Effects of Cell Motility Properties on Cell ... - ACS Publications

12

Effects

of

Cell

Motility

Properties

on

Cell

P o p u l a t i o n s in E c o s y s t e m s

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DOUGLAS A. LAUFFENBURGER University of Pennsylvania, Department of Chemical Engineering, Philadelphia, PA 19104

Because most natural ecosystems cannot r e a l l y be considered well-mixed, conclusions regarding dynamics of c e l l population growth and i n t e r a c t i o n s drawn from well-mixed, chemostat studies may not necessarily be v a l i d . Many microbial species, in f a c t , possess sophisticated movement behavioral properties by which d i s t r i b u t i o n of a population in space i s influenced by concentrations and gradients of chemicals commonly present in their environment. These chemotactic and chemokinetic properties can require s i g n i f i c a n t devotion of genetic information and sometimes s i g n i f i c a n t energy expenditure, yet are of little apparent use i n artificial well-mixed systems. However, in non-mixed environments, the e f f e c t s of chemosensory movement properties may be extremely important in determining the a b i l i t y of a species to grow, or in deciding the outcome of competitive i n t e r a c t i o n s . This paper summarizes the a v a i l a b l e experimental evidence that t h i s i s indeed the case, that movement properties can have c r u c i a l e f f e c t s in microbial ecosystems. We then present some mathematical models that help explain and predict these effects.

0097-6156/83/0207-0265 $08.00/0 © 1983 American Chemical Society

Blanch et al.; Foundations of Biochemical Engineering ACS Symposium Series; American Chemical Society: Washington, DC, 1983.

266

BIOCHEMICAL ENGINEERING

S t u d i e s o f m i c r o b i a l p o p u l a t i o n dynamics have focused p r i m a r i l y on well-mixed, macroscopically homogeneous s y s t e m s . T h i s emphasis i s e a s i l y d i s c e r n i b l e from t h e c o n t e n t s o f t h i s symposium volume. However, t h e s i t u a t i o n i n most n a t u r a l l y o c c u r i n g m i c r o b i a l systems i s f a r from i d e a l l a b o r a t o r y c o n d i t i o n s , so that understanding gained a b o u t w e l l - m i x e d s y s t e m s may n o t p r o v i d e t h e appropriate insight into ecological situations. Environments which a r e not well-mixed can allow formation o f s p a t i a l gradients o f chemical concentrat i o n s and c e l l d e n s i t i e s . Also, chemical d i f f u s i o n and c e l l m o t i l i t y ( i . e . , s e l f - p r o p e l l e d movement r e q u i r i n g energy expenditure) can replace convection a s t h e d o m i n a n t mode o f t r a n s p o r t . According to common c a t e g o r i z a t i o n , h a l f t h e o r d e r s o f b a c t e r i a c o n t a i n a t l e a s t o n e m o t i l e s p e c i e s (1), including many o f t h e commonly o c c u r i n g s p e c i e s . F u r t h e r , most m o t i l e b a c t e r i a e x h i b i t c h e m o t a x i s , w h i c h i s most s i m p l y d e f i n e d a s c e l l movement t o w a r d o r away f r o m chemicals ( 2 ) , o r a s p r e f e r e n t i a l c e l l movement t o w a r d h i g h e r o r lower c o n c e n t r a t i o n s o f a c h e m i c a l stimulus {3). A c t u a l l y , t h e r e a r e a number o f d i f f e r e n t types o f movement r e s p o n s e s w h i c h l e a d t o b e h a v i o r o f t h i s g e n e r a l d e s c r i p t i o n ( 4 ) . For peritrichously flagellated bacteria, w h i c h a p p e a r t o be t h e m o s t c o m m o n l y e n c o u n t e r e d group (and o n w h i c h t h i s p a p e r w i l l a c c o r d i n g l y f o c u s ) , k l i n o k i n e s i s ( i n which the t u r n i n g frequency o f swimming b a c t e r i a i s m o d u l a t e d b y s t i m u l u s c o n c e n t r a t i o n ) a p p e a r s t o be c l o s e s t t o o b s e r v e d b e h a v i o r ( 5 ) . T h i s i s i l l u s t r a t e d i n F i g u r e 1. In the absence o f a chemical stimulus gradient, these b a c t e r i a swim i n roughly s t r a i g h t l i n e steps c a l l e d "runs" f o r a short time (about 1 second) and t h e n s t o p and change d i r e c t i o n , o r " t u m b l e " , f o r a b o u t 1/10 s e c o n d ( 6 ) . The d i r e c t i o n c h a n g e i s p u r e l y r a n d o m , b u t t h e p r o b a b i l i t y o f tumbling i s constant d u r i n g a r u n ,so t h a t t h e r u n time d i s t r i b u t i o n i s P o i s s o n i a n ( 6 ) . T h i s movement b e h a v i o r i s t e r m e d r a n d o m m o t i l i t y . In the presence o f a g r a d i e n t , t h e d i r e c t i o n change r e m a i n s random b u t t h e t u m b l i n g p r o b a b i l i t y decreases f o r a c e l l swimming t o w a r d h i g h e r a t t r a c t a n t c o n c e n t r a t i o n s o r lower r e p e l l e n t c o n c e n t r a t i o n s (6), l e a d i n g to n e t m i g r a t i o n i n t h e d i r e c t i o n o f t h e g r a d i e n t . T h i s mechanism p r o v i d e s v e r y e f f i c i e n t response (]_) ; i n an o p t i m a l g r a d i e n t t h e n e t m i g r a t i o n v e l o c i t y i s r o u g h l y h a l f t h e l i n e a r c e l l swimming s p e e d ( 8 , 9 ) .


12.

LAUFFENBURGER

Cell

267

Motility

RANDOM MOTILITY

CHEMOTAXIS

Figure 1. Illustration of typical movement of peritrichously flagellated bacteria. The left-handfigureshows movement of a cell in the absence of a chemical stimulus concentration gradient. The right-handfigureshows that the run length is increased when the cell moves in the direction of increasing attractant concentration, toward the top of the figure. The angles between respective runs in the two figures are identical. The increase in run length results in a greater drift in the direction of increasing attractant concentration for chemotactic movement than for random movement.


268


S t i m u l u s c o n c e n t r a t i o n s a r e m e a s u r e d by o c c u p a n c y o f c e l l membrane r e c e p t o r s w h i c h c a n r e v e r s i b l y b i n d stimulus molecules according to Michaelis-Menten enzyme k i n e t i c s w i t h d i s s o c i a t i o n c o n s t a n t . G r a d i e n t s a r e d e t e c t e d by a t e m p o r a l s e n s i n g m e c h a n i s m by w h i c h r e c e p t o r o c c u p a n c y o v e r t h e e n t i r e c e l l i s m o n i t o r e d d u r i n g a r u n (1Λ) . A spatial sensing m e c h a n i s m by w h i c h d i f f e r e n c e s i n r e c e p t o r o c c u p a n c y a c r o s s the c e l l l e n g t h i s measured i s i m p r a c t i c a l f o r s u c h r a p i d l y swimming c e l l s , b e c a u s e o f f a l s e a p p a r e n t g r a d i e n t s due t o c e l l m o t i o n i t s e l f (12).

A l a r g e v a r i e t y o f c h e m i c a l s c a n s e r v e as chemotactic s t i m u l i . Approximately 20 a t t r a c t a n t r e c e p t o r s a n d 10 r e p e l l e n t r e c e p t o r s h a v e b e e n i d e n t i f i e d f o r E s c h e r i c h i a c o l i and Salmonella t y p h i m u r i u m , t h e two m o s t w i d e l y - s t u d i e d s p e c i e s ( Γ 3 ) . T h e s e r e c e p t o r s a r e s p e c i f i c f o r one o r two chemicals at h i g h a f f i n i t i e s (low K ^ ) , but w i l l a l s o b i n d a range o f r e l a t e d m o l e c u l e s w i t h lower a f f i n i t y (2). Table 1 l i s t s a number o f t h e w e l l - k n o w n s t i m u l i f o r a v a r i e t y of species, although t h i s i s c e r t a i n l y i n c o m p l e t e a s new r e s p o n s e s a r e b e i n g d i s c o v e r e d almost d a i l y . I n g e n e r a l , s u b s t a n c e s b e n e f i c i a l t o an o r g a n i s m s u c h as n u t r i e n t s and o x y g e n ( i f a e r o b i c ) s e r v e as a t t r a c t a n t s w h i l e t o x i c compounds o r compounds c a u s i n g pH e x t r e m e s a c t a s r e p e l l e n t s ( 2 ) . Some a m i n o a c i d s a r e a t t r a c t a n t s and o t h e r s r e p e l l e n t s , d e p e n d i n g upon t h e s p e c i e s . A l t h o u g h t h e r e i s n o t an e x a c t correspondence between m e t a b o l i z a b l e compounds and a t t r a c t a n t s n o r b e t w e e n u n f a v o r a b l e c o m p o u n d s and r e p e l l e n t s (L4), the observed responses can g e n e r a l l y be r a t i o n a l i z e d i n t e r m s o f p a r t i c u l a r s p e c i e s b i o c h e m i c a l p a t h w a y s and by r e c o g n i t i o n o f some p u z z l i n g s t i m u l i as a n a l o g u e s o f o t h e r s t i m u l i (2). Speculation regarding a possible survival advantage o f chemotaxis i s thus not s u r p r i s i n g . Approximately 40 g e n e s a r e d e v o t e d s p e c i f i c a l l y t o t h e c h e m o t a c t i c r e s p o n s e i n E ^ c o l i and S^ t y p h i m u r i u m (15) and a s i m i l a r number o f g e n e s may be d e v o t e d t o t h e m o t i l i t y apparatus. S u c h an i n v e s t m e n t m u s t p r o v i d e some b e n e f i t i n t h e c o m p e t i t i v e w o r l d o f m i c r o b i a l ecology. However, f u n d a m e n t a l u n d e r s t a n d i n g o f the c i r c u m s t a n c e s i n w h i c h a s i g n i f i c a n t a d v a n t a g e due t o m o t i l i t y a n d c h e m o t a x i s w i l l a c t u a l l y be p r e s e n t i s l a c k i n g , a s a r e e s t i m a t e s o f t h e m a g n i t u d e o f s u c h an advantage. This understanding i s l a c k i n g even f o r a s i n g l e s t i m u l u s , and t h e p r o b l e m b e c o m e s e v e n more complex i n any n a t u r a l e n v i r o n m e n t i n w h i c h m u l t i p l e

y


12.

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269

Table 1. Classification of Weil-Known Bacterial Chemotactic Responses Genus

Classes o f Attractants

Classes o f Repellents

Escherichia

sugars amino acids C>

pH extremes a l i p h a t i c alcohols

Pseudomonas

sugars amino acids nucleotides vitamins

inorganic ions pH extremes amino acids

°2 . . ammonium ions Bacillus

Salmonella

sugars amino acids °2 sugars amino acids

inorganic ions pH extremes metabolic poisons a l i p h a t i c alcohols

0„ Vibrio

amino acids

Spirillum

sugars amino acids

inorganic ions pH extremes

Rhodospirilium

nucleotides s u l f h y d r y l compounds

pH extremes poisons

Clostridium Bdellovibrio

amino acids

Proteus

sugars amino acids °2 sugars

Erwinia

inorganic acids pH extremes inorganic ions pH extremes inorganic ions pH extremes

Sarcina Serratia

sugars amino acids

inorganic ions pH extremes

0„ Bordetella Pasteurella Marine b a c t e r i a

algal culture f i l t r a t e s

heavy metals t o x i c hydrocarbons

Source: Refs. 36-38.


270


s t i m u l i , p r o b a b l y b o t h a t t r a c t a n t s and r e p e l l e n t s , a r e present. M u l t i p l e s i g n a l s a r e a d d i t i v e i n some s e n s e ( 1 6 ) , and p r e s u m a b l y t h e b a c t e r i a a t t e m p t t o o p t i m i z e t h e i r growth through p r e f e r e n t i a l m i g r a t i o n . F r o m an e n g i n e e r i n g p e r s p e c t i v e , t h e r e i s no q u a n t i t a t i v e b a s i s f o r p r e d i c t i o n o f the e f f e c t s o f c e l l m o t i l i t y and c h e m o t a x i s o n m i c r o b i a l p o p u l a t i o n d y n a m i c s i n a n y given s i t u a t i o n at present. T h i s paper i s devoted to r e v i e w o f the s m a l l body o f i n f o r m a t i o n a v a i l a b l e i n t h i s area at t h i s time. Experimental

Observations

T h e r e e x i s t s o n l y a s m a l l number o f p u b l i s h e d experiments p e r t a i n i n g t o the e f f e c t s o f c e l l m o t i l i t y on p o p u l a t i o n g r o w t h . A well-known m i c r o b i o l o g y t e x t g i v e s one example w i t h o u t d o c u m e n t a t i o n : the c o m p e t i t i o n b e t w e e n an a e r o t a c t i c ( i . e . , c h e m o t a c t i c a l l y a t t r a c t e d by o x y g e n ) P s e u d o m o n a s s p e c i e s a n d an i m m o t i l e A c i n e t o b a c t e r s p e c i e s , f o r o x y g e n ( 1 7 ) . When t h e g r o w t h medium i s w e l l - a e r a t e d t h e A c i n e t o b a c t e r predominate, thus showing s u p e r i o r g r o w t h k i n e t i c s on t h e r a t e - l i m i t i n g s u b s t r a t e (presumably oxygen) s i n c e c e l l m o t i l i t y i s a p p a r e n t l y irrelevant. But i n a non-mixed c u l t u r e the Pseudomonas p r e d o m i n a t e . The c h e m o t a c t i c a b i l i t y of the Pseudomonas s p e c i e s p r o v i d e s , i n t h i s s i t u a t i o n , enough o f a b e n e f i t t o overcome i t s growth k i n e t i c inferiority. The f i r s t l i t e r a t u r e r e p o r t i n t h i s a r e a was by S m i t h a n d D o e t s c h (]J*) , who s t u d i e d c o m p e t i t i o n b e t w e e n a e r o t a c t i c P s e u d o m o n a s f l u o r e s c e n s and an i m m o t i l e m u t a n t s t r a i n o f t h e same s p e c i e s , f o r o x y g e n (see Figure 2 ). In a e r a t e d mixed c u l t u r e b o t h s t r a i n s g r e w t o a r o u g h l y 1:1 r a t i o o v e r a 2 4 - h o u r p e r i o d i n d i c a t i n g t h a t t h e i r g r o w t h k i n e t i c p r o p e r t i e s were i d e n t i c a l as e x p e c t e d . In n o n - a e r a t e d media, the a e r o t a c t i c s t r a i n outgrew the mutant t o a f i n a l ratio o f o v e r 10:1 a f t e r 24 h o u r s . U n f o r t u n a t e l y , the a u t h o r s c r e d i t e d m o t i l i t y per se f o r t h i s a d v a n t a g e , e v e n t h o u g h i t i s not c l e a r whether random m o t i l i t y without chemotaxis i s n e c e s s a r i l y always b e n e f i c i a l . In the c o u r s e o f s t u d y i n g t h e r o l e o f f i m b r i a e i n b a c t e r i a l g r o w t h , O l d and D u g u i d l o o k e d a t c o m p e t i t i o n f o r o x y g e n b e t w e e n two n o n f i m b r i a t e s t r a i n s o f Salmonella typhimurium: one a e r o t a c t i c and one immotile (_19) (see Table 2 ) . In a e r o b i c shaken b r o t h , t h e a e r o t a c t i c s t r a i n m u l t i p l i e d by a f a c t o r o f 46 w i t h i n 48 h o u r s , w h i l e t h e i m m o t i l e s t r a i n m u l t i p l i e d by a f a c t o r o f 52. Again the growth k i n e t i c


12.

LAUFFENBURGER

Cell

Time (hr)

211

Motility

Time (hr)

Figure 2. Multiplication of aerotactic (O) and immotile (Φ) strains of Pseudo monas fluorescens in two different experiments, each in aerated mixed culture (left) and nonaerated mixed culture (right). Reproduced, with permission, from Ref. 18. Copyright 1969, Society for General Microbiology.



2.

0.15 2.91 3.00 2.94

0.15 0.38 0.55 0.69

0.19 0.35 0.46 0.48 0.66 1.41

0.16 3.27 3.48 3.61

0.16 0.37 0.71 1.84

typhimurium

Society

94 6,900 3,800 4,900

94 380 500 540

65 270 195 86 105 190

130 4,200 4,600 4,700

130 530 380 1,030

Challenged bacteria

f o r Microbiology.

0.00024 0.016 0.011 0.011

0.00024 0.00080 0.0016 4.5

0.00017 0.00080 4.8 74 185 450

0.00033 0.0012 0.0029 0.0093

0.00033 0.0053 165 880

Challenging bacteria

Viable Count (10 bacteria)/ml o f

1970, A m e r i c a n

0 6 24 48

Aerobic shaken broth

Copyright

0 6 24 48

0 6 24 48 72 96

0 6 24 48

Aerobic s t a t i c broth


Aeiobic shaken broth


Conditions o f Growth

Amt of Growth

Strains of Salmonella

Time o f Incubation hr 0 6 24 48

of P a i r s of Variant

f r o m R e f e r e n c e 19.

S6353, fim f l a

S6351, fim f l a

Reproduced w i t h p e r m i s s i o n

S6355, fim f l a

+

S6351, fim f l a

+

S6358, f i m f l a

S6352, fim f l a

Challenged (rha-)

i n Mixed C u l t u r e s

Challenging (rha+)

Growth

COMPETING STRAINS

Table

W

9

-J to

12.

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Cell

Motility

273

p r o p e r t i e s appeared n e a r l y the same. In a e r o b i c s t a t i c b r o t h , however, the a e r o t a c t i c s t r a i n m u l t i p l i e d by a f a c t o r o f 18,000 i n 48 hours w h i l e the immotile s t r a i n m u l t i p l i e d by a f a c t o r o f 6. Interp r e t a t i o n o f t h i s c a s e i s not s t r a i g h t f o r w a r d , t h o u g h , because p e l l i c l e f o r m a t i o n was noted a f t e r 24 h o u r s , c o m p l i c a t i n g the s i t u a t i o n . More r e c e n t l y , P i l g r a m and W i l l i a m s s t u d i e d a s l i g h t l y d i f f e r e n t c a s e — c o m p e t i t i o n between c h e m o t a c t i c P r o t e u s m i r a b i l i s and a nonchemotactic but randomly m o t i l e mutant o f the s p e c i e s (20) (see F i g u r e s 3 and 4 ) . In both pure and mixed c u l t u r e s o f p e r i o d i c a l l y a g i t a t e d amino a c i d b r o t h , the two s t r a i n s grew t o a 1:1 r a t i o a f t e r 14 h o u r s . On the o t h e r hand, i n both pure and mixed c u l t u r e s o f s e m i s o l i d agar the r a t i o o f c h e m o t a c t i c to randomly m o t i l e s t r a i n s was g r e a t e r than 5:1 a f t e r 14 h o u r s . The f i n a l e x p e r i m e n t a l r e p o r t s were by F r é t e r et_ al. (21, 22, 2 3 ) , who s t u d i e d growth o f V i b r i o c h o l e r a e i n mouse and r a b b i t l a r g e i n t e s t i n e (see Figure 5 and Table 3. Here t h r e e s t r a i n s were compared: the c h e m o t a c t i c w i l d t y p e , an immotile mutant s t r a i n , and a nonchemotactic but randomly m o t i l e mutant strain. In w e l l - s t i r r e d c o n t i n u o u s flow c u l t u r e , a l l t h r e e s t r a i n s grew i n p r o p o r t i o n . In the i n t e s t i n a l l o o p s , the nonchemotactic s t r a i n was r a p i d l y d i s p l a c e d by the c h e m o t a c t i c w i l d t y p e . Most i n t e r e s t i n g l y , i n another experiment the randomly m o t i l e s t r a i n was a l s o r a p i d l y d i s p l a c e d by the immotile s t r a i n . A p p a r e n t l y , i n t h i s s i t u a t i o n at l e a s t , m o t i l i t y w i t h o u t chemot a x i s was a l i a b i l i t y f o r the c e l l s . It i s e v i d e n t t h a t t h e o r e t i c a l a n a l y s i s o f m o t i l i t y and chemotaxis i s n e c e s s a r y i n o r d e r t o p r o v i d e q u a n t i t a t i v e i n t e r p r e t a t i o n o f these r e s u l t s , and even q u a l i t a t i v e e x p l a n a t i o n o f the l a s t , perhaps c o u n t e r - i n t u i t i v e , o b s e r v a t i o n by F r é t e r et al_. This w i l l be the c o n c e r n o f the next s e c t i o n o f t h i s p a p e r . But i s i m p o r t a n t to emphasize at t h i s p o i n t t h a t the e x p e r i m e n t a l r e s u l t s c i t e d here demonstrate t h a t the e f f e c t s o f c e l l movement p r o p e r t i e s can c l e a r l y be s i g n i f i c a n t , and even dominant, i n d e t e r m i n i n g the c o m p e t i t i v e a b i l i t i e s o f b a c t e r i a l p o p u l a t i o n s i n nonmixed e n v i r o n m e n t s . Theoretical

Analyses

E a r l y attempts at t h e o r e t i c a l a n a l y s i s o f the e f f e c t s o f c e l l m o t i l i t y on p o p u l a t i o n growth c e n t e r e d on uptake o f n u t r i e n t by a s i n g l e c e l l i n a medium o f i n f i n i t e e x t e n t (12, 24, 25, 2 6 ) . These a n a l y s e s have


274


6

J8

10

12

J H

Hours

6

8 10

12

H 2L

Hours Figure 3. Growth of pure cultures of chemotactic (O) and nonchemotactic ( O strains of Proteus mirabilis in periodically shaken amino acid broth (top) and soft-agar amino acid medium (bottom). Reproduced, with permission, from Ref. 20. Copyright 1976, National Research Council of Canada.


12.

LAUFFENBURGER

Cell

Motility

ο

I

1

ι

ι

ι

ι

ι

0

2

4

6

8

10

12

IK

ι

Η 24

Hours Figure 4. Growth of mixed cultures of chemotactic (O) and nonchemotactic ( Q ) strains of Proteus mirabilis in periodically shaken amino acid broth (top) and soft-agar amino acid medium (bottom). Reproduced, with permission, from Ref. 20. Copyright 1976, National Research Council of Canada.


BIOCHEMICAL

276

c

ENGINEERING

100 η

I 90 5 80 Ο

^ 70 ° ~ 60 5 2 50 6 30ζ
1 a r e m e r e l y extrapolations. N u m e r i c a l c o m p u t a t i o n s h a v e shown t h e p e r t u r b a t i o n r e s u l t s t o be a c c u r a t e up t o a t l e a s t 6 = 1.1, however ( 3 1 ) . An i n t e r e s t i n g i n f e r e n c e w h i c h c a n be d r a w n f r o m F i g u r e 12 i s t h a t t h e r e i s a minimum v a l u e o f 6 that m u s t be e x c e e d e d i n o r d e r f o r m o t i l i t y t o c o n f e r an advantage i n t h i s c o n f i n e d growth s i t u a t i o n . I f λ r e p r e s e n t s t h e B r o w n i a n m o t i o n c o e f f i c i e n t f o r an 2

1

χ


LAUFFENBURGER

Cell

Motility


283

284


Figure 9. Plot of dimensionless total steady state bacterial density vs. 8 for single chemotactic populations. Extrapolations of perturbation computations beyond 8 = 1 shown ( ). Asymptotic values for δ = 0 shown (- · - · -).


12.

LAUFFENBURGER

Figure 10.

Cell

Motility

285

Typical steady state profiles of dimensionless bacterial density, v.


286


i m m o t i l e s p e c i e s (_33) , t h e n a c h e m o t a c t i c s p e c i e s m u s t have a l a r g e r v a l u e , say λ > λι. By i t s e l f t h i s w o u l d y i e l d a s m a l l e r v a l u e o f B. Increasing 6 from 0 i n c r e a s e s B, and t h e r e w i l l be a c r i t i c a l v a l u e , 6*, a t w h i c h Β f o r λ2 and 6 b e c o m e s e q u a l t o Β f o r λ O n l y f o r 6>6* w i l l t h e c h e m o t a c t i c s t r a i n o u t g r o w t h e immotile s t r a i n . Thus, s p e c u l a t i o n t h a t a m o t i l e c h e m o t a c t i c s t r a i n s h o u l d a l w a y s be s u p e r i o r t o an immotile s t r a i n i s not n e c e s s a r i l y t r u e . T h e s e s i n g l e p o p u l a t i o n r e s u l t s s u g g e s t some i m p o r t a n t i m p l i c a t i o n s f o r c o m p e t i t i o n b e t w e e n two o r more p o p u l a t i o n s g r o w i n g t o g e t h e r o n a s i n g l e r a t e limiting nutrient. I f one s p e c i e s h a s s u p e r i o r g r o w t h k i n e t i c p r o p e r t i e s but t h e o t h e r has s u p e r i o r m o t i l i t y p r o p e r t i e s , we m i g h t e x p e c t c o e x i s t e n c e t o o c c u r . A n a l y s i s o f t h e t h i r d c a s e , c o m p e t i t i o n b e t w e e n two r a n d o m l y m o t i l e p o p u l a t i o n s , shows t h a t t h i s i s i n d e e d possible. T h e r e a r e now a c t u a l l y t h r e e p e r m i s s i b l e steady s t a t e s : 1) c o e x i s t e n c e , 2) s p e c i e s 1 o n l y , a n d 3) s p e c i e s 2 o n l y . For sake o f c l a r i t y , l e t the two s p e c i e s have i d e n t i c a l p r o p e r t i e s e x c e p t t h a t ki > k2« Then s p e c i e s 1 w i l l have a g r e a t e r growth rate at a l l n u t r i e n t concentrations. In a d d i t i o n , the t h r e s h o l d c o n c e n t r a t i o n f o r net growth o f s p e c i e s 2 must be g r e a t e r t h a n t h a t o f s p e c i e s 1; i . e . , c i * We c a n i m m e d i a t e l y see t h a t a n e c e s s a r y c o n d i t i o n f o r coexistence i s that ω * · The v a l u e s o f ω f o r e a c h s p e c i e s a r e d e t e r m i n e d by t h e same e q u a t i o n a s i n t h e s i n g l e p o p u l a t i o n c a s e , u n a f f e c t e d by t h e p r e s e n c e o f the other s p e c i e s . We c a n , t h e r e f o r e , move d i r e c t l y to a g r a p h i c a l d e s c r i p t i o n o f the steadys t a t e behavior f o r our c o m p e t i t i o n model. F i g u r e 11 shows i s o c l i n e s o f ω i n t h e p l a n e o f (κ,λ) v a l u e s . If we s p e c i f y v a l u e s λ χ and κι f o r p o p u l a t i o n 1, t h i s y i e l d s a value for ωι. We c a n t h e n immediately d i s c o v e r t h e p e r m i s s i b l e s t e a d y - s t a t e s f o r any s p e c i e s 2 with parameter values λ and κ (see Figure 1 2 ) . If κ < K o n l y s p e c i e s 1 can s u r v i v e , u n l e s s λ2 i s such t h a t o)2 > ω — which would a l l o w c o e x i s t e n c e . If κ > Κ χ , o n l y s p e c i e s 2 can s u r v i v e , u n l e s s λ is s u c h t h a t α)2 < ω ι , w h i c h a g a i n a l l o w s c o e x i s t e n c e . S o , t h e c o m p e t i t i o n o u t c o m e c a n be p r e d i c t e d f r o m t h e s i n g l e p o p u l a t i o n r e s u l t s , w i t h one m i n o r modification. Remember t h a t t h e ω c r i t e r i o n i s o n l y a necessary condition for coexistence. I t turns out t h a t a s e c o n d , s l i g h t l y more r e s t r i c t i v e c o n d i t i o n i s a l s o r e q u i r e d , t o e n s u r e t h a t t h e c e l l d e n s i t i e s and n u t r i e n t c o n c e n t r a t i o n r e m a i n p o s i t i v e e v e r y w h e r e (3JL) . The d i f f e r e n c e b e t w e e n t h e two c o n d i t i o n s i s s m a l l 2

1 #

u

>

c

>

ω

2

2

2

l

2

r

χ

2

2


2

u

LAUFFENBURGER

Figure 11.

Cell

Motility

Sample plot of curves of constant ω in plane of (Χ, λ).


288


Ω

10

!

!

e

I

10*

ΙΟ

1

COEXISTENCE

j

2

SPECIES 1 ONLY ι

! !

ι

/

/

ι

8

•

32

!

λ, =10"' •*•

Χ

• /

ΙΟ" 2

II I Ι

/

,ο

4

SPECIES 2 ONLY

ι

/

s

/

ι

-/ / -ι

I ι Ι

! COEXISTENCE j ίο j

J

6

I

K

1 1 =

12

Figure 12. Illustration of predicted results for competition between two randomly motile populations with identical properties except for Κ and λ.


12.

LAUFFENBURGER

Cell

Motility

289

for t y p i c a l parameter v a l u e s , however, so t h a t g i v e n the a p p r o x i m a t i o n i n v o l v e d i n the model i t s e l f , we n e e d n o t be t o o c o n c e r n e d w i t h i t . When t h e c e l l d e n s i t i e s a r e c o m p u t e d f o r a c o e x i s t e n c e s t a t e , t h e s p e c i e s w i t h s m a l l e r k c a n grow to a g r e a t e r p o p u l a t i o n s i z e than the s p e c i e s w i t h l a r g e r k, i f i t s m o t i l i t y p r o p e r t i e s a r e s u f f i c i e n t l y superior. T h i s g i v e s an e x p l a n a t i o n t o t h e r e s u l t c i t e d by S t a n i e r e t a l (1_7) . S i m i l a r r e s u l t s a r e e x p e c t e d when c h e m o t a x i s i s present i n competing p o p u l a t i o n s , although the a n a l y s i s has not y e t been c a r r i e d o u t . Chapman (5) f o r m u l a t e d a m o d e l f o r c o m p e t i t i o n o f two chemotactic s p e c i e s i n a t r a v e l i n g band, which p r e d i c t e d t h a t superior chemotaxis could allow a species with i n f e r i o r growth r a t e c o n s t a n t to exclude the other species. T h i s r e s u l t i s c o n s i s t e n t w i t h our e x p e c t a t i o n s ; the model, however, i s r a t h e r e m p i r i c a l . Conclusions R e v i e w o f t h e l i t e r a t u r e r e v e a l s a s m a l l number o f e x p e r i m e n t s t h a t show t h a t c e l l m o t i l i t y p r o p e r t i e s c a n h a v e d r a m a t i c e f f e c t s o n p o p u l a t i o n g r o w t h and c o m p e t i t i o n i n non-mixed systems. Simple mathematical models have been d e v e l o p e d w h i c h p r o v i d e q u a l i t a t i v e e x p l a n a t i o n f o r a l l t h e o b s e r v e d p h e n o m e n a , and yield q u a n t i t a t i v e p r e d i c t i o n of the magnitude of e f f e c t s w h i c h m i g h t be e x p e c t e d i n a v a r i e t y o f s i t u a t i o n s . Ac k nowledgme n t s T h i s work h a s b e e n p a r t i a l l y s u p p o r t e d by NSF C h e m i c a l and B i o c h e m i c a l P r o c e s s e s Program G r a n t CPE80-06701. D.A.L. w o u l d a l s o l i k e t o t h a n k P a t Thompson f o r t y p i n g t h i s m a n u s c r i p t and R e n a t e S c h u l t z f o r many o f t h e f i g u r e s .

L i t e r a t u r e Cited 1.

Breed, R. S.; Murray, E . D. G . ; Smith, N. R. "Bergey's Manual of Determinative Bacteriology"; Williams and w i l k i n s : Baltimore, 7th e d i t i o n , 1957.

2.

Koshland, Jr., D. E. "Bacterial Chemotaxis as a Model Behavioral System"; Raven Press: New York, 1980.


290


3.

Lauffenburger, D. A. "Effects of M o t i l i t y and Chemotaxis in C e l l Population Dynamical Systems"; PhD Thesis, University of Minnesota, 1979.

4·

Fraenkel, G. S.; Gunn, D. L . "The Orientation of Animals"; Dover: New York, 1961.

5.

Chapman, P. A. "Chemotaxis of Bacteria"; PhD Thesis, University of Minnesota, 1973.

6.

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7.

Macnab, R. M . ; Koshland, Jr., D. E . J. Mechanochem. C e l l M o t i l . 1973, 2, 141-148.

8.

Macnab, R. M. In "Biological Regulation and Development", v o l . 2; R. Goldberger, e d . ; Plenum Press: New York, 1980, pp. 377-412.

9.

Dahlquist,

J.

F . W.; E l w e l l , R. Α . ; Lovely, P. S.

Supramol. S t r u c t .

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10.

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11.

Macnab, R. M . ; Koshland, Jr., D. E . Proc. N a t l . Acad. S c i . USA 1972, 69, 2509-2512. Berg, H. C.; P u r c e l l , E . M. Biophys. J. 1977, 20, 193-219.

12. 13.

Hazelbauer, G. L.; Parkinson, J. S. In "Receptors and Recognition: Microbial Interactions"; J. R e i s s i g , e d . ; Chapman and Wall: London, 1977; pp. 60-80.

14.

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16.

Tsang, N . ; Macnab, R. Μ., Koshland, Science 1973, 181, 60-63.

Jr.,

Jr.,

D. E .

17.

Stanier, R . ; Adelberg, E.; Ingraham, J. "The Microbial World"; P r e n t i c e - H a l l : Englewood Cliffs, New Jersey, 1976. 18.

Smith, J. L.; Doetsch, 1969, 55, 379-391.

R. N. J. Gen. M i c r o b i o l .


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20.

Pilgram, W. R.; Williams, F. D. Can. J. M i c r o b i o l . 1976, 22, 1771-1773.

21.

F r e t e r , R.; A l l w e i s s , B . ; O'Brien, P. C. M . ; Halstead, S. A. In "Proceedings of the 13th Joint Conference on Cholera", U.S. - Japan Cooperative Medical Sciences Program; U.S. Govt. Printing O f f i c e , Washington, D . C . ; pp. 153-181.

22.

F r e t e r , R.; O'Brien, P. C. M . ; Halstead, S. A. Adv. Exp. Med. B i o l . 1978, 107, 429-437.

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F r e t e r , R.; O'Brien, P. C. M . ; Macsai, M. S. Am. J. C l i n . Nutr. 1979, 32, 128-132.

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Carlson, F . D. In "Spermatozoan M o t i l i t y " ; D. W. Bishop, ed.; AAAS Publication No. 72: Washington, D. C., 1962.

25.

Koch, A. L . Adv. Microb. P h y s i o l .

1971,

1970,

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147-217. 26.

Brunn, P. O. J. Biomech. Eng. 1981,

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27.

P u r c e l l , Ε. M. Am. J. Physics 1977,

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28.

Lauffenburger, D. Α . ; A r i s , R.; K e l l e r , Κ. H. Microb. E c o l . 1981, 7, 207-227. Lauffenburger, D. Α . ; A r i s , R.; K e l l e r , Κ. H. Biophys. J . (submitted for p u b l i c a t i o n ) .

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RECEIVED

June 1, 1982


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