Somdatta Sinha

Mathematical models of malaria - a review

Mathematical models have been used to provide an explicit framework for understanding malaria transmission dynamics in human population for over 100 years. With the disease still thriving and threatening to be a major source of death and disability due to changed environmental and socio-economic conditions, it is necessary to make a critical assessment of the existing models, and study their evolution and efficacy in describing the host-parasite biology. In this article, starting from the basic Ross model, the key mathematical models and their underlying features, based on their specific contributions in the understanding of spread and transmission of malaria have been discussed. The first aim of this article is to develop, starting from the basic models, a hierarchical structure of a range of deterministic models of different levels of complexity. The second objective is to elaborate, using some of the representative mathematical models, the evolution of modelling strategies to describe malaria incidence by including the critical features of host-vector-parasite interactions. Emphasis is more on the evolution of the deterministic differential equation based epidemiological compartment models with a brief discussion on data based statistical models. In this comprehensive survey, the approach has been to summarize the modelling activity in this area so that it helps reach a wider range of researchers working on epidemiology, transmission, and other aspects of malaria. This may facilitate the mathematicians to further develop suitable models in this direction relevant to the present scenario, and help the biologists and public health personnel to adopt better understanding of the modelling strategies to control the disease

Network properties of protein structures

Protein structures can be studied as complex networks of interacting amino acids. We study proteins of different structural classes from the network perspective. Our results indicate that proteins, regardless of their structural class, show small-world network property. Various network parameters offer insight into the structural organisation of proteins and provide indications of modularity in protein networks.

Using genomic signatures for HIV-1 sub-typing

BackgroundHuman Immunodeficiency Virus type 1 (HIV-1), the causative agent of Acquired Immune Deficiency Syndrome (AIDS), exhibits very high genetic diversity with different variants or subtypes prevalent in different parts of the world. Proper classification of the HIV-1 subtypes, displaying differential infectivity, plays a major role in monitoring the epidemic and is also a critical component for effective treatment strategy. The existing methods to classify HIV-1 sequence subtypes, based on phylogenetic analysis focusing only on specific genes/regions, have shown inconsistencies as they lack the capability to analyse whole genome variations. Several isolates are left unclassified due to unresolved sub-typing. It is apparent that classification of subtypes based on complete genome sequences, rather than sub-genomic regions, is a more robust and comprehensive approach to address genome-wide heterogeneity. However, no simple methodology exists that directly computes HIV-1 subtype from the complete genome sequence.ResultsWe use Chaos Game Representation (CGR) as an approach to identify the distinctive genomic signature associated with the DNA sequence organisation in different HIV-1 subtypes. We first analysed the effect of nucleotide word lengths (k = 2 to 8) on whole genomes of the HIV-1 M group sequences, and found the optimum word length of k = 6, that could classify HIV-1 subtypes based on a Test sequence set. Using the optimised word length, we then showed accurate classification of the HIV-1 subtypes from both the Reference Set sequences and from all available sequences in the database. Finally, we applied the approach to cluster the five unclassified HIV-1 sequences from Africa and Europe, and predict their possible subtypes.ConclusionWe propose a genomic signature-based approach, using CGR with suitable word length optimisation, which can be applied to classify intra-species variations, and apply it to the complex problem of HIV-1 subtype classification. We demonstrate that CGR is a simple and computationally less intensive method that not only accurately segregates the HIV-1 subtype and sub-subtypes, but also aid in the classification of the unclassified sequences. We hope that it will be useful in subtype annotation of the newly sequenced HIV-1 genomes.

Complex Behaviour of the Repressible Operon

The repressor-mediated repression process in bacteria is modelled using a gene-enzyme-endproduct control unit. A combined analytical-numerical study shows that the system, though stable normally, becomes unstable for super-repressing strains even at low values of the cooperativity of repression, provided demand for the endproduct saturates at large endproduct concentrations. In addition the system also shows bistability, i.e., the co-existence of a stable steady-state and a stable limit cycle. The tryptophan operon is used as a model system and the results are discussed in the light of differential regulation of gene expression in lower organisms, especially in mutant strains.

On the dynamics of controlled metabolic network and cellular behavior

The existence of elaborate control mechanisms for the various biochemical processes inside and within living cells is responsible for the coherent behaviour observed in its spatio-temporal organisation. Stability and sensitivity are both necessary properties of living systems and these are achieved through negative and positive feedback loops as in other control systems. We have studied a three-step reaction scheme involving a negative and a positive feedback loop in the form of end-product inhibition and allosteric activation. The variety of behaviour exhibited by this system, under different conditions, includes steady state, simple limit cycle oscillations, complex oscillations and period bifurcations leading to random oscillations or chaos. The system also shows the existence of two distinct chaotic regimes under the variation of a single parameter. These results, in comparison with single biochemical control loops, show that new behaviours can be exhibited in a more complex network which are not seen in the single control loops. The results are discussed in the light of a diverse variety of cellular functions in normal and altered cells indicating the role of controlled metabolic network as the underlying basis for cellular behaviour.

Controllability of spatiotemporal systems using constant pinnings

Most natural spatiotemporal systems are an organized ensemble of dynamical subsystems whose behaviour is regulated by nonlinearly coupled multi-variable processes. The response of each of these variables and/or combinations of them to any single or composite external perturbation can influence its spatiotemporal behaviour in a complex and non-intuitive manner. This paper attempts to study the dynamical response of two coupled map lattice systems having local dynamics described by (a) coupled discrete maps, and (b) coupled differential equations, to external perturbation (or “pinning”) applied to each variable individually or simultaneously. We show that the response of different variables to external pinning is quite different. Our results indicate that, though this pinning approach is useful in controlling complex dynamics both globally and locally, enhancing complexity in dynamics (“anti-control”) is variable dependent. Thus complete controllability (i.e., control and anti-control) of dynamics in these spatially extended systems having coupled multi-variable processes does not only depend on the sign and strength of the perturbation, but also on the specific variable being pinned. This also implies that external manipulation of state variables in natural systems can lead to quite different responses depending at the level of action of the perturbation.

Mining of protein contact maps for protein fold prediction

The three‐dimensional structure of proteins is useful to carry out the biophysical and biochemical functions in a cell. Approaches to protein structure/fold prediction typically extract amino acid sequence features, and machine learning approaches are then applied to classification problem. Protein contact maps are two‐dimensional representations of the contacts among the amino acid residues in the folded protein structure. This paper highlights the need for a systematic study of these contact networks. Mining of contact maps to derive features pertaining to fold information offers a new mechanism for fold discovery from the protein sequence via the contact maps. These ideas are explored in the structural class of all‐alpha proteins to identify structural elements. A simple and computationally inexpensive algorithm based on triangle subdivision method is proposed to extract additional features from the contact map. The method successfully characterizes the off‐diagonal interactions in the contact map for predicting specific ‘folds’. The decision tree classification results show great promise in developing a new and simple tool for the challenging problem of fold prediction.

Theoretical investigation of malaria prevalence in two Indian cities using the response surface method.

BackgroundElucidation of the relationships between malaria incidence and climatic and non-climatic factors in a region is of utmost importance in understanding the causative factors of disease spread and design of control strategies. Very often malaria prevalence data is restricted to short time scales (months to few years). This demands application of rigorous statistical modelling techniques for analysis and prediction. The monthly malaria prevalence data for three to five years from two cities in southern India, situated in two different climatic zones, are studied to capture their dependence on climatic factors.MethodsThe statistical technique of response surface method (RSM) is applied for the first time to study any epidemiological data. A new step-by-step model reduction technique is proposed to refine the initial model obtained from RSM. This provides a simpler structure and gives better fit. This combined approach is applied to two types of epidemiological data (Slide Positivity Rates values and Total Malaria cases), for two cities in India with varying strengths of disease prevalence and environmental conditions.ResultsThe study on these data sets reveals that RSM can be used successfully to elucidate the important environmental factors influencing the transmission of the disease by analysing short epidemiological time series. The proposed approach has high predictive ability over relatively long time horizons.ConclusionsThis method promises to provide reliable forecast of malaria incidence across varying environmental conditions, which may help in designing useful control programmes for malaria.

Stability of synchronization in a multi-cellular system

Networks of biochemical reactions regulated by positive- and negative-feedback processes underlie functional dynamics in single cells. Synchronization of dynamics in the constituent cells is a hallmark of collective behavior in multi-cellular biological systems. Stability of the synchronized state is required for robust functioning of the multi-cell system in the face of noise and perturbation. Yet, the ability to respond to signals and change functional dynamics are also important features during development, disease, and evolution in living systems. In this paper, using a coupled multi-cell system model, we investigate the role of system size, coupling strength and its topology on the synchronization of the collective dynamics and its stability. Even though different coupling topologies lead to synchronization of collective dynamics, diffusive coupling through the end product of the pathway does not confer stability to the synchronized state. The results are discussed with a view to their prevalence in biological systems.

Dynamics of simple one-dimensional maps under perturbation

It is known that the one-dimensional discrete maps having single-humped nonlinear functions with the same order of maximum belong to a single class that shows the universal behaviour of a cascade of period-doubling bifurcations from stability to chaos with the change of parameters. This paper concerns studies of the dynamics exhibited by some of these simple one-dimensional maps under constant perturbations. We show that the “universality” in their dynamics breaks down under constant perturbations with the logistic map showing different dynamics compared to the other maps. Thus these maps can be classified into two types with respect to their response to constant perturbations. Unidimensional discrete maps are interchangeably used as models for specific processes in many disciplines due to the similarity in their dynamics. These results prove that the differences in their behaviour under perturbations need to be taken into consideration before using them for modelling any real process.