as equações diferenciais associadas a tais mecanismos devem ser instáveis. A instabilidade pode ser estudada usando o algoritmo de análise de redes estequiométricas. Tal algoritmo facilita o trabalho matemático e pode ser implementado num programa de computador, o que simplifica a análise e ajuda a entender e melhorar os mecanismos de reação. No entanto, e apesar da redução na complexidade que é alcançada usando a análise de redes estequiométricas, a complexidade e comprimento dos polinômios envolvidos fazem que, nos casos mais complexos e de maior envergadura, a solução dos mesmos não seja possível. Neste trabalho, o problema foi superado, parcialmente, adicionando à matriz estequiométrica um conjunto de linhas que codificam a relação de dualidade entre as diferentes reações presentes na rede química dada como entrada ao programa. Desta forma foi possível analisar 28 modelos diferentes de homoquiralidade biológica, extraindo deles o conjunto de requisitos necessários para ter um modelo cinético e termodinamicamente consistente.
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Derechos de autor 2022 Jesús Alberto Ágreda Bastidas, Juan Andrés Montoya Arguello, Carolina Mejía
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