Physical models have been in use in biological studies since long. A prominent example would be the double helical model of DNA proposed by Watson and Crick. Mathematical models go a little beyond this reality expressing the relationships of independent variables as functions.

Now modeling techniques are used to create environments similar to the real situation and to interpret the data and predict the outcomes based on the existing patterns of relations. The question is why mathematics? Everything in nature essentially follows a mathematical pattern from petals of a flower, scales on a fish etc to the arrangement of leaves of a tree in tropical forests. Hence modeling can be most accurate in biological models.

This is essentially an interdisciplinary approach requiring the application of mathematics through statistical principles on biological phenomena. The statistical applications analyze the trends and distribution of data. The advantage is that the entire data relation can be fitted in a single equation depicted as a line or graph.

Uses of mathematical models

1. Prediction of trends and data patterns in future.

For example consider that we need to know the average size of population of a particular species after a couple of years. From the growth rate and other parameters we can define the increase of growth of population size as a function and use the mathematical function of regression to describe the trend of population growth. A graphical representation of the function gives a line or curve from which the average population size can be calculated. This gives the flexibility of extrapolating the results after the last time point. The data description also gets clearer and easier for analysis with such models.

2. Understanding processes within the system

From the observed data, it is possible to construct models that define interspecies and intra species interactions and this can be used to predict the finer relations between the data values.

3. Identification of variables

Sometimes such models allow us to identify variable which were not known at the time of data collection but were found to have a significant effect on the data patterns. This is used in designing experimental approach and in identification of bias and errors. In existing experimental designs, the approach can be used to decide which of the variables need to be randomized in order to improve the significance level.

4. Confirmation of Data accuracy

The observed spatial and temporal relations between data are sometimes known from previous models or hypothesis. The reverse way of testing the data to whether it fits the processes in question can be used to find the accuracy of data. For example, survival rates and migration effects follow a specific pattern and the existing relation can be applied to the data in question whether it follows the generality or not. The results are finer allowing the researcher to study the reasons for such variations.

Mathematical tools used in modeling

Regression and correlation techniques were used widely for construction of accurate mathematical models in biology.

Correlation is measure of the strength of relationship between two variables in a system. Regression essentially determines that relation. Regression is used where you suspect the presence of a dependent and independent variable. Changes in the independent variable are observed against the changes in the dependent variables over some parameter.
For example rate of a chemical reaction can be determined by varying the temperature of the system.
The data from dependent variable is plotted on y axis and the independent variables on the x axis. The regression analysis of the data tries to represent the relation between the data of the two axes in an equation which can be represented as a single line.

For a straight line or linear relation, the equation will be
y = mx+ c

y - Dependent variable
x - Independent variable
m - Gradient and
c - Intercept

Significance testing can be applied in the various regression analyzes to determine the changes if any in the dependent variable relative to the changes in independent variable. The basic method tends to minimize the squares of errors between data and predicted line. The portions of data value which cannot be explained by the regression analysis form the residual error.

Correlation coefficient is calculated from the data values to measure the strength of a relation. There are different methods such as Pearson's, Spearman's etc to calculate the coefficient. More sophisticated models are based on techniques such as multiple regressions where the effect of two or more non correlated variables on the system needs to be analyzed.
With the advent of statistical software and complexity of databases, there is an urgent need to construct simple models which can be used to stimulate and predict the outcome. In fact, this science has been the basis of neural networks and artificial intelligence.

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