Microb Ecol (1987) 13:31--45

MICROBIAL ECOLOGY @Springer-VerlagNew York Inc. 1987

A Mathematical Model for the Growth of Bacterial Microcolonies on Marine Sediment A. Michael Davidson and John C. Fry* Departments of Mathematics and *Applied Biology,University of Wales Institute of Scienceand Technology,P.O. Box 13, CardiffCF1 3XF, United Kingdom

Abstract. Counts of bacterial microcolonies attached to deep-sea sediment particles showed 4-, 8-, 16-, a n d 32-celled microcolonies to be very rare. This was investigated with a mathematical model in which microcolonies grew from single cells at a constant growth rate (g), detached from particles at constant rate (h), and reattached as single cells. Terms for attachment o f foreign bacteria (a) and death of single cells (d) were also included. The best method of fitting the model to the microcolony counts was a weighted least-squares approach by which ),(0.83 hour -l) was estimated to be about 20 times greater than ~t(0.038 hour-'). This showed that the bacteria were very mobile between sediment particles and this mobility was explained in terms o f attachment by reversible sorption. The implications o f the results for the frequency of dividing cell method for estimating growth rates o f sediment bacteria are discussed. The ratio o f ~ and/z was found to be very robust both in terms o f the errors associated with the microcolony counts and the range o f microcolony sizes used to obtain the solution.

Introduction Sediments support large populations o f bacteria [21], most o f which seem attached to the surface of sediment particles [22]. This is true for both static sediments and those washed into the water column in tidal estuaries [8, 9]. Thus any mathematical model describing the growth o f bacteria in sediment should be based on surface growth equations. Attached bacteria will grow and form microcolonies on surfaces, and a model describing the growth o f such microcolonies has recently been published [I 5]. This model described the growth o f 1-, 2-, 4-, 8-, and 16-celled microcolonies. It considered simultaneous and continuous growth and attachment o f microorganisms, and assumed that the growth and attachment rates were constant. The model predicted that numbers of all sizes o f microcolonies would eventually reach the same population density on the surface. If this idea is carried to its logical conclusion for the larger sizes o f microcolonies, it is clear that the model predicts a continuous covering o f bacteria. This is plainly not the case for sediment particles. They are never completely covered by bacteria and they

32

A.M. Davidson and J. C. Fry

normally have only small numbers of bacteria growing on them, although occasionally more densely covered particles are observed. Two-celled microc o l o n i e s h a v e b e e n r e p o r t e d in m a r i n e s e d i m e n t s d u r i n g e x p e r i m e n t s t o e s t i m a t e g r o w t h r a t e s b y t h e f r e q u e n c y o f d i v i d i n g cell ( F D C ) t e c h n i q u e [2, 24]. L a r g e r m i c r o c o l o n i e s c o n t a i n i n g u p t o 100 cells h a v e b e e n o b s e r v e d o n s a n d g r a i n s [19] o n w h i c h 5- t o 2 0 - c e l l e d m i c r o c o l o n i e s a r e c o m m o n b u t l a r g e r m i c r o c o l o n i e s a r e r a r e [29]. T h u s , as b a c t e r i a g r o w in s e d i m e n t t h e y m u s t detach from the particles to prevent a continuous covering of microcolonies. A model describing microcolony growth on sediment should therefore contain t e r m s for d e t a c h m e n t , l i k e t h o s e u s e d t o c a l c u l a t e t h e g r o w t h r a t e o f e p i p h y t i c b a c t e r i a in t h e R i v e r F r o m e , U n i t e d K i n g d o m [12]. M o s t s e d i m e n t s a r e r e l a t i v e l y s t a t i c a n d t h e b a c t e r i a in t h e o v e r l y i n g w a t e r a r e far less n u m e r o u s t h a n t h o s e i n t h e s e d i m e n t , so it s e e m s s e n s i b l e t h a t m o s t d e t a c h i n g b a c t e r i a w i l l r e a t t a c h to o t h e r s e d i m e n t p a r t i c l e s , a l t h o u g h a s m a l l i n p u t f r o m t h e s u r f a c e w a t e r is likely. B a c t e r i a d o n o t l i v e i n d e f i n i t e l y , g r o w t h r a t e s in s e d i m e n t s a r e p r o b a b l y l o w [2, 24], a n d b a c t e r i a g r o w i n g s l o w l y in p u r e c u l t u r e h a v e l o w v i a b i l i t y [27]. C o n s e q u e n t l y a d e a t h t e r m s h o u l d a l s o b e i n c l u d e d i n a s e d i m e n t m i c r o c o l o n y g r o w t h m o d e l . I d e a l l y , a p r e d a t i o n t e r m s h o u l d b e i n c l u d e d as well. H o w e v e r , v e r y little is k n o w n a b o u t t h e r e m o v a l o f s e d i m e n t b a c t e r i a b y t h e p r e d a t i o n o f e i t h e r p r o t o z o a o r l a r g e r i n v e r t e b r a t e s [6]. It is t h e r e f o r e p r o b a b l y b e s t t o o m i t a specific p r e d a t i o n t e r m a n d a s s u m e it is i n c o r p o r a t e d into the death term, which thus represents death by predation and other means. L i t t l e is k n o w n o f t h e p o p u l a t i o n d y n a m i c s o f s e d i m e n t b a c t e r i a , so it w a s t h e a i m o f t h i s p a p e r to c o n s t r u c t a m a t h e m a t i c a l m o d e l t o d e s c r i b e t h e g r o w t h o f m i c r o c o l o n i e s a t t a c h e d to s e d i m e n t p a r t i c l e s u s i n g t h e i d e a s d i s c u s s e d a b o v e . I t w a s a l s o i n t e n d e d to t e s t t h e u s e f u l n e s s o f t h e m o d e l b y a p p l y i n g it t o c o u n t s of microcolonies from a deep-sea sediment.

Methods

Sampling and Enumeration of Sediment Bacteria Sediment samples were taken from 2,010 m in the Porcupine Seabight in the northeast Atlantic Ocean southwest of Ireland (51"0.3'N; 12~ on August 13, 1984 with a multiple corer [ 1]. After sampling, the top 1-2 mm of sediment was removed by suction and preserved in 2% (w/v) formaldehyde. Preserved sediment (25 #1) was stained for 3 rain in 15-ml membrane filtered 2% (w/v) formaldehyde containing acridine orange (5 mg liter-~) and filtered onto a 25-mm polycarbonate membrane filter (0.2 #m), which had been prestained for 3 min in 2 g liter-~ irgalan black in 2% (v/v) acetic acid. The filter was mounted in a minimum of sterile paraffin oil under a coverslip and viewed by epifluorescence microscopy [7]. This procedure avoided homogenization, which would disturb the bacteria attached to sediment panicles. The sediment particles and bacteria were evenly spread on the membrane filter, and when viewed, about 50% of the membrane was occupied by a thin layer of particles. Bacteria were counted as brightly green-fluorescing bacterial-shaped objects with a sharp outline. About 400 single bacteria were counted on each membrane filter in randomly selected squares of a 10 • 10 eyepiece graticule. Two-celled microcolonies were defined as either two closely spaced cells of identical morphology or as cells showing a clear invagination; larger microcolonies were defined similarly but for 4, 8, 16, or 32 identical cells. Intermediately sized microcolonies were rarely observed, and when they were, inspection of the shape of the individual

Model for Microcolony Growth

on

Sediment

33

mxteraal at tachmeat

.~yh~ §

2..tt.oh..,c

-

6 I

~ I

I

i

milratloB

Fig. 1. Overall diagrammatic representation of the model describing the growth of bacterial microcolonies on sediment particles, c, is the number of microcolonies containing 2 n cells, thus co = number of individual cells, c~ = number of 2-celled microcolonies, and so on.

cells enabled them to be classified as 2, 4, 8, 16, or 32 cells, according to the size they most closely resembled. Microcolonies were counted in circular areas o f membrane filter (70-102 #m diameter) defined by the iris diaphragm on the epifluorescence illuminator. Counting was continued until about 65 two-celled microcolonies had been seen. In this sediment using these methods about 97% of the bacteria appeared on the sediment particles. Thus, for the calculation of the total counts, equal numbers of bacteria were assumed to be visible on the surface and invisible behind the opaque sediment particles [8]. Counts of single cells were made with a 100 x planachromat objective lens (numerical aperture, NA = 1.25) and 8 x eyepieces (type = KPI). Microcolonies were counted with a 63 x planapochromat objective lens (NA = 1.4) and 16 x eyepieces (KPI). The growth rate of the sediment bacteria was estimated with the FDC technique and formula derived for oceanic water [ 10].

Description of the Model The number ofmicrocolonies containing 2" cells at time t will be denoted by c,(t). Thus the number of single-celled microcoionies is co, the number of 2-celled microcolonies el, and so on (Table 1). The mechanisms producing change in the c. to be included in the model are growth, death, detachment and, for the single-celled microcolonies, external attachment and the reattachment o f ceils lost from larger microcolonies by detachment. The model is assumed to be a linear one, with constant rates for growth, death, and detachment. The overall structure of the model is shown in Fig. 1. The equation governing the change in Cn with time is of the form:

Rate of growth growth change = from - into - death - detachment of

Cn

Ca - I

Ca+ I

34

A . M . D a v i d s o n a n d J. C. Fry

to

a

c I

x (2-ceii|)

c2 (4-celll)

(3-ceH.)

xr a

~c~

-"-to

(4-cellm)

co

{l-cell)

b

to r

X

2n~_t c n

ur n - t

2-"~'_tc.

2" cn 4

~eg

-c n

2B

-*~to c o

e~§

from e n , ~

Fig. 2. D i a g r a m s to illustrate details o f the c o n s t r u c t i o n o f the m o d e l describing t h e growth o f bacterial microcolonies on s e d i m e n t particles: (a) d e t a c h m e n t o f single cells f r o m a 4-celled m i c r o c o l o n y (c2) a n d (b) the c o m p l e t e inputs a n d o u t p u t s for a cn microcolony. W h e r e c. is the n u m b e r o f microcolonies containing 2" cells, X is the d e t a c h m e n t rate o f single cells (hour-t), d is t h e d e a t h rate (cell hour-t), a n d ~ is the specific growth rate (hour-O.

T h e form o f the 4 t e r m s on the f i g h t - h a n d side can n o w be considered:

Growth. A c o n s t a n t growth rate ~ h o u r -~ is a s s u m e d so that the growth t e r m s are ~c.-1 a n d - ~ c . . Death. A c o n s t a n t d e a t h rate d cell h o u r ~ is a s s u m e d . T h i s gives a t e r m - d c 0 in the co equation, % cl in the cl equation and, in general, -~

co in the c. equation.

Detachment. A c o n s t a n t d e t a c h m e n t rate X hour-* is a s s u m e d . A c. m i c r o c o l o n y that loses one cell by d e t a c h m e n t separates into the single cell, which is counted as a co a n d a unit o f 2" - 1 cells. T h i s second unit will be classified in the c o u n t i n g process a n d it is a s s u m e d that it will be c o u n t e d as a c., which it m o s t closely resembles, or as a c._ 1. F o r example, a 4-celled m i c r o c o l o n y after d e t a c h m e n t leaves a 3-celled unit that could be c o u n t e d either as a 4-celled or as a 2-celled m i c r o c o l o n y a n d it is a s s u m e d , for m a t h e m a t i c a l c o n v e n i e n c e , that these possibilities are equally likely. T h e overall effect is s h o w n in Fig. 2a. M i g r a t i o n proceeds at rate hc2 to Co by d e t a c h m e n t . T h e 3-celled units divide equally into cl a n d c2, so there is a m i g r a t i o n at rate ~ c2 to cl a n d ~ c2 back to c2, giving a net loss at rate ~ c2 f r o m cv In the general case o f c . , the 2" - 1 celled unit will m o r e closely resemble a c. t h a n a c. 1, a n d generalizing the a s s u m p t i o n for n = 2, the probabilities o f the 2" - 1 celled unit being counted as a cn_, or a c., are taken as 1/2. ~ a n d 1 - '/2. ~, respectively. T h e net loss to c. will therefore proceed at a rate o f

(l A s s e m b l i n g all t e r m s together, the gai n a n d loss in cn will be as s h o w n in Fig. 2b. Note that there is a feedback c o n t r i b u t i o n to c. d u e to d e t a c h m e n t o f single cells from c.+1. T h e rate o f change equation for c. can n o w be written as the differential e q u a t i o n

Model for Mierocolony G r o w t h on Sediment

35

The first and last equations (i.e., for Co and cN) m u s t be considered separately. The equation for eo contains terms from the feedback o f single cells and also from the external a t t a c h m e n t term, which is a s s u m e d to be a constant rate o f cell g-~ h o u r L F r o m Fig. 1 the equation is seen to be d ~Co=-dco-

/~co+ kc~ + k [ c t + cz + . . . c r ~ ] + a

where microcolonies o f size up to 2 N are being included. In reality, single cells on sediment particles should detach and reattach to other s e d i m e n t particles at the same rate as individual ceils in microcolonies. However, -Xco for d e t a c h m e n t a n d +he0 for reattachment cancel each other out, and so are not needed in this equation or in the m o d e l diagram (Fig. 1). For cN the d e t a c h m e n t feedback term is omitted, giving c~=~c~_,-

+ ~ + ~

c~

The mathematical m o d e l has thus been written as a set o f N + 1 linear first-order differential equations in the variables co(t), e~(O. . . . . cN(t). Introducing the vector C(t) o f microcolony populations

F (d

Lc oj the model can be written in matrix form d ~C=BC+A dt where A is the column matrix

and B is the (N + 1) x (N + 1) matrix --[d+a]

2X

-

0

+,.+X

t.,

-

,k

h

. . . X

~

0 ...

h

0

0

+.+

B=

0

0

0

;~

2 N- i

. . .

-

-

+/~+

36

A . M . Davidson and J. C. Fry

The behavior of such a set of equations is easily analyzed mathematically [14]. Provided that the system is stable (which requires the eigenvalues of B to have negative real parts), the value of C(0 will, as t increases, approach asymptotically the equilibrium solutiom Co given by BCo+ A = 0 or

BCo --- - A The rate of approach to this solution will be governed by the size of the eigenvalues of B. Thus, for a given set of u, k, a, and d, it is possible to calculate the equilibrium sizes of microcolonies given by the elements of Co.

F i t t i n g t h e M o d e l to t h e D a t a The model developed above enables the vector of equilibrium populations Co to be calculated for any given set of k, a, and d, with u regarded as a known constant. The values of k, a, and d that gave the best fit to the vector of observed populations of microcolonies C were calculated. The goodness of fit was measured in the following ways:

1. Least squares (LS). Minimization of the error criterion

~

[C.o - e.l 2

a-o

where c.o is the calculated equilibrium value of c. and 4. is the observed value of c..,

2. Least squares of ratio (LSR). Minimization of

.-o k

c.o

1

3. Least squares of logarithm (LSL). Minimization of

n-0

n-0

4. Weighted least squares (I/VLS). Minimization o f ~ adCoo - e,l 2 n-O

with the weights an suitably chosen. LS and LSR are special cases o f WLS, with a~ = l and an = V~.02,respectively. LS is appropriate if the absolute errors are evenly distributed, while LSR and LSL are appropriate if the relative errors are evenly distributed. WLS was used with the an chosen to take account of the observed variation in experimental errors with n and is explained in detail later. The model was fitted to the data by minimization of the nonlinear error criterion with respect to the three parameters X, a, and d. The calculations were done using the NAG (Numerical Algorithms Group) routine EO4CCF [5] on a VAX 11/780 computer.

Results Counts and Estimation of Growth Rate The numbers of single cells and microcolonies present in the sediment are s h o w n i n T a b l e 1, f r o m w h i c h it is c l e a r t h a t l a r g e m i c r o c o l o n i e s w e r e r a r e l y

Model for Microcolony Growth on Sediment

37

Table 1. Numbers of microcolonies of different sizes

counted on deep-sea sediment particles ~

Cnh

co c~ c2 c3 c4 c5

Number of cells Number in micro- actually colonies counted 1 2 4 8 16 32

2,833 377 10 8 2 1

Number (x 107 g ~; dry weigh0 in sediment _ 95% confidence intervals 2,194 89.5 2.2 1.9 0.6 0.2

+ 287 + 24.5 + 1.8 --+ 1.6 - 0.9 --+ 0.5

Values are means of counts from 7 replicate membrane filters Notation used for number of cells in microcolonies where c, is the number of microcoionies containing 2" cells seen. T h e m e a n F D C a n d its 95% c o n f i d e n c e i n t e r v a l for this s e d i m e n t was 5.82 +__ 0.68%, w h i c h c o r r e s p o n d s to a specific g r o w t h rate (/z) o f 0 . 0 3 8 h o u r -1, or a g e n e r a t i o n t i m e o f 18 h o u r .

Qualitative Features of the Model Before p r o c e e d i n g to c o n s i d e r the actual fitting p r o c e d u r e there are t w o p o i n t s w h i c h e m e r g e f r o m the m a t h e m a t i c a l f o r m o f the m o d e l . A l t h o u g h the m o d e l c o n t a i n s f o u r p a r a m e t e r s , the e q u i l i b r i u m v a l u e s Co will d e p e n d o n l y o n the ratios o f these p a r a m e t e r s , thus r e d u c i n g the n u m b e r o f i n d e p e n d e n t p a r a m e t e r s to 3. I n the calculations d o n e , t* was fixed, a n d h, a, a n d d calculated for the best fit. T h e m a g n i t u d e o f / ~ (and h e n c e the o t h e r p a r a m e t e r s ) will affect the eigenvalues o f B a n d h e n c e the rate o f a p p r o a c h to equilibrium. F r o m the f o r m o f the m a t r i x B it can be s h o w n t h a t as n increases the r a t i o o f the e q u i l i b r i u m values cn_ ~/cn t e n d s to 1. It can also be s h o w n t h a t cn-~ > c,, so the e x p e c t e d f o r m for c, is for the ratios b e t w e e n successive cn to decrease as n increases. T h i s is in q u a l i t a t i v e a g r e e m e n t with the e x p e r i m e n t a l d a t a (Table 1).

Comparison of the Use of Different Error Criteria T h e results o b t a i n e d using the different criteria (LS, L S R , LSL) a n d a fixed set o f starting values Q, = 1.0 h o u r -1, d = 0.1 cell h o u r - l ; a = 50.0 x 107 cell g-t h o u r -t ) are given in T a b l e 2. It can be seen t h a t all m e t h o d s give b r o a d l y s i m i l a r results f o r ~, d, a n d a, a n d also give g o o d fits to the o b s e r v e d n u m b e r s o f m i c r o c o l o n i e s . All these s o l u t i o n s c o r r e s p o n d to stable m o d e l s , as can be seen by calculating the eigenvalues o f the c o r r e s p o n d i n g B matrices. I n e a c h case

38

A.M. Davidson and J. C. Fry

Table 2. Microcolony growth model solutions for deep-sea sediment using various error criteria~

c,b

Observed number of microcoionies (x i0 Tg-~; dry weight)

Co c~ c2 c3 c4 cs

2,194.0 89.5 2.2 1.9 0.6 0.2

Calculated values c (hour-*) d (cell hour -*) a (cell g-* hour-t; x 107) Minimum value of error criterion

LS 2,194.0 88.7 7.2 1.1 0.3 0.1 0.904 0.068

Calculated values ofc~ using LSR LSL 2,035.3 96.4 9.1 1.7 0.5 0.2

1,557.9 75.4 7.3 1.4 0.4 0.2

0.764 0.073

WLS 2,155.0 95.1 8.4 1.4 0.4 0.2

0.745 0.076

0.827 0.064

63.84

69.68

58.48

53.55

25.81

0.64

1.77

20.77

- LS, least squares; LSR, least squares of ratio; LSL, least squares of logarithm; and WLS, weighted least squares See Table l "k, detachment rate; d, death rate of single cells; a, external attachment rate

the v a l u e o f the m i n i m u m e r r o r c r i t e r i o n is a l m o s t e n t i r e l y a c c o u n t e d for b y the e r r o r i n c2, w h i c h is o u t o f l i n e w i t h the o t h e r v a l u e s . It c a n easily be seen t h a t LS a t t a c h e s e q u a l w e i g h t i n g to a b s o l u t e differences a n d is t h e r e f o r e n o t a p p r o p r i a t e for s u c h a w i d e r a n g e o f m i c r o c o l o n y c o u n t s , w h i l e L S L ( a n d to a lesser e x t e n t L S R ) w e i g h t the fitting o f the s m a l l e r , p o o r l y k n o w n c o u n t s . W L S is t h e r e f o r e suggested as a c o m p r o m i s e w i t h the w e i g h t s c a l c u l a t e d as follows. F r o m the 95% c o n f i d e n c e i n t e r v a l s i n T a b l e 1 it c a n b e seen t h a t the r e l a t i v e e r r o r i n the o b s e r v a t i o n s i n c r e a s e s b y a factor o f a b o u t 2 w i t h each i n c r e a s e i n n. T h u s the r e l a t i v e e r r o r i n co is 13.1%, t h a t i n cl is 27.4%, a n d so o n u p to c5 w h e r e t h e r e l a t i v e e r r o r is 250%. It was felt a p p r o p r i a t e to c h o o s e w e i g h t i n g factors i n the fitting c r i t e r i o n to reflect this b y a t t a c h i n g m o r e i m p o r t a n c e to t h e larger, m o r e a c c u r a t e l y k n o w n , v a l u e s o f cn. T h i s has b e e n d o n e b y t a k i n g t h e r a t i o o f the e r r o r i n c5 to t h a t i n Co (19.1) a n d c h o o s i n g c o n s t a n t s bn i n a g e o m e t r i c p r o g r e s s i o n w i t h the r a t i o o f b0 to b5 (19.1) 2. T h i s is a c h i e v e d b y setting b5 = 1, b4 = b, b 3 = b 2 . . . b0 = b 5 with b = (19.1) 2:s = 3.25 T h e w e i g h t i n g factors an i n t h e e r r o r c r i t e r i o n are n o w t a k e n as an=b./c.o 2

Model for Microcolony Growth on Sediment

39

This criterion n o w gives equal weighting to a 13.1% error in co and a 250% error in c5. The results with WLS are shown in the final c o l u m n in Table 2. It can be seen that as expected the value o f X does indeed lie between the values obtained with LS and LSL. WLS is consequently the m o s t appropriate m e t h o d for solving the m o d e l and is used in all the subsequent calculations.

Robustness of the Model Solutions In an a t t e m p t to d e t e r m i n e the sensitivity o f the results to errors in the data, the calculations were repeated with cn at the limits o f their possible ranges. T h e results show that the values o f a and d v a r y little, but values for h vary from about 0.6 to 1.2 h o u r -1 (Table 3).

Truncation Effects In the calculations the model has been used with N = 5 since data was available for up to 32-celled microcolonies. Although larger units were not observed, there is no reason why they should not be included in the model. T o investigate this effect, the model was used with N = 9, 14, 19, but using results only up to c5 in the error criterion. T h e results are shown in Table 4 and it is easily seen that the truncation effect is negligible. It is therefore entirely acceptable to use the N = 5 model which involves solving smaller sets o f equations. As an extension o f this, the model was then used with N = 4, 3, 2, 1--this time using only the calculated co in the error criterion. T h e results (Table 4) are in remarkably good agreement with the larger model. In particular, a m o d e l containing only co and ct is very successful.

Existence of Multiple Solutions It is possible for the optimization procedure to converge to other solutions, with the particular solution found depending on the starting values used in the optimization. T h e results given a b o v e used a particular set o f starting values. Equally good fits, in the sense o f similar values o f the error criterion, can be found and in Table 5 some are given. Although these solutions have very different values o f d and a, it can be seen that the value o f X changes very little. This is to be expected from the m a t h ematical f o r m o f the model. I f the first equation is r e m o v e d from the model, the remaining N equations in the N + 1 u n k n o w n s Co, c~. . . . . cN d e t e r m i n e the ratios o f the equilibrium populations and, from the form o f the matrix B, it is the value o f ~ that principally determines the solution. T h e value o f d is a factor, but o f m u c h less importance. Thus h is essentially d e t e r m i n e d by these equations. The remaining first equation then calculates the values o f a to give the correct value for Co. This can be d e m o n s t r a t e d by rewriting the first equation as - ( d + ~)Co + 2hot + (cz + c3 + . . . + CN) = --a

40

A . M . Davidson and J. C. Fly

Table 3.

Model solutions by the weighted least-squares approach using different values for number of microcolonies Calculated values using values for c, b at the limits of their 95% confidence intervals obtained experimentally (Table 1) by setting All at their All at their lower limit upper limit or zero High/low Low/high (high) (low) alternately alternately

Value a calculated and units h (hour -~) d(cells hour -~) a(cellsg-thour~; • )

0.638 0.064

1.23 0.067

59.48

0.781 0.062

56.64

0.587 0.069

55.22

56.84

a See Table 2 h See Table 1 Table 4.

Model solutions by the weighted least-squares approach using different numbers of equations Calculated value ~ of

Nb

(hour -j )

d (cell hour -~)

a (cell g-I hour-l; • l0 T)

19 14 9 5 4 3 2 1

0.833 0.833 0.834 0.827 0.810 0.815 1.039 0.865

0.0651 0.0649 0.0646 0.0643 0.0649 0.0645 0.0630 0.0570

54.95 54.97 54.59 54.58 55.85 55.77 56.22 53.44

a See Table 2 b The size of the largest microcolonies used in the models were 2N and the models contain N + 1 equations and, .defining the constant e by e = (#co -

2Xct -

Xc2...

-XcN)

the equation reduces to a

= cod + e

T h i s s u g g e s t s t h a t s o l u t i o n s w i l l e x i s t f o r all v a l u e s o f a a n d d l y i n g o n t h e straight line described by this equation. However, the ratio structure does d e p e n d w e a k l y o n d, s o s o l u t i o n s w i l l o n l y o c c u r at s o m e p o i n t s a l o n g t h i s l i n e . F o r t h e v a l u e s u s e d a b o v e , e h a s t h e v a l u e - 8 6 . 6 a n d t h e v a l u e o f Co i n t h e W L S s o l u t i o n is 2 , 1 5 5 , w h i c h is i d e n t i c a l t o t h e v a l u e i n T a b l e 1. T h e l i n e is therefore a = 2155d

- 86.6

Model for Microcolony Growth on Sediment

41

T h i s l i n e is s h o w n in Fig. 3 a n d a l s o t h e s o l u t i o n s in T a b l e 4 w h i c h c a n b e all seen to lie n e a r t h i s line. A l l t h e s e s o l u t i o n s h a v e s i m i l a r v a l u e s o f t h e e r r o r c r i t e r i o n , so a r e e q u a l l y g o o d m a t h e m a t i c a l l y , p r o v i d e d t h a t d is l a r g e e n o u g h t o e n s u r e s t a b i l i t y . T h i s is t r u e i n all c a s e s f o u n d a b o v e . A l l t h e s e s o l u t i o n s will t h e r e f o r e p r o d u c e t h e s a m e set o f e q u i l i b r i u m p o p u l a t i o n s . T h e y will, h o w e v e r , d i f f e r in t h e i r r a t e s o f a p p r o a c h to e q u i l i b r i u m , w i t h l a r g e r v a l u e s o l d p r o d u c i n g a f a s t e r a p p r o a c h .

Discussion T h e m o s t i m p o r t a n t f a c t t h a t e m e r g e s f r o m t h e r e s u l t s is t h a t t h e e s t i m a t e d d e t a c h m e n t r a t e (X) w a s h i g h (ca 0 . 8 3 h o u r -1) c o m p a r e d w i t h t h e specific g r o w t h rate (/~; 0 . 0 3 8 h o u r - J ) ; t h u s ?~ w a s n e a r l y 22 t i m e s h i g h e r t h a n #. T h e c a l c u l a t i o n

Table S. Some stable model solutions using weighted least squares obtained with different starting values of h, d, and aa Calculated value of a

(hour -t )

d (cell hour -l)

(cell g-i hour-~; x 107)

0.840 0.840 0.837 0.834 0.827 0.814

0.0406 0.0413 0.0464 0.0514 0.0643 0.0871

0.94 2.56 13.94 25.40 54.59 105.98

~ See Table 2

1:20

"7 4= "7 D}

80

Fig. 3. The relationship between the external attachment rate (a) and the death rate of single cells (d) for multiple stable solutions to the microcolony growth model. The continuous line represents the theoretical relationship about which some actual values are plotted (0).

(b 40

o d

0

0.06

0.03

d,

cell

h -1

0.09

42

A.M. Davidson and J. C. Fry

of/~ from FDC data is likely to give an overestimate because bacterial productivities in sediment seem unreasonably high when FDC is used [2, 24]. However using our model,/~ and • will always vary in a similar proportion; thus, if # is in error the absolute value o f X might be wrong, but not the ratio of/~ and X. So we can be reasonably certain that given our data, ~ will be about 20 times higher than/~. Unfortunately, we have no better way o f estimating ~t at present, so the best estimate o f X must remain at about 0.83 hour -1. This means that sediment bacteria are a very mobile population and the average time spent on a sediment particle will be 1/h or about 1.2 hour. This contrasts greatly with the results obtained for bacteria epiphytic on the freshwater plant L e m n a sp. [12] where the experimentally determined detachment rate (0.0031 hour -I) was lower than the growth rate (0.0049 hour -~) and the population was basically static (1/X = 322 hour). There is some supporting evidence for active motility at sediment surfaces. For example, one copiotrophic marine Vibrio sp. was motile for longer and grew more quickly under low-nutrient conditions at a solid-water interface than it did in the liquid phase [16]. It is clear that the data fit the model reasonably well. The main source of error using the weighted least-squares method, which was best for this data, was in the value for 4-celled microcolonies. There are two possible reasons for this large error. First, the data might be collected with too much error for all sizes of microcolonies to be predicted accurately. This could well be the case as very few microcolonies o f 4-cells or larger were counted. Second, the assumption that bacteria in all microcolonies grow and detach at the same rate might be wrong--~t and ), might be different for different sizes of microcolonies. This is also possible as organic nutrients are known to adsorb to surfaces [3], and hence detachment might be reduced by chemotaxic attraction [26]. Also, once attached to a sediment particle rich in nutrients growth might be increased, so larger microcolonies could have higher growth rates. Although the microcolony counts for larger microcolonies were not very accurate, it is unlikely that either X, or the ratio of), and #, are radically wrong. There are several reasons for this. The value for X was fairly stable witla the different methods of calculation and with slight variations in the numbers of different sizes of microcolonies. The ratio of X and # only varies between 15.5 and 32.4 for the most extreme values calculated (Table 3). Only an increase in the numbers o f larger microcolonies would cause ~ to decrease. For example, if k and # were both 0.038 hour -~ and a = 100 x 107 cell g-~ hour -~, the predicted number of 1-32-celled microcolonies would be 2,427, 972, 569,424, 363, and 327 x 107 g - i respectively. Thus in all sediments with small numbers o f large-celled microcolonies the model we have used predicts that ~ is large compared with tz. The large value of ~ predicted here fits in well with the idea o f reversible sorption proposed by Marshall, Stout, and Mitchell [18]. This type of attachment results in bacteria being held 5-10 nm away from a surface [28]. It is a likely mechanism for temporary attachment in marine sediments for many reasons. Reversible sorption is enhanced in seawater [ 18] and it occurs quickly when bacteria approach a surface, but it is weak enough to allow the organisms to swim away by flagellar action. Studies with interference reflection microscopy have suggested that slow-growing bacteria at 4~ may attach weakly to surfaces

Model for MicrocolonyGrowth on Sediment

43

[4], and such a low temperature is similar to the deep-sea environment. Also, large numbers of indigenous bacteria from a seawater lagoon sediment could be removed by gentle washing [25]. Pure cultures of a Vibrio sp., which was incapable of irreversible sorption, was able to take up nutrients bound to surfaces [11]. Chemotaxis might encourage movement between sediment particles because flagellar action uses considerable amounts of energy in slowgrowing organisms in nutrient-poor environments and marine pseudomonads have very low thresholds for chemotactic responses [28]. Reversible sorption of aquatic bacteria might be a general phenomenon in aquatic habitats. The rapid attachment and loss o f bacteria from marine copepod fecal pellets [13] could be explicable in these terms. Also, the unrealistically high populations of bacteria predicted to be attached to particles in m0del simulations of a salt-marsh estuary ecosystem [30] could be due to the authors' failure to recognize the role of reversible sorption in their model. They even mention this as a possibility in their discussion. Some sediment bacteria are more firmly fixed because in some enumeration studies homogenizing and sonication have been required to remove the majority of the bacteria [20,24]. It might be that bacteria finding nutrient-rich sediment particles would become irreversibly sorbed by adhesive polymer production [4, 18]. This took several hours to occur but was enhanced by 7 mg liter -2 glucose [18]. There is, however, no definitive evidence yet for this type of attachment in finely divided sediment [17] like the one examined in the present work. The results show that many solutions to the model are possible with different values for external attachment and death, but that a and d always increase or decrease together. This is biologically logical, as if sediment bacteria constantly grow then, even if migrating between particles, they must die or leave the sediment to prevent overpopulation. Thus d really represents the sum of death and emigration completely away from the sediment, and at any given u the values of d and a will be dependent on each other. As yet no estimates of a or d can be made for sediment bacteria. However, deep-sea sediment is relatively stable and the overlying water must contain relatively few bacteria, so the entry of foreign bacteria into the sediment will be rare. This means that the most sensible solution for the model is with the minimum, stable value for a and hence d, which will give a slow rate of approach to the equilibrium solution. This implies that changes in populations of microcolonies in sediments will be slow. Although we have only used the model to calculate the equilibrium solutions, the model is dynamic and so could be used to investigate changes in numbers of different sizes of microcolonies with time. This work has several implications for the use of the FDC method for estimating growth rates of sediment bacteria. The technique might overestimate growth rates [2, 24] because it has been implied [23] that firmly attached bacteria remain as 2-celled microcolonies for longer than they would in water, from which the FDC prediction equations are derived. However, as the estimated value of~ is large and as it has been argued earlier that most sediment bacteria are mobile, it is unlikely that 2-celled microcolonies remain together for much longer than they do in water. It has also been found that daughter cells of some attached bacteria swim from the surface immediately after cell division [ 16].

44

A . M . Davidson and J. C. Fry

So the basic formulae used for water must be inapplicable in sediment for some other reason. Our model predicts that/z is dependent on both ~ and the microcolony ratios. Thus # could never be calculated solely from the microcolony ratios (as in the FDC method) if ~, varied either temporally or spacially in sediments. Thus the FDC method might never give good estimates of/~ in sediments. We believe this paper raises several new issues about the dynamics of sediment bacteria. The model suggests many further routes for research, some o f which have been discussed. In particular, there is a need for further counts of microcolonies in sediments to explore the use of the model with more sets o f data. Acknowledgments. We thank Dr. R. S. Lampitt and other staff at the Institute of Oceanographic Sciences, Wormley, Surrey, United Kingdom, for collecting the sediment samples.

References 1. Barnett PRO, Watson J, Connelly D (1984) A multiple corer for taking virtually undisturbed samples from shelf, bathyal and abyssal sediments. Oceanol Acta 7:399-408 2. Fallon RD, Newell SY, Hopkinson CS (1983) Bacterial production in marine sediments: Will cell-specific measures agree with whole system metabolism'?. Mar Ecol Prog Ser 11:117-119 3. Fletcher M~ Marshal~ KC ( l 982) Are s~lid surfaces ~f ec~l~gjcal signi~cance t~ aquatic bacteria? Adv Microb Ecol 6:199-236 4. Fletcher M, McEldowney S (1984) Microbial attachment to nonbiological surfaces. In: Klug M J, Reddy CA (eds) Current perspectives in microbial ecology. American Society for Microbiology, Washington DC, pp 124-129 5. Fox L (1982) Numerical algorithms group library manual Mark 10; routine document NAG FLIB: 1265/187. Numerical Algorithms Group Ltd, Oxford 6. Fry JC (1982) Interaction between bacteria and benthic invertebrates. In: Nedwell DB, Brown CH (eds) Sediment microbiology. Academic Press, London, pp 171-201 7. Fry JC, Zia T (1982) Viability of heterotrophic bacteria in freshwater. J Gen Microbiol 128: 2841-2850 8. Goulder R (1977) Attached and free bacteria in an estuary with abundant suspended solids. J Appl Bact 43:399-405 9. Goulder R, Bent EJ, Boak AC (198 l) Attachment to suspended solids as a strategy ofestuarine bacteria. In: Jones NV, WolffWJ (eds) Feeding and survival strategies ofestuarine organisms. Plenum Press, New York, pp l-15 10. Hanson RB, Sharer D, Ryan T, Pope DH, Lowery H K (1983) Bacterioplankton in Antarctic ocean waters during late austral winter: Abundance, frequency of dividing cells and estimates of production. Appl Environ Mierobiol 45:1622-1632 11. Hermansson M, Marshall KC (1985) Utilization of surface localised substrate by non-adhesive marine bacteria. Microbiol Ecol 11:9 l-105 12. Hossell JC, Baker JH (1979) Estimation of the growth rates of epiphytic bacteria and Lemna minor in a river. Freshwater Biol 9:319-327 13. Jacobsen TR, Azam F (1984) Role of bacteria in copepod faecal pellet decomposition: Colonization, growth rates and mineralization. Bull Mar Sci 35:495-502 14. Jones DS, Sleeman BD (1983) Differential equations and mathematical biology. George Allen and Unwin, London 15. Kieft TL, Caldwell DE (1983) A computer simulation of surface mierocolony formation during microbial colonisation. Microb Ecol 9:7-13 16. Kjelleberg S, Humphrey BA, Marshall KC (1982) Effect of interfaces on small, starved marine bacteria. Appl Environ Microbiol 43: I 166-1172

Model for Microcolony Growth on Sediment

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17. Marshall KC (1980) Adsorption of microorganisms to soils and sediments. In: Bitton G, Marshall KC (eds) Adsorption of microorganisms to surfaces. John Wiley and Sons, New York, pp 317-329 18. Marshall KC, Stout R, Mitchell R (1971) Mechanism of the initial events in the sorption of marine bacteria to surfaces. J Gen Microbiol 68:337-348 19. Meadows PS, Anderson JG (1966) Microorganisms attached to marine and freshwater sand grains. Nature (London) 212:1059-1060 20. Meyer-Reil LA (1983) Benthic response to sedimentation events during autumn to spring at a shallow water station in the Western Kiel Bight, II. Analysis of benthic bacterial populations. Mar Biol 77:247-256 21. Meyer-Rei• LA ( • 984) Bacteria• bi•mass and heter•tr•phic activity in sediments and •v•r•ying water. In: Hobbie JE, Williams PJ, Le B (eds) Heterotrophic activity in the sea. Plenum Press, New York, pp 523-546 22. Meyer-Reil LA, Dawson R, Liebezeit G, Tiedje H (1978) Fluctuations and interactions of bacterial activity in sandy beach sediments and overlying waters. Mar Biol 48:16L-171 23. Moriarty DJW (1984) Measurements of bacterial growth rotes in some marine systems using the incorporation of tritiated thymidine into DNA. In: Hobble JE, Williams PJ, Le B (eds) Heterotrophic activity in the sea. Plenum Press, New York, pp 217-231 24. Newell SY, Fallon RD (1982) Bacterial productivity in the water column and sediments of the Georgia (USA) coastal zone: Estimates via direct counting and parallel measurement of thymidine incorporation. Microb Ecol 8:33-46 25. Roper MM, Marshall KC (1974) Modification of the interaction between Escherichia coli and bacteriophage in saline sediment. Microb Ecol h 1-13 26. Rowbury R3, Armitage JP, King C (1983) Movement, taxes and cellular interactions in the response of microorganisms to the natural environment. Syrup Soc Gen Microbiol 34: 299-350 27. Tempest DW, Herbert D, Phipps PJ (1967) Studies on the growth ofAerobacter aerogenes at low dilution rates in a chemostat. In: Powell EO, Evans CGT, Strange RE, Tempest DW (eds) Microbiol physiology and continuous culture. HMSO, London, pp 240-254 28. Wardell JN, Brown CM, Flannigan B (1983) Microbes and surfaces. Syrup Soc Gen Microbiol 34:351-378 29. Weise W, Rheinheimer G (1978) Scanning electron microscopy and epifluorescence investigation of bacterial colonization of marine sand sediments. Microb Ecol 4:175-188 30. Wetzel RL, Christian R R (1984) Model studies on the interactions among carbon substrates, bacteria and consumers in a salt marsh estuary. Bull Mar Sci 35:601-614

A mathematical model for the growth of bacterial microcolonies on marine sediment.

Counts of bacterial microcolonies attached to deep-sea sediment particles showed 4-, 8-, 16-, and 32-celled microcolonies to be very rare. This was in...
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