While these methods are precise for a full power law distribution, they do not converge to the proper value when the distribution is truncated either from above or from below (Fig 1A).

This truncated power law estimate does converge to the y >|< — .

Also, as noted above, the power law eXponent of yH(x) is 1 unit greater than the power law eXponent of VH(x)since these two quantities share the properties linking CDFs to PDFs, namely that a PDF is proportional to the first derivative of its CDF.

Formally, the expected number of unique haplotypes (U(R)) in a sample size (R) can be estimated, assuming a truncated power law formalism as defined above, to be (see Sl—S3 Texts):

For most populations, the haplotype frequency distribution follows a power law with eXponent or of the PDF between 1.4 and 1.9.

Further, estimates of the power law eXponent converged before the full sample size was analyzed (Table 3 and 83—88 Figs).

Occupancy distributions [7]offer a more natural representation of our data sampling process and use the mathematical concept of regular variation to extrapolate A and H using a power law function that describes the distribution of allele and haplotype frequencies in the population [7,8].

The power law relationship was constructed under the assumption that an infinite number of categories (or kinds) exist in the population; so for our purposes we apply a boundary constraint to estimate a finite number of categories under this general framework.

We generated H haplotypes with relative frequencies (pj ’ 5) taken from a pure power law distribution with exponent of a.

A similar analysis with a truncated power law gave similar results.

We then compared the convergence rate of our method with other frameworks for estimating H. We used a simulated population from a truncated power law , as described above and formed sub samples of incremental size.

The equilibrium constants of the binding follow a log-normal distribution around the mean and a power law distribution in the tail.

Furthermore, the kinetics of binding follows a log-normal distribution near the mean and a power law distribution at the tail.

On the other hand, since partition function Z has an asymptotic form and a power law distribution at the tail, the associated free energy is therefore exponentially distributed at the tail (x—>oo)(See details in SI Text).

So, the distribution function f of the equilibrium constants K or the affinity can be shown to be: which is a log-normal distribution near the mean above TC While which shows a power law decay near the low K value tail of the distribution (near or below TC).

This power law behavior is familiar in the physics and chemistry community as a signature of the long range order and self similarity often appeared in the critical phenomena of phase transition, fractals, turbulence and earthquakes etc.

The parameters, ps, 3, pc, 5, p5, T and p6, T, are power law parameters that account for a nonlinear increase in susceptibility and transmissibility as animal numbers on a farm increase.

Previous work has indicated that a model with power laws provides a closer fit to the 2001 data than when these powers are set to unity [43,51,52].

Region-specific transmissibility and susceptibility parameters (and associated power laws ) capture specific epidemiological characteristics and policy measures used in the main hot spots of Cumbria, Devon and the Welsh and Scottish borders.