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Conceived and designed the experiments: KAS. Performed the experiments: KAS. Analyzed the data: KAS. Contributed reagents/materials/analysis tools: KAS. Wrote the paper: KAS. Designed and wrote computer programs used in data transformation and analysis: KAS.

The author has declared that no competing interests exist.

Human polymorphonuclear leucocytes, PMN, are highly motile cells with average 12-15 µm diameters and prominent, loboid nuclei. They are produced in the bone marrow, are essential for host defense, and are the most populous of white blood cell types. PMN also participate in acute and chronic inflammatory processes, in the regulation of the immune response, in angiogenesis, and interact with tumors. To accommodate these varied functions, their behavior is adaptive, but still definable in terms of a set of behavioral states. PMN morphodynamics have generally involved a non-equilibrium stationary, spheroid

Human white blood cells, polymorphonuclear leucocytes (PMN), were microscopically imaged and analyzed as single living cells. PMN are generally observed in a spheroid

Polymorphonuclear neutrophil granulocytes (PMNs) are the body's most abundant class of white blood cells. At circulating cell levels of ∼10^{11}, PMNs make up about 62% of all human white cells

Once activated, idling PMNs exploit shape change dynamics as they tether to the endothelium of post-capillary venules, making and breaking bonds while rolling along the venule's endothelial bed of selectins. Following concentration gradients of inflammatory ligands, they wriggle through the vessel wall and migrate to the site of the initiating inflammation. The motions accompanying this migration exploit the PMN's cytoskeletal, actin polymerization-depolymerization cycle that configures the dynamics of shape change and translation

The behavioral dynamics of two of the shape changing metastable states and their associated translational motions are well established. PMNs manifest these metastable shape-motional states

The complexity of the scenario described above and the necessity of as many as 100 distinct protein/protein interactions to coordinate the actin cytoskeletal apparatus alone, prompts both a phenomenological approach and the consideration of a potentially larger set of shape-motional states which may be transitional, non-stationary and not easily quantified.

In addition to the metastable spherical-round,

We remind ourselves that there are neurobiological limits on the both the resolution afforded by empirically meaningful partitions

The issue more generally is the selection of an appropriate sample size of an intrinsically non-stationary system. Counter-intuitively, it has been shown that under certain conditions of limited information, repeated too-short sample lengths come to be computationally superior globally

In the past I have dealt with this problem by studying repeated time series derived measures yielding populations of not necessarily convergent estimates

It is difficult to find a global quantitative measure on the dynamics of emergent phenomenon with the nice properties of additivity, continuity and differentiability

I have examined the autonomous, time-dependent shape changes of individual _{1}/_{2} is computed as the ratio of the radius of the cell assumed to be an ideal circle, _{1} = p/2π_{2} = (A/π)^{0.5} using the sum of the pixels within the cell's silhouette as the cell's _{1}(t)/_{2}(t) = {^{0.50}}(t) computed at each time step. If both _{1} and _{2} were derived from an abstract, idealized circle, _{1}/_{2} = 1.0, such that _{1}/_{2}} = 0, the characteristic lower limit of a generic order parameter. Deviations from this reference characterize changes in state

One hundred and eighty-nine PMNs from fifty-three peripheral blood samples were collected from 18 healthy adult volunteers, aged 26 to 72. The blood samples were allowed to sediment gravitationally for 40 minutes at room temperature. A population of PMNs (and other white cells and platelets) were removed from the buffy coat by micropipette and, along with associated plasma, placed within a 12 mm ring painted on a glass slide, forming a ∼20–25 µm deep well, compared to the average 5.7 µm vertical space between a plain slide and its cover slip

PMN autonomous motions were observed using an Olympus BX41 microscope fitted with CytoViva dark field and fluorescent optical illumination systems, which includes a unique, high-aperture, cardioid annular condenser (

Images were collected every 2 sec for 30 min using an Optronics Microfire 1200×1600 CCD array camera

In light of the above discussion of biological constraints on sample length and the intrinsic non-stationarity of the PMNs shape motion series, statistical distributions of often individually non-convergent measures made on each of the cells, serve as the basis for comparisons of

On the _{1} and standard deviation, _{2}, as well as the skewness, _{3}, indicating the asymmetry of the density distribution of _{3}_{3} = _{3}/var^{3/2}_{4}, of _{4} = m_{4}/variance^{2} -3 _{1}

To visualize the phase space behavior I used relatively denoised, three dimensional Broomhead-King, B/K, eigenfunction, Ψ_{i} embedding of the _{1}, Ψ_{2}, and Ψ_{3} with respect to each other _{M}, matrix was computed, with M = 8, a typical correlation decay interval. C_{M} was then decomposed into its eigenvalue-ordered eigenvectors. The eigenvectors associated with the three largest eigenvalues were each composed with the original _{1}, Ψ_{2}, and Ψ_{3}. These formed the axes of the B/K eigenspace reconstruction. Because

Another graphical representation of the orbital behavior of _{1}, _{2}, Recurrence Plot, RP[_{i,j}, introduced by Eckmann _{i,j} are two dimensional lattices, each point computed as RP_{i,j} = Θ(ε_{i,j = 1…N}^{2} represents the location of the orbit in phase space at time _{i,j} has been used to discriminate among three characteristic patterns of intermittency

There were highly significant differences between the measures _{2}_{3}_{1}, and α that were made on the _{1} or _{4}.

Idling (n = 10) | Treadmilling (n = 10) | t(df); ρ |

Mean = 2.935 | Mean = 2.932 | t_{(18)} = 0.004; ρ<0.4983 |

SD = 0.1023 | SD = 0.3882 | t_{(18)} = 4.816; ρ<0.0001 |

Skew = 0.2993 | Skew = 0.7273 | t_{(18)} = 2.419; ρ<0.0132 |

Kurtosis = 0.7229 | Kurtosis = 1.165 | t_{(18)} = 0.652; ρ = 0.2612 |

λ_{1} = 0.6242 |
λ_{1} = 0.5066 |
t_{(18)} = 2.592; ρ<0.0090 |

α = −0.4561 | α = −1.0290 | t_{(18)} = 10.600; ρ<0.0001 |

The qualitative differences in the shape-motional patterns implied by the statistically significant differences in the measures in _{2} _{3}, and the increase in shape-motional order in _{1}, the leading Lyapounov index of expansive, orbital mixing

Consistent with the differences in behavior described by direct observation and the aggregate of measures (see _{1}, Ψ_{2}, Ψ_{3}}_{1…900} B/K eigenfunctions embedding of four representative

Another geometric, graphical treatment of the cell's shape motional behavior is displayed in _{i,j} of the four representative PMNs in the _{i,j} of the _{i,j} of cells in the _{i,j}. _{i,j} portraits are consistent with recurrence patterns of intermittency

ε = 1 for all plots.

Four of the seven order parameter measures demonstrated statistically significant differences between the

The upper panel shows the

Idling | Treadmilling |

Mean = 2.0969 | Mean = 2.1614 |

SD = 0.0348 | SD = 0.1620 |

Skew = −0.1942 | Skew = 2.0076 |

Kurtosis = 2.8552 | Kurtosis = 8.0831 |

λ_{1} = 0.6520 |
λ_{1} = 0.498 |

α = −0.5681 | α = −1.0486 |

There are established physiological mechanisms and behavior that are consistent with both our qualitative microscopic observations and quantitative aggregate measure descriptions. PMNs are known to oscillate on multiple time and space scales, from 7 sec, 70 sec, and 260 sec membrane potential fluctuations

The slowest Fourier mode in _{ω}_{ω}

As listed in

Many characteristics of the changes in measures in the distinct single state observations and in the computable, real-time transition from _{2} and _{3} of the cell shape fluctuations; (2) Decreased leading Λ_{1} becoming less positive in the direction of zero, shadowing the leading eigenvalue of the unknown underlying partial differential equation; (3) An increase in the log-log power spectral scaling index, α, reflecting a “less white” spectral pattern of _{1}, Ψ_{2}, Ψ_{3}}_{i} demonstrated directly the space-time morphogenic transformation undergone by _{i,j} depicted increased phase space clustering consistent with the more hierarchical, intermittent dynamics of the _{i,j}. It should be noted that the action spaces of less uniform intermittency and those of more uniform transitivity reflect common metastable alternatives in the dynamics of some biological sciences

Finally, I have spent hundreds of hours microscopically tracking 189 individual PMN cells in the hopes of answering these questions about state and state transitions. While only one such transition was recorded with sufficient observations in both the

FromTo | Idling | Treadmilling | Translocating |

Idling | |||

Treadmilling | |||

Translocating |

I wish to thank Prof. Arnold Mandell and Dr. Paul Gailey for their helpful comments and suggestions.

_{1}muscarinic receptor-targetted hydrophobic eigenmode matched peptides as functional modulators.