INTRODUCTION

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION REFERENCES

Many bacteria swim by rotating their helical flagellar filaments and/or helical cell bodies. It has been reported that some bacteria swim well in viscous media (Shoesmith, 1960; Schneider and Doetsch, 1974; Kaiser and Doetsch, 1975; Strength et al., 1976; Greenberg and Canale-Parola, 1977a,b). For example, the swimming speed of Pseudomonas aeruginosa (single polar flagellation) in polyvinylpyrollidone (PVP) solutions increases with viscosity up to a characteristic point and thereafter decreases, as shown in Fig. 1 a (Schneider and Doetsch, 1974). The swimming speed of Leptospira interrogans (helical cell body without any external flagella) monotonically increases with viscosity in medium supplemented with methylcellulose until the viscosity exceeds 300 mPa · s (Kaiser and Doetsch, 1975). Other flagellation types of bacteria (Bacillus megaterium, peritrichous; Escherichia coli, peritrichous; Sarcina ureae, one flagellum/cell; Serratia marcescens, peritrichous; Spirillum serpens, bipolar; and Thiospirillum jenense, polar) also exhibit an increase in swimming speed in more viscous solutions (Schneider and Doetsch, 1974). No effect of temperature or buffer composition on the phenomenon was detected and different viscous agents, PVP and methylcellulose, produced a similar phenomenon (Schneider and Doetsch, 1974).

View larger version (10K): [in this window] [in a new window]   FIGURE 1   Bacterial swimming speeds, flagellar rotation rates, and -f ratios as a function of viscosity. (a) Experimental data of bacterial swim- ming speed. Bacterial species is Pseudomonas aeruginosa and viscous agent is polyvinylpyrollidone. The data by Schneider and Doetsch (1974) are redrawn. (b) Calculated swimming speeds as a function of viscosity. (c) Calculated flagellar rotation rates. (d) Calculated -f ratios. In b, c, and d, thick (thin) lines refer to the modified (traditional) RFT. Values in Table 1 were used in the calculation.

No such phenomena have been reported in larger organisms, and the traditional theories for bacterial motion predict that the swimming speed monotonically decreases with viscosity (Holwill and Burge, 1963; Chwang and Wu, 1971; Azuma, 1992; Ramia et al., 1993) (see Fig. 1 b).

Berg and Turner (1979) suggested that the above-mentioned phenomenon is caused by loose and quasi-rigid networks consisting of long, linear polymer molecules such as PVP and methylcellulose (e.g., see the textbook by Strobl, 1997), and recommended Ficoll, which is a highly branched polymer, as a simple means of increasing the viscosity. This suggestion has guided many researchers in the experimental analysis of bacterial motion. However, they did not express it mathematically, and no quantitative analysis based on the suggestion has been made.

In this study we interpreted the suggestion by Berg and Turner and mathematically developed it with regard to the motion of a single-polar-flagellated bacterium such as P. aeruginosa and Vibrio alginolyticus. In addition, we showed that the obtained equations quantitatively explained the peculiar phenomenon of bacterial motion in polymer solutions.

	RESULTS

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION REFERENCES

Introduction of two apparent viscosities

Our purpose is to mathematically develop the suggestion by Berg and Turner (1979). Their suggestions are as follows: 1) solutions of long linear polymers are highly structured (gel-like); 2) the solute forms a loose, quasi-rigid network easily penetrated by particles of microscopic size; 3) the network can exert forces normal to a segment of a slender body; and 4) therefore, traditional hydrodynamic treatments do not apply to the motion of microorganisms (or of cilia and flagella) in solutions containing viscous agents.

Fig. 2 a is a schematic drawing expressing their suggestion. It is assumed that the length of a slender body such as a bacterial flagellar filament is much larger than the mesh size of a polymer network, and that the motion of the slender body is faster than the polymer network. Consequently, the network exerts force on an element of the slender body mainly in the normal direction. In other words, the network forms a virtual tube around the slender body, and the body moves easily in the tube, but with difficulty outside the tube. This idea is almost the same as the "reptation model" in polymer physics (Strobl, 1997).

View larger version (28K): [in this window] [in a new window]   FIGURE 2   Motion model of free-swimming bacterial cell based on the modified RFT proposed in this study. (a) Schematic drawing expressing the concept of apparent viscosity for a slender body. (b) Schematic drawing expressing the concept of apparent viscosity for a spheroid body. (c) The force acting on a flagellar element ds. dFT and dFN are tangential and normal components of force acting on ds. is the pitch angle of the flagellar helix. (d) The forces acting on a cell body and a flagellar filament. The symbols used here are defined in Table 1.

The simplest way to express the idea mathematically is to introduce two apparent viscosities, µ^*_T and µ^*_N, referring to the motions of a microscopic body in the tangential and normal directions to the surface of the body. When a microscopic body moves in the virtual space that is surrounded with the polymer network, µ^*_T is assumed. Conversely, µ^*_N is assumed when the body goes outside the virtual space; in other words, when the polymer network must be reconstructed as the virtual space is moving. For example, when a microscopic sphere with radius a moves at a speed of  in a polymer solution, the drag force is assumed to be 6µ^*_Na because the sphere moves to the outside of the original virtual space (see Fig. 2 b). In contrast, when the same sphere rotates at a rate of , the drag torque is assumed to be 8µ^*_Ta³ because the motion is in the virtual space. When a solution does not contain any polymer molecules, µ^*_N equals µ^*_T and traditional hydrodynamics is valid. As the polymer concentration increases, the network becomes denser and the ratio of µ^*_N to µ^*_T becomes larger.

Mathematical expression of a modified resistive force theory

The following simple idea has often been adopted to obtain hydrodynamic force acting on a moving helix (Fig. 2 c). 1) The hydrodynamic force and torque acting on the whole helix can be obtained by integrating the force acting on a small element of the helix; 2) the force acting on the element is a composition of forces in the directions parallel and perpendicular to the element.

The original idea was adopted mainly for analyzing the motion of spermatozoa (Hancock, 1953; Gray and Hancock, 1955). Holwill and Burge (1963) applied it to the motion of flagellated bacteria, and the method is presently called resistive force theory (RFT). To derive a new theory of bacterial motion in a polymer solution, we modify the traditional RFT using apparent viscosities µ^*_T and µ^*_N.

The mathematical expression of the traditional RFT is summarized as follows (Magariyama et al., 1995). Here, the object is a bacterial cell that has a single-polar flagellum such as P. aeruginosa and V. alginolyticus, and the symbols are summarized in Table 1 (see also Fig. 2 d).

Equations of motion

(1)

(2)

(3)

Drag force and torque acting on a cell body

(4)

(5)

Drag force and torque acting on a flagellar filament

(6)

(7)

Torque generated by a flagellar motor

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

A prediction based on an approximation

To calculate the bacterial swimming speed in polymer solutions, we assumed an extreme case in which the polymer network does not affect the tangential motion of a microscopic slender body but does affect the normal motion in the same way as the motion of a macroscopic body. In this extreme approximation, µ^*_T equals the viscosity of the solution without polymer, µ₀, and is independent of the polymer concentration. Furthermore, µ^*_N is the same as the macroscopic viscosity of the polymer solution, µ, and increases with the polymer concentration. The values of the other parameters used in the calculation are summarized in Table 1.

The calculated swimming speed based on this approximation increased up to a characteristic viscosity and then decreased with viscosity, but the decrease was less than that predicted by the traditional RFT, as shown in Fig. 1 b. This result is similar to the experimental one. The calculated flagellar rotation rate monotonically decreased with viscosity in a way similar to the prediction based on the traditional RFT, as shown in Fig. 1 c.

The ratio of swimming speed to flagellar rotation rate (-f ratio) refers to the distance progressed per flagellar revolution, namely a kind of propulsion efficiency of the bacterial flagellar system. The -f ratio calculated by the modified RFT increased with viscosity, although that by the traditional RFT was constant independent of viscosity, as shown in Fig. 1 d. This means that the improvement of the propulsion efficiency of the bacterial motion made up for a decline in flagellar rotation rate due to polymer addition and caused an increase in swimming speed with viscosity.

Peak position of swimming speed

The polar flagellum of Vibrio alginolyticus is driven by sodium ions, and the motor activity decreases with decreasing sodium concentration below 20 mM (Liu et al., 1990; Kawagishi et al., 1995). Atsumi et al. (1996) reported that the peak position of swimming speed of V. alginolyticus shifts toward high viscosity in PVP solutions as the sodium concentration is lowered, and that the difference between the swimming speeds in different sodium concentrations becomes smaller at higher viscosity. The modified RFT could also explain this experimental result, supposing that the decrease of sodium concentration reduces ₀. Fig. 3 a shows calculated swimming speeds as a function of viscosity with four different values of ₀, where the same values as Table 1 were used except ₀. As ₀ decreased, 1) the curve of swimming speed moved downward; 2) the peak position shifted toward the higher viscosity, and 3) the peak became broader.

View larger version (20K): [in this window] [in a new window]   FIGURE 3   Peak positions of swimming speed calculated based on the modified RFT. (a) Swimming speeds as a function of viscosity. The swimming speeds were calculated for four different values of 0, 8500, 1700, 850, and 340 rpm. The values of the other parameters are in Table 1. The upper curve was calculated for larger values of 0. (b) Peak position of swimming speed as a function of T0 and 0.

To further examine the peak shift caused by changes in the characteristics of motor torque, T₀ and ₀, we calculated µ_peak, which satisfies /µ = 0, with the values of bacterial parameters and water viscosity fixed to those in Table 1. The obtained equation is

(28)

and µ_peak as a function of T₀ and ₀ is shown in Fig. 3 b. With decreasing T₀ or increasing ₀, µ_peak decreased; that is, the peak position shifted toward the low viscosity, suggesting it become harder to detect the peak of swimming speed experimentally because the peak position becomes near or below the water viscosity. The exact torque characteristics of the V. alginolyticus motor, however, have not been measured, especially the change with sodium concentration. To confirm the validity of the present model further, it is desired to obtain such information.

	DISCUSSION

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION REFERENCES

The suggestion by Berg and Turner (1979) was mathematically expressed by introducing two apparent viscosities into the traditional RFT, and the peculiar phenomenon was successfully interpreted. The peak-position shift in swimming speed of V. alginolyticus with the sodium concentration was also explained, assuming that ₀ decreased with the concentration.

The following observations have been reported about the motions of microscopic bodies in polymer solutions.

• The diffusion and sedimentation of microspheres in polymer solutions generally follow "Stokes-Einstein behavior," i.e., f = 6µ^*_transa, where f is a modified Stokes friction factor, a is the radius of the sphere, and µ^*_trans is the effective viscosity that is dependent on the concentration, molecular weight and morphology of the polymer, the sphere size, and so on (Kluijtmans, et al., 2000; Turner and Hallett, 1976; Yang and Jamieson, 1988; Brown and Rymden, 1988; Phillies et al., 1989; Onyenemezu et al., 1993; Bu and Russo, 1994);
• The rotations of bacterial tethered cell in methylcellulose solutions are faster than those in Ficoll solutions at the same viscosities (Berg and Turner, 1979). The measurement of the rotational Brownian motion of globular proteins in dextran solutions indicates that the effective viscosity is expressed as µ^*_rot = µ₀(µ/µ₀)^q, where µ and µ₀ are the macroviscosities of polymer solution and water as mentioned above, and q is smaller than unity and is distinctly polymer- and protein-dependent (Lavalette et al., 1999).

Generally, µ^*_trans and µ^*_rot have different values in polymer solutions. In other words, these observations indicate that a motion of a microscopic body in a polymer solution can be decomposed into two kinds of motions to which the Stokes-Einstein expression applies using the respective apparent viscosities. Of course, our assumption and/or deduced results should be experimentally verified in the next stage because the above reports are not direct evidence of two effective viscosities.

The modified RFT predicted an increase in the -f ratio (improvement of propulsion efficiency) with viscosity (polymer concentration) while the calculated -f ratio based on the traditional RFT was constant. To examine the validity of the modified RFT, we can simultaneously measure flagellar rotation rates and swimming speeds in polymer solutions. Such experiments in solutions containing no viscous agents have been carried out by using laser dark-field microscopy (LDM) (Kudo et al., 1990; Magariyama et al., 1995, 2001). We are planning to measure the -f ratio in polymer solutions by LDM.

The apparent viscosities depend on the characteristics of the polymer network, which are affected by the properties of the polymer such as the length (molecular weight), the morphology (linear or spherical), and the interaction between the molecules. Although the motion of a microscopic slender body in the tangential direction was not assumed to be affected by the polymer network in this study, the actual motion must be affected by the network because it moves to some degree. In addition, although we assumed that the change in sodium concentration altered only the motor torque, it may affect the characteristics of the polymer network as well. It is necessary to determine theoretically and/or experimentally how the hydrodynamic force acting on a flagellar element depends on the properties of a polymer in addition to the concentration.

To determine the values of apparent viscosities, we must estimate the hydrodynamic forces acting on a microscopic slender body in the normal and tangential directions. A conceivable experimental way is to measure the Brownian motion of a microscopic slender body in polymer solutions and determine its diffusion constants in the normal and tangential directions. We are planning to measure the Brownian motion of straight flagellar filaments or particles of tobacco mosaic virus (TMV) in polymer solutions and estimate the two apparent viscosities.

Most bacterial cells live in sticky conditions, e.g., surfaces of other organisms and biofilm, rather than in mobile liquids, e.g., marine water. In other words, they are usually living in various polymer solutions. The modified RFT indicates that a long, thin, helical shape is suitable for maintaining its swimming speed in viscous conditions containing polymer molecules because of the improved propulsion efficiency. The propulsion system of spirochetes, in which a helical or flat-sine wave of the cell body moves backward, may be an ultimate mechanism for swimming in highly viscous conditions.