It is generally believed that eukaryotic cells are large enough to detect and then move toward a signal by counting receptor occupancy. This can work because the relatively large eukaryotic cell is not subject to Brownian motion and can therefore use spatial differentiation to detect the direction of the signal over the relevant time scale. How this spatial differentiating is accomplished is an active area of research in fibroblasts, neutrophils and Dictyostelium, where the major components of the chemotactic response system are well known.
Of comparable interest is the behavior of these cells in the absence of a chemotactic (or other) signal. We might imagine, for example, that cells move about randomly in such a situation (), or that they have evolved a strategy that somehow optimizes their chances of finding the source of the signal, even when they cannot sense it ().
The search problem and search models.
Is there indeed such a thing as an optimum search strategy? Recent theoretical work has suggested that a Lévy walk is the optimum for revisitable targets, that is, targets that repopulate at the same location after a period of time 
. A Lévy walk is a special class of random walks whose step lengths (l
) are best described by a power-law: N
<3. Thus there is no intrinsic scale to the step lengths, and very long steps can occur (). Although there was thought to be experimental evidence for Lévy walk behavior in animal populations, a recent reanalysis of the data makes this unlikely 
, but see also 
. In a search for non-replenishable targets, where, like hide and seek, each target can be found only once, it has been suggested that a two-state model optimizes the search 
(). A searcher alternates between a local random search and a fast linear relocation. Target detection does not occur during the linear phase, both phases stop at random times, and each new phase is initiated in a random direction. It has been suggested that such intermittent behavior may be used by foraging animals 
Do any of these processes describe the behavior of single cells searching for a hidden target? A great deal is known about how neurons 
, amoebae 
, and fibroblasts 
find their targets once the signal has been sensed. In all three cases, more or less linear trajectories with variable low amplitude random behavior is the likely rule once the target is in range. But before pioneer neurons sense and begin to move up (or away from) a graded signal, do they send out filopodia at random, or do they bias their search to enhance the chances of finding the as yet undetected target? Do neutrophils wander at random before they detect bacterial peptides, or do they bias their motion in some fashion that provides a more efficient search algorithm?
We studied these questions by placing well-separated Dictyostelium amoebae on an agar surface free of food at a density of ~1 cell/cm2, ~1000 cell diameters between cells, a distance chosen so that the amoebae could not sense and signal to each other. Our results are of two kinds: First, these cells show long directional persistence. They bias their motion by making turns every 1–2 min, remembering their last turn and turning away from it in a zigzag fashion. Similar results were obtained with the distantly related slime mold Polysphondylium. We provide a model that satisfactorily captures the turning bias of freely moving cells, and links short and long-term cell motion persistence times. Second, although we cannot say that this behavior has been optimized by selection, we do demonstrate that it is only somewhat less efficient than straight-line behavior.
We believe this is the first experimental evidence for a biased walk in a foraging eukaryotic cell in the absence of spatial and temporal cues. Because the machinery underlying eukaryotic cell motion has been so highly conserved during evolution, we think it is likely that similar behavior is characteristic of other target-seeking eukaryotic cells.