A THEORY OF THE SURVIVAL VALUE OF MOTILITY 143 



(D n + A' • v), where the constant A' will depend on the kind of motion 

 executed by the particle. 



THE DEPENDENCE OF ENERGY EXPENDITURE ON THE 

 AVERAGE VELOCITY OF THE CELL 



Now comes the question of the dependence of the energy expended 

 by the cell velocity. A consideration of the hydrodynamics of the mo- 

 tion of microorganisms reveals, as was pointed out by Taylor (1951), 

 that these motions are characterized by low Reynolds numbers. The 

 Reynolds number provides an estimate of the relation of the inertial 

 stresses in the fluid surrounding the moving organism to the viscous 

 stresses and is given by, I ■ v • p/n, where / is length of the particle, v 

 is its velocity, p is the density of the fluid, and n is the viscosity of the 

 fluid. For most microorganisms this quantity is less than 10~ 2 or 10~ 3 . 

 This means that the inertial stresses are less than 1% of the viscous 

 stresses. For such a situation one can neglect the inertial forces in 

 the hydrodynamic equations of motion and consider only the viscous 

 forces. It has been shown quite generally by Lamb (1932) that mo- 

 tions dominated by viscous forces dissipate energy in proportion to 

 the square of the strain rate and hence to the square of the velocity, 

 that is, the rate of energy dissipation, P m , due to the motion at low 

 Reynolds number, is given by a relation of the form, P m = A ■ v 2 , 

 where A is a constant that will depend on the particular motile proc- 

 ess involved. 



To determine the net rate of energy dissipation by the cell, allow- 

 ance must be made for the fact that the motility mechanism is not 

 100% efficient. For every unit of energy converted to motile energy, 

 and thence dissipated as heat in the surroundings, there is a certain 

 amount of energy lost directly as heat owing to the inefficiency of 

 the motility machine. In addition to this energy dissipated as a result 

 of moving, there is also the basal rate of energy expenditure which is 

 essential for survival of the cell. If we let P () equal the basal rate of 

 energy expended by the cell, and let E designate the efficiency of 

 the motility mechanism, then the total energy dissipated by the 

 moving cell becomes 



