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present the estimators of the variance of dilution errors. Next we present the analysis of triples and discuss the statistical test for the normal distribution function of them. Finally we analyze outliers in the data and estimate the dilution errors when the outliers were eliminated. The proofs of theorems are presented in the Appendix.

2. D ata. We test our model of random errors in the analysis of popu- lation growth on the data concerning the Shigella flexneri bacteria used by Lachowicz et al. (1995). We use 24 of the 8-hour passage cultures of two strains incubated in broth separately (Lachowicz et al. (1995) considered also the co-cultures of two strains for the analysis of its mutual influence, which are for simplicity not considered here). Each new passage of the cul- ture starts from the 1:100 dilution of the end state of the previous passage.

The strains were cultivated in 100 ml of nutrient, and the population size were determined by sampling and dilution on the basis of the number of colonies grown on the agar plates.

Let us introduce the notations: i = 1 ,2 , . .. - index of time, ti , - sampling time, di - dilution in time U, j = 1, 2,3 - repetition (triple) of observations, Xij - number of colonies observed on the plate. Instead of the dilution di we use the fraction pi — 1 /di. Since 0.1 ml of diluted sample is spread on the plate 0.1 Pi is the probability that the bacteria cell from the culture is chosen. Let us denote the mean and the variance of the triple by

1 3 i 3

(1) x i = ^ x i,j> si = 2 'S ^2t{x i,j ~ x i ) •

L 3 =1 J=1

3. Param eters o f Verhulst grow th. It is stated in the paper of La- chowicz et al. (1995) that the first hour of the population growth should be omitted, and the subsequent is (by many authors, see Doucet and Sloep (1992) for example) assumed that the population size satisfies the Verhulst equation. It means that the number N = N (t ) of bacteria cells in the ap- pointed volume of the culture (here 100 ml) in time t satisfies the following differential equation

~ N = aiV(l - yN ).

dt v k '

Its famous explicit solution has the following form

N(t) = k

1 +

AT(0) exp (—at)

t < 0.

The parameters of the growth curve are denote, respectively: a is the

growth rate, k = AT(oo) is the stationary state, and N(0) is the initial

size of the population. Our goal is to estimate for each passage separately

Random errors in analysis of population growth

9 using the triple means. Let us mention here that the growth parameters are optimum if

E 8 ^min,

where

(2) Xi == 0.1piN(ti).

4. Standard test o f ran dom errors m odel. According to the stan- dard description of the test for the model of population growth it is assumed that Xij is the sum of x\ and the random error. Taking into account the rep- etitions we can assume that the observed random variables are the form

(3) X ij — Xi +

where all used random variables are mutually independent identically normally distributed N ( 0,cr2), o 2 > 0. The test is based on the following Fisher statistic

p 48£f=i(gt ~ Xi)2 5,16

For the example estimation of parameters and to test the standard model of random errors the data from the first observed culture will be used. On the basis of the data given in column 6 of Table 1 the following values of

Figure 1. The estimation of the growth curve of Shigella flexneri.

Table 1. Test of the Verhulst model. Example.

Incubation time in hours

Number of colonies on 3 plates

Dilu- tion

Mean number of colonies

Verhulst function

Expected number of colonies

1 2 3 4 5 6 7 8

1 74 74 70 5 x l0 4 72.7 3.16x l07 63.3

2 151 161 130 lx lO 5 147.3 1.59xl08 159.4

3 131 144 155 5 x l0 5 143.3 7.84x10s 156.8

4 429 389 374 lx lO 6 397.3 3.46 xlO9 345.9

5 171 186 178 5 x l0 6 178.3 1.06xl010 211.5

6 194 183 190 lx lO 7 189.0 1.78xl010 178.0

7 215 220 194 lx lO 7 209.7 2.06x l010 205.8

8 36 34 57 5 x l0 7 42.3 2 .12x l010 42.5

the estimators were obtained: a = 1.623, k = 2.14xl010, N ( 0) — 6.25xl06.

The column 7 gives the expected number of bacteria cells in 100 ml culture from the solution of the Verhulst equation and the column 8 - the expected number of colonies on the plate computed from (2).

Figure 1 presents log (N (t)) curve depending upon t as well as the em- pirical points in the equivalent scale. The goodness of fit of the curve and the empirical points seems to be satisfactory, but this supposition obviously contradicts the result of the test.

The data to the verification of the model are given in Table 1. Thus both the Verhulst model of the growth of the S. flexneri and the assump- tion of random errors should be rejected (F^ iq = 12.53, significance level 0.01, critical value 4.44). We obtained a similar result 18 times on 24 tested cultures.

In practice (see for example Battes and Watts (1988)) various modifi- cations of the assumptions are used to fit the model and the experiment.

Let us assume, modifying (3), that the dispersion of the random errors is proportional to Xi,

(5) X = Xi(\ -{- £zj),

where all considered random variables are mutually independent iden-

tically normally distributed iV(0, <r2), a2 > 0. Under this assumption let

us consider the analysis of variance for log (X ij). Now, for the examined

culture, we obtain = 3.72, and this result does not allow to reject

the model on the significance level 0.01. However in the whole data set the

assumption (5) was rejected 13 times on 24 tested passages.

Random errors in analysis of population growth

11

5. Model of random errors. Recall that N = N(t) denotes the popu- lation size in time t and let P be the fraction of the population spread on the plate. A priori three random mechanisms may be included into the model:

the stochastic process of population growth, i.e. N(t), t > 0, is a random process, the errors of dilution, i.e. P is the random variable, and finally the sampling from the population before spreading on the plates. Implicite the uniform distribution of bacteria cells in the broth is assumed.

We describe the mechanisms by one operator Rp which transforms the integer-valued random variables, where P characterizes the dilution. We assume that the number of colonies on the plate is equal to

N

(6) X = R p (N) = J 2 ^ (P ),

k

= 1

where <5fc(P), k = 1 ,2 ,... denote the mixed Bernoulli trials, which under the condition P — p have the success probability P(<5fc(P) = 1 |P = p) = p.

Note that if Po, Pi, N are mutually independent, then the convolution of the operators is equal to

(7) Rp0(Rpj(N)) = Rp0p1(N).

P

1. Let N = n and P = p be deterministic. Then X has the binomial, approximately the normal N(np , np( 1 — p)) distribution function.

T

1. Let 6k(P), k — 1 ,2 ,... under P = p, N = n be mutually independent. Then

E (X ) - E{NE(P\N)),

Var(X) = E {{N(N - 1)E{P2\N)) + E(X) - (E (X ))2.

T

2. Let N = n be deterministic. IfE(P) = p , then (8) E(X) = np , Var(X) = n(n - 1) Var(P) + np{ 1 - p).

Let

= Rp pu)(n), where the random variables P q , P^ \ j = 1,2,3, are mutually independent. Let X = E(Po) = po, E (P ^ ) = p\.

Then E (X ) = npopi,

-i cy

Var(X) = -n (n — l)Var(PoP][1')) + -n (n — l)piVar(Po)

o o

2 2 1

+ -n p 0(l - Po)Pi + 7fipw i(l - p0pi).

R

. Consider the birth and death process N = N(t), t > 0,

which corresponds to the considered deterministic Verhulst growth. Under

the suitable varying intensities of the process it was proved (see Ethier and

Kurtz (1986)) that E (N) -> oo, E(1V2)/(E(1V))2 -> 1. Assume that the

dilutions in the population size estimation are essential e.g. the number of

colonies on the plate in expectation belongs to the essential interval (the dilutions used here provides from 40 to 400 colonies on the plate).

C ^{orolla ry} 1. ^If E(N) ^-► oo, E{N2)/E(N))2 -> 1 ^{, P =} P*/(EN), and P* and N are mutually independent, then

Var(X) = Var(P*).

6. In d ex o f triple. In the experiment the viable count is observed in such a way that 3 times 0.1 ml of culture is drawn from the sample and spread on the plates. Let us suppose that the sample contains n bacteria cells and from it the fraction p is drawn. It follows from Proposition 1 that the number of colonies on the plate is approximately normal N(np, np{ 1 — p)).

Let X \,X ² ,X% be the 3-element sample from the absolutely continuous distribution function Fx and density fx - Let X\$ < X ² $ < X ^{^ 3} denote the order statistics in the sample. Let us define the random variable

j j _ * 2,3 ~ * 1,3

“ * 3 ,3 - * 1 ,3

and call it the index of the triple.

If n and p are unknown then the hypothesis of normality, without the knowledge of position and scale parameter may be tested using the following theorem.

T ^{he o r em} 3. If Fx is a normal distribution function then U has the truncated Cauchy distribution with the density of the form

(9) fu(u) J _ _ _ 3 V 3 _ _

2tt (u - \ ) 2 + f ’

0 < u < 1.

Table 2. The distribution function of the empirical index of the triples.

k fk fk k fk fk k fk fk k fk fk

0 9.2 8 5 11.2 19 10 11.9 9 15 10.8 9

1 9.6 6 6 11.4 10 11 11.9 17 16 10.4 7

2 10.0 7 7 11.7 12 12 11.8 9 17 10.0 8

3 10.4 13 8 11.8 19 13 11.4 15 18 9.2 8

4 10.8 11 9 11.9 12 14 11.2 10 19 9.2 7

Legend: fk - number of observed U which belong to the interval [ ^ , ^ 1 ) , fk = 216(irt/(^bł) — F

(-^

)) - the expected value of fk-

The test of normality for 216 triples is given in Table 2. The domain

[0,1] of U was divided into 20 parts numbered from 0 to 19. The chi-square

Random errors in analysis of population growth

13 test does not al low to re ject the normality of the distribution function (x2 = 20 .55 , degrees of freedom 19 , s ignif icance leve l 0 .05 , critical value 30 .14) .

7 . Analysis of triples . The three plate technique of viable counting enab les the description of the var iance of the volume of the samp le of the culture which is drawn to spread . It is possible that the sampl ing precision depends on the volume of the last samp le . Hence let us divide the data set on the strata of the same dilution , and introduce the notation : s - number of strata , k - index of stratum , mk - s ize of stratum , m = J2k= i mk Now • we introduce the notations of the mean x^k and var iance s2k of the l -th tr ip le for the A : -th stratum analogously to (1) , in which we add the index k of stratum .

Let nitk denote the number of bacteria ce l ls in the sample before spread- ing the l -th triple for the k -th stratum . Assume that the number of colonies on the plate is the random var iab le Xij^ — Rp i f c ( i+e* ó k )(nl ,k ) i where pi^

characterizes the expected dilution , j = 1,2,3, are the mutually in- dependent random var iab le N (0 ,crfk ) , afk > 0 . In virtue of (8) we have E (X ij'k) = nitkP i ,k , Var(X/J ) fc ) = Us ing the moment method of estimation from the equal it ies nitkP i ,k — ^ i ,k i kP i kal k = slkwe obtain the estimators

■° l ,k — Sfk/xfk-

For the estimator of the variance under the assumption afk = ak we use

(10) = m< fnZ\slk/slk -

The estimator of the common var iance under the assumption ak = a2 we use

(11) d2 =

The va lues of (10) computed for our data are presented in Table 3 . It suggests a increasing tendency of the var iance with increasing dilution . The test of ser ies does not prove this tendency . From (11) we obtain s2 = 0 .0097 .

Table 3. The estimation of the variance of the dilution error .

k d fc mk

k dk mk ak ^{i .2} k dk mk dl

1 lxlO4 3 0 .0112 5 2xl05 6 0 .0072 9 5 xlO6 22 0 .0120

2 2xl04 1 0 .0056 6 5x 105 22 0 .0074 10 1 xlO7 38 0 .0060

3 5xl04 9 ’ 0 .0055 7 lxlO6 16 0 .0076 11 2 xlO7 16 0 .0065

4 lxlO5 8 0 .0065 8 2xl06 8 0 .0136 12 5 xlO7 67 0 .0172

8. Error of dilution. In viable count technique the samples are diluted before spreading on the plates in the proportion 1:100 by drawing 1 ml sample of culture and adding it to 99 ml of broth. For higher dilutions the standard operation is repeated. The last dilution may be residual, (l:Vo), which is realized by drawing 1 ml sub-sample and adding it into 0 < Vo < 100 ml broth. Let p be defined by

(12) p — (0.01)cp*, where c is integer, 0.01 < p* < 1.

The dilution d = 1/p is realized as the composition of c standard dilutions and the residual one in which Vo = 1/p*. Since 0.1 ml sample is drawn from the last dilution and spread on the plate, thus in the terms of the operation R using (7), the number of colonies on the plate is equal to

(13) X #o.ip*(-R(o.oi)c(n)) — Ą).iP(n), where n is the population size in 100 ml of the culture.

Let us assume that the error of dilution arises from the inaccurate draw- ing of a suitable volume of culture. Because in the experiment were drawn 1 ml in the standard dilution or 0.1 ml for spreading let us assume that, these volumes are drawn with the relative random errors eo and ei, respec- tively. In virtue of (12), for some dilution p, let us define the random variable P = , where

i>(0) = (0.01)c(l + e g ) ... (1 + eg), = 0.1p*(l + ew ),

and where e^d ... e[cd are the probabilistic copies of eo and e ^ i, j = 1,2,3, are the probabilistic copies of e\.

Recall that in time ti the Verhulst growth is N(U) and the dilution d{ = 1/pi is used. For pi define Ci and p* according to (12) (for simplicity the index i will be further omitted). We assume that

(14) X i,j = R p W p (i)(N (ti)).

T ^{h eor em} 4. Let N(t), t > 0 be (nonrandom) population growth. If X ij

is defined by (14) and Xi — t 3 = are

mutually independent, then E(Xij) = E(Xj) = X{, (15) V a r ( ^ - ) = ad + H-0o<7i,

(16) Var( ^ ) - ao + ^ ? ( 1 + *o)-

where aft = 0.01_2cVar(P/°^), (j2 = (0.1p*)~2V a r(P ^ ).

Random errors in analysis of population growth

15 R

. In virtue of Proposition 1 the random variable X^j defined by (14) conditioned by P = = p is normally distributed and also for the simple dilution error the normal distribution may be supposed. It is ob- vious that the mixed normal distribution if it mean is normally distributed (with common variance) is also normal. Although for X ij the second as- sumption for P may be satisfied in approximately only, a mixture of normal distributions is approximately allowed.

Let us consider the possibility of occuring of outliers in the data set.

Table 4. The distribution of the residuals.

Dilution

- 5 - 4 - 3 - 2

Residua]

- 1 0 1 2 3 4 5 >6

(')

5 x l0 4 0 0 0 0 0 1 3 2 0 0 0 0 6

lx lO 5 0 0 0 0 3 2 0 0 0 0 0 0 5

5 x l0 5 0 0 0 2 4 4 2 2 2 0 0 0 16

lx lO 6 0 0 1 0 4 4 4 0 0 0 1

14

2 x l0 6 0 0 0 3 3 0 2 0 0 0 0 0 8

5 x l0 6 1 0 0 2 7 3 4 2 1 0 0

22

lx lO 7 0 0 1 3 5 8 10 6 1 2 1

38

2 x l0 7 0 0 0 5 1 5 5 0 0 0 0 0 16

5 x l0 7 0 0 3 6 12 17 14 9 4 1 0

67

Total 1 0 5 21 39 44 44 21 8 3 2 4 192

(1) - the stratum size.

Table 5. The distribution of the residuals. Data corrected.

Dilution

- 3 - 2

Residual

- 1 0 1 2 3 4

(*)

5 x l0 4 0 0 0 1 3 2 2 0 6

lx lO 5 0 0 4 1 0 0 0 0 5

5 x l0 5 0 2 4 5 3 2 1 0 16

lx lO 6 0 0 4 5 4 0 0 0 13

2 x l0 6 0 3 3 0 2 0 0 0 8

5 x l0 6 0 1 7 4 4 2 1 0 19

lx lO 7 0 1 6 9 10 6 1 2 35

2 x l0 7 0 5 1 5 5 0 0 0 16

5 x l0 7 4 5 14 17 13 10 4 0 67

Total 4 17 43 46 44 22 7 2 185

(1) - the stratum size.

Define the standardized residuals

(17) ^%i,j Xi ¹⁰

The residuals (17) computed for our data and grouped in the strata are presented in Table 4. The result suggests the existence of outliers in the classes of with the size written in bold digits. The significance problem of the distinguished outliers was omitted. Let us withdraw the outliers from the data set, estimate on the abridged data the parameters of the Verhulst growth and compute the new residuals. The distribution of corrected resid- uals is presented in Table 5. The repeated estimation of the parameters implies shifts of some points in the parameter space. Since these shifts are not characteristic the results are omitted.

9. Errors of dilution. Continuation. Let us consider the variance of the dilution errors after the elimination of the outliers. We estimate the pa- rameters (7 q o =Var(eo) and a\ =Var(ei) by the minimum chi-square method requiring for the standarized residuals

Vij = (Xij - xi))/(Var(Xij ))1''2 Vt = (Xi4 - rti)/(Var X^ 2 the optimum fitting of the distribution function N ( 0,1). The results of com- putations of the distribution function of the standarized residuals based on (15) and (16) are given in Table 6. Now we obtain S q , o = 0-00897, sf — 0.00954, which are comparable with those from section 7. The chi- square test does enables the rejection of the Verhulst growth curve and random error model. It suggests that the S. flexneri growth should be de-

Table 6. The distribution of the residuals Vij and VL

Class of residual

Variable V C) (2)

Variable V C1) (2)

Class of residual

Variable V i1) (2)

Variable V i1) (2)

-1 .6 37.1 33 12.4 14 0.2 43.3 45 14.4 18

-1 .4 16.6 9 5.5 3 0.4 40.8 37 13.6 14

-1 .2 21.6 14 7.2 3 0.6 36.9 39 12.3 14

-1 .0 26.9 27 9.0 6 0.8 32.1 28 10.7 15

-0 .8 32.1 31 10.7 14 1.0 26.9 21 9.0 7

-0 .6 36.9 43 12.3 12 1.2 21.6 18 7.2 2

-0 .4 40.8 44 13.6 8 1.4 16.6 15 5.5 5

-0 .2 43.3 46 14.4 18 1.6 37.1 59 12.4 18

0 44.2 47 14.7 15

C) - computed size of class, (2) - observed size of class.

Random errors in analysis of population growth 17

scribed by the Verhulst equation assuming the random errors defined by (14) and after elimination of the outliers in the data set.

Appendix

A. Proof of Theorem ^{1 .} For X defined by (6) it follows that

N ^ N

E(X) = E( £ 6k(Pj) = E(E( Sk(P)\N)) = E(NE(P\N)),

k=l k=l

E(X2) = ^e ( £ £ Sk(P)6i(P))+E(Y,Sk(P))

k = l l = l l j L k k =

1

- E(1V(JV - 1)E(P2|1V)) + E{NE{P\N)).

Now we find Var(A) = E (X 2) - (E (X ))2.

B. Proof of Theorem 2. From Theorem 1 we obtain E(X) = riE(P) = np ,

Var(X) = n(n — 1)E(P2) + np — n2p2 = n(n — l)Var(P) + np{ 1 — p).

Note that

X ! = RPoPm(n) = j 2 6k(Po)Sik1)(Pil1)),

1 Jfe=l

x 2 = P w (n) = £ > ( P o )4 2>(Pi2)h

1

k=l

where ^ (P ), ^ ^ (P '), k = 1,2, . . . , n under the condition P = p, P' = p', P" — p" are mutually independent. Then we have E (X i) = E (A 2) = npopi ,

E(X,X2) = e ( £ V f 0))

k = 1 l—l

= nE(<5i(Po)41)(^ i1)) 4 2)(A (2))) +n(n - 1)E(,5 i (P o )152(P o )4 1)(^(2))42V i (2)))

= nE(P0)E(P1<1))E(pf)) + n(n - 1)E(P02)E(P1(1))E(P1(2))

= npopj + n(n - l)p2E(P02), Cov(Xi,X2) = E(X ⁱ X2) - E(X ⁱ )E(X2)

= npoPi + n(n - l)pfE(P02) - n2plp\

= n(n - l)p2Var(Po) + npo(l - po)Pi-

Hence

Var(X) = ^Var(X1) + ?C o v (X i,X 2)

= |n(re - l)Var(Po-Pi) + |n(re - l)pfVar(Po)

2 1

+ -np0(l - Po)Pi + ^np0pi(l - popi).

C. Proof of Theorem 3. In a standard way we obtain the probability distribution function and the density of U:

Fu(u) — ^ J f x { x ) f x ( x + y )(l - Fx ( x + ^ dydx, (18) fuiu) = \ f f f x { x ) f x { x + y ) f x ( x + - \ d y d x .

U J—oo J 0 \

The distribution function of U does not depend upon the position and the scale parameters of X. For the proof the substitution of 6 + aX for X for each b and a > 0 is sufficient.

In virtue of (18) for 0 < u < 1 we obtain

fi r°° r°° / 1 / „ . / v\2' fu(u) _{(27r)3/ 2}

(27r)3/ 2w"

£ L f ° exp ( - K *2 + ( x + y)2+ 9 + 9 ) ) ^ dy dx

poo poo

y ex:

J 0 J—oo

3 / 1 / l ' 2 2 ( x + 3 n 1 + «

x exp

V 2 V 3 ; u + i ) 2 + i +

n / u? ) ) dxdy

D . P rop osi tion 2. If e^, k = 1, 2 , ... are independent random variables and E(cfc) = 0, £/ien Var(n£=1(l + e*)) = n j =1(l + Var(efc))) - 1.

E. P r o o f o f T h eorem 4. Here N = N(ti ) is nonrandom. From (14) and (11) we have

E(Xitj) = NE{P}0)P $) = JV0.1p*(0.01)c = NO.lpi = x<.

Random errors in analysis of population growth

19

Proposition 2 implies Var (Pl 0 )P $ )

= (E(/><1 ))2V ax(/f >) + (E (ff >))2Var(i$>) + V a r(/f V a r ^ ) , and from Theorem 2 we have ^

Var(Xjj) = ^^ ² (Va^(^( ⁰ )f^<J,) + o(l)) = ^x ?(< ^t (S ^{+ <Ą +} + °M)- Var(Xy) = N tQ varipW plf) + |p?Var(ą<0)) + o(l))

= (*o2 + (1 + (Ą) + o(l)), which is easily seen to be (15) and (16).

Acnowledgement. The author would like to thank Tadeusz M. La- chowicz for providing used in the paper the experimental data realized in the Institute of Microbiology, Wroclaw University.

References

D. M. B ates and D. G. W a tts, Nonlinear Regression Analysis and Rs Applications, Wiley, New York, 1988.

P. D oucet and P. B. Sloep, Mathematical Modelling in the Life Sciences, Ellis Horwood, New York, 1992.

S. N. E thier and T. G. K u rtz Markov Processes, Wiley, New York, 1986.

T. M. Lachow icz, D. D ziadkow iec, I. K opociń ska and B. K opociński, Population dynamics in the co-culture of Shigella flexneri lb original strain and its antigenic 3b mutant carrying a prophage, Microbios 83 (1995), 89-106.

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