IGEM:IMPERIAL/2008/Prototype/Drylab/Data Analysis/Alt Models
The following table describes the alternative models created for characteristics of bacteria motility.
Parameter Estimation Methods
The method of maximum likelihood involves the determination of the parameter which maximises the likelihood of given data samples. A mathematical explanation can be found here.
The experimental data obtained does not give us access to the entire underlying distribution but we hope that the data us representative of the underlying distribution. The size of the data set used to estimate the parameters is therefore a crucial factor in the accuracy of the outcome. By applying the relevant estimator to the synthetic dataset we generated, we can see that increasing the size of the data set increases the accuracy with which we can estimate the parameter. The order of the data set does not influence the likelihood. Advantages and disadvantages of the MLE approach to parameter estimation are summarized here.
The nth moment of a distribution is defined by: . Take a look at this site for detailed explanations. The centred moment at n=1 is defined as the mean of the distribution. By taking moments with respect to the mean, we can obtain the shape of the graph with respect to the average of the distribution. This is convenient for common distributions such as the Gaussian and Maxwell-Boltzmann distributions, among many others. Take a look at this site for more details on central moments. Centred moments of 2nd order is the Variance, 3rd order refers to the Skewness and 4th order refers to the Kurtosis of the distribution.
The Maxwell-Boltzmann distribution is commonly used to describe molecular speeds, which are under the influence of brownian motion. Although bacteria size does not come close to that of small molecules and in general bacteria motility is controlled by beating flagellar, we cannot ignore the effects of colliding molecules on the micro-sized bacteria.
The Gaussian or Normal distribution is the most common distribution and is used as the first level of assumption on the distribution of bacteria bacteria characteristics.
Most Likely Estimators: and
The von Mises distribution is a continuous distribution defined on the
The von Mises distribution is a continuous probability distribution on the range 0≤x<2π. It may be thought of as the circular analogue of the normal distribution. It is used where a distribution of angles are the result of the addition of many small independent angular deviations, such as target sensing. Since bacteria use various types of chemoreceptors to pick up chemo attractants and repellants, we may assume that the tumbling angle which causes the bacteria to change its direction of motion in response to its environment follows a von Mises distribution.
Run and Tumbling Duration
The exponential distribution is the only continuous memoryless random distribution. If we assume that both the run and tumbling durations are memoryless processes, then they are probably exponentially distributed.
Most Likely Estimator:
The following table describes the various types of distributions which bacteria motility characteristics may follow.