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Correlation Functions for the Chemokinetic Movement of Granulocytes and Tumor Cells

We discuss and interpret the chemokinetic mechanism of cell movement as a stochastic process in two ways. First we apply the Langevin approach and secondly we make use of the Fokker-Planck equation in order to calculate the velocity correlation function and the mean square displacement of a cell treated as a Brownian particle moving in a solvent of a molecular fluid. Using Kubo's formulation of the fluctuation-dissipation-theorem relating the diffusion constant D to the velocity correlation function we relate D to the thermal velocity and to the correlation time. Finally, we compare our theoretical predictions with experiments carried out on granulocytes and tumor cells. Introduction Motile microorganisms like bacteria or cells are far more complex than atoms or molecules. They move in a complicated way, and it is generally accepted by experimental observations [1-4] that once a cell starts moving in a certain direction it persists with constant velocity in that direction for some characteristic time t before it changes direction. Cell movement is usually devided into three different mechanisms: (i) Locomotion which means a variation of cell shape but no movement of the center-of-mass of the cell, (ii) Chemokinesis by which is meant a nondirected (isotropic) movement of the center-of-mass of the cell where the cell becomes elongated in the direction of the movement, and (iii) Chemotaxis which is an anisotropic (directed) movement of the elongated cell caused by a chemotactic gradient. Very recently [1—3] some successful attempts have been reported to put the experimental results of cell movement into a phenomenological formalism in order to get a better understanding and interpretation of the mechanisms. The phenomenological concept which has been applied [1-3] in order to analyse the chemokinetic movement of granulocytes is based on Fürth's [4] investigation treating the cells as Brownian particles. Fürth discussed two asymptotic formulas 120 T. F. Nonnenmacher for the mean square displacement, which in our notation have the form (one dimensional case)  ô ' ' (la) and = vt for t < ô (lb) where Ä÷(ß) = ÷ (t) — x0 is the deviation of ÷ at the time t from x0 = x(t = 0). Eq. (la) is valid for observation times t large compared to some characteristic reference time ô and prescribes a "pure" random process, while Eq. (1 b) is approximately valid for t < ô. There are three parameters: D (diffusion coefficient), ô (some reference time, which is called "internal clock" in Refs. [1-3]) and the "track velocity" vc. In a phenomenological description these parameters are not further specified: They are free in order to allow the best fit to experimental observations. In a theoretical concept, however, based on a molecular theory there is no place for unspecified parameters. Consequently, one motivation of our current investigation is to find relations connecting these parameters, and to express the parameters D, ô and vc through molecular data. A second motivation of this paper is to elevate the phenomenological concept just described to a molecular level by making use of the theory of stochastic processes which is the basic theory for Brownian particles moving in a fluid background system. Our main goal is to calculate the velocity correlation function  f° tk chemokinetic cell movement. The knowledge of this correlation function enables us to construct an exact formula for the experimentally observed [1—3] mean square displacement <^jc(i)>Going to the limits (i) t> ô ana (ii) t < ô we just recover F rth's asymptotic expressions (la) and (lb). In addition, the application of Kubo's version of what is called a fluctuationdissipation-theorem