A power-law summarizes uptake dependence on host receptors

The approximately linear relationship between the logarithm of uptake and host receptors indicates the power-law dependence.

Sequential perturbation of Zipper model parameters shows that these changes can be mapped to changes in both power-law parameters (84C Fig).

Surprisingly, we found that the uptake probability of a single bacterium follows a simple power-law dependence on the concentration of integrins.

Furthermore, the value of a power-law parameter depends on the particular host-bacterium pair but not on bacterial concentration.

This power-law captures the complex, variable processes underlying bacterial invasion while also enabling differentiation of cell lines.

A detailed but unwieldy mechanistic model describing individual host-pathogen receptor binding events is captured by a simple power-law dependence on the concentration of the host receptors.

The power-law parameters capture characteristics of the host-bacterium pair interaction and can differentiate host cell lines.

Here, to characterize the fundamental property of bacterial uptake, we employ kinetic modeling and experiments that distill a simple power-law , relating uptake probability—the amount of bacteria per host cell scaled by the bacteria concentration—with host receptor levels.

We describe how different hosts and bacterial strains translate into different power-law parameters which serves as the basis of a novel, operational definition of cell type.

We, therefore, have developed a power-law based estimator to measure allele and haplotype diversity that accommodates heavy tails using the concepts of regular variation and occupancy distributions.

Finally, we compared the convergence of our power-law versus classical diversity estimators such as Capture recapture, Chao, ACE and Jackknife methods.

This suggests that power-law based estimators offer a valid alternative to classical diversity estimators and may have broad applicability in the field of population genetics.

We here use a power-law methodology that accommodates heavy-tails to estimate both the population coverage by ethnicity in the US and the genetic diversity of alleles and haplotypes.

Therefore, we built our model around a truncated power-law for estimating the properties of infinite discrete distributions with regularly varying heavy tails [7,8,14].

However, in heavy tailed distributions (such as power-law distributions) the vast majority of haplotypes are very rare.

This power-law framework is useful for modeling the probability mass (and number) of A and H that are unseen in the sample and represented by the “invisible” tail of the distribution.

Last, we discuss broader applications of the power-law methodology outside HSCT for modeling species richness in the field of ecology, which is characterized by similar heavy-tailed distributions.

Capture-recapture and power-law estimates were found to converge to the true value of H, but the capture-recapture method required a very deep sample of the population to attain accuracy whereas the power-law method converged quickly and offered accurate estimates, even with limited sampling (Fig 1B).

Thus, our current power-law methodology appears flawed for providing accurate estimates for the number of unique alleles and requires future modifications to accommodate the mixed data sources.

Furthermore, over the range of 1—60 spikes/ s, the f-chrve of the eLIF using a value of AT of 15 mV can be accurately fit with a power-law function with an exponent of 1.78, which is within the range of values observed experimentally for stellate cells (Fig 3F).

For power-law and Boltzmann fits, we also confirmed fits in Origin 8.5 (OriginLab, Northampton, MA).

For the f-V curve, experimental and modeling results were fit using a power-law function: where f is the firing rate, p is the exponent of the fit reported in the results section, a and b are positive constants and VC is the minimal voltage required to elicit spike generation.

Power-law scaling of stellate cell f-V curve without voltage fluctuations

In the visual system, a power-law scaling between spike firing rate and membrane voltage of layer 11 pyramidal neurons is critical for gain control and contrast invariance [12,37].

Modeling has shown that a power-law scaling with an exponent near 2 between spike firing rate and voltage can arise from the combination of an intrinsic, steep and linear f-Vrelationship, and smoothing through Gaussian-distributed voltage fluctuations [3,19,34,47].

Line indicates fit to a power-law function of the form shown in panel inset.

Surprisingly, we found that f-V curves were nonlinear and could be fit With a power-law function (Fig 2D; mean r2 = 0.95 i 0.02, range: 07—099) using an exponent (p) of 1.45 i 0.08 (Fig 2D; n = 19, range: 0.45—2.0, 17/19 had p values >1).

Contrary to previous assumptions [12,37], therefore, neuronal f-V curves can eXpress significant power-law scaling in the absence of any fluctuation-based smoothing.