kinetics of biocide kill

International Biodeterioration 26 (1990) 111-125

Kinetics of Biocide Kill

R. N. Smith

Biodeterioration Centre, Hatfield Polytechnic, Hatfield, Hertfordshire ALl0 9AB, UK

A B S T R A C T

The efficient use of biocides to control microbial contamination is dependent upon selecting the most potent agent at the anticipated end-use concentration. This is based upon an accurate determination of two basic parameters:

(1) The time taken by the biocide to achieve a total kill (death rate or decimal reduction time).

(2) The effect of biocide concentration on the death rate or decimal reduction time.

The time taken to achieve a total kill can be calculated from the death rate. In the simplest case a plot of the natural logarithm of survivors declines linearly when plotted against time and the slope of that line is the death rate. However, the plot of the line of survivors against time is frequently non-linear. Concave curves may result from attempts to control a mixed population with different degrees of tolerance to the biocide; the shape of the curve being a combination of two or more different linear declines. Convex curves, or curves with a shoulder, may be due to one of three phenomena. The target organisms may adhere together in clumps of two or more," the nature of the reaction of biocide with the target organism is one where the organism first changes from a resistant to a susceptible state," or the nature of the biocide molecule is such that uptake is relatively slow and death only commences when a critical concentration has accumulated within the cell. With all such convex curves the decline eventually becomes linear and again the slope is the death rate.

The relationship between death rate and biocide concentration is rarely proportional and usually exponential. Thus halving the concentration may cause a disproportionate increase in the decimal reduction time. This relationship is controlled by the concentration coefficient or concentration

111

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112 R, N. Smith

exponent. When the logarithm of concentration is plotted against the log of death rate or log of decimal reduction time, a linear relationship is obtained whose slope is the concentration coefficient. Thus if the death rates at two or more concentrations are known, the concentration coefficient can be determined and the effect of dilution predicted.

The mathematical interpretation of these phenomena and their influence on biocide treatment are discussed.

I N T R O D U C T I O N

The efficient use of biocides requires that their respective activities be accurately interpreted and assessed. A proper biocide strategy requires far more than the m i n i m u m inhibitory concentrat ion. The death rate, decimal reduct ion time and the effects of temperature, concentrat ion and time on these properties must also be established.

Microbial death invariably appears as a first order reaction when the behaviour of a biocide at a part icular concentra t ion is investigated. However, when different concentrat ions are examined it becomes apparent that the biocide-cell reaction is much more complex than a simple first order reaction.

The particular properties of certain biocides or the target organisms are often responsible for deviations from the expected pattern of normal first order kinetics. This can lead to misinterpretat ion or rejection of experimental data. Many of these p h e n o m e n a have been discussed by Meynell & Meynell (1970) and Russell et al. (1982). In this paper the major factors responsible for deviations from the normal first order kinetics, the impl icat ion of h igher order reactions and the interpretat ion of such data are discussed.

F IRST O R D E R BIOCIDE KILL

A first order reaction may be def ined as the process whereby the popula t ion is mult ipl ied by a constant factor, less than one, for each unit of time. When a microbial popula t ion is exposed to a lethal biocide concentrat ion death occurs progressively according to first order reaction kinetics, where K is the death rate.

Viable K, Dead

The above equat ion does not include the biocide and it has been suggested that in certain circumstances the role of the biocide is not to

Kinetics of biocide kill

TABLE 1 Decl ine in Viable Nu mb e r s of a Popula t ion in Which 90% of the

Cells are Kil led Each H o u r

113

Time (h) Number of Proportion (%) surviving survivors at the end of each hour

o lO ~ 1 105 10 2 104 l0 3 103 10 4 102 10 5 10 10 6 ! 10 7 0-I 10 8 0'01 10

participate in the death reaction but rather to render conditions within the cell hostile and so the cell death becomes a first order reaction.

In the example given in Table 1 a population exposed to a biocide solution may decline by 90% every hour, or more correctly the population is multiplied by 0.1 every hour. This demonstrates the three important characteristics of exponential death.

(i) Log - l inear decline If the log of the numbers is plotted against time then a straight line is obtained whose slope is -ve , i.e. coming down to and eventually crossing the X axis (Fig. 1).

1 4 -

O

7

._s

I I 0 5 10 15

T ime (h)

Fig. 1. Normal first order (exponent ia l ) decline of survivors against time. Init ial popula t ion , 1 × 10 6 ml -~ viable cells; death rate, K = 1 h -t.

114

(ii)

(iii)

R. N. Smith

Size of the surviving popula t ion The n u m b e r of surviving cells in the popula t ion is dependen t u p o n the size of the initial populat ion. Probabili ty of achieving sterility - - commercia l sterility Theoretically with a logari thmic decline the popula t ion never reaches zero since there is always a small propor t ion of a cell remain ing viable. However, it is clearly impossible for 0.1 or 0.01 of a cell to remain viable and this fraction can be properly interpreted as the probabil i ty of one cell surviving in the populat ion, or the chance of the sample being unsterile. Thus in the example given in Table 1 there is a 1 in 10 chance of the sample being non-sterile after 7 h, and a 1 in 100 chance after 8 h. Since there is a ten-fold reduct ion every hour, after 12 h the chance of the sample being con tamina ted will be 1 mill ion to 1. Thus a manufacturer , by increasing the contact time, can select a level of commercia l sterility for his product such that the chance of any item being non-sterile is 1 billion to 1.

KINETICS OF FIRST O R D E R BIOCIDE KILL

This may be represented by the reaction:

Viable K, Dead

In this case the decline in survivors follows the first order kinetics as described above and gives the usual log- l inear decline.

When the log of survivors in Table 1 is plotted against t ime we see that the decline is cont inuous and as such is the result of an infinite n u m b e r of infinitely small reductions.

This may be represented by the formula:

X, = Xo X (l + l / n ) - "

where:

n

(1 + 1/n) -n =

X. = initial popula t ion

Xt = survivors

n u m b e r of reductions

the factor by which the popula t ion is reduced when a complete set of (n), (I + 1/n) reductions has occurred

Kinetics of biocide kill 115

As the value of n increases the value o f (1 + 1/n) -n decreases but the decrease in value gets progressively less until it reaches a limit value below which it does not fall. This limit value is 2.718 -~ or 0,3678. Thus when a popula t ion has been reduced to 0.3678 or 36. 78%, there has been one set compr is ing an infinite n u m b e r o f infinitely small reductions. The dea th rate K is the n u m b e r o f 36-78% reduct ions which take place in one hour.

Thus if K = 2 then the reduct ion in n um ber s over three hours will be:

o r

)(1 = X0 X 2.718 -ut

Xt = X0 X 2.718 -23

Xt = X0 X 0.36786

)(1 = X0 X 0.000 248

Since 2.718 is also the base o f natura l logari thms, conver t ing the surviving popula t ion to na tura l logar i thms gives:

lnXt = l n X 0 - ( K X t )

o r

a n d

In (Xt/X0) = - K . t

- K = In (Xt/Xo)" 1/t

Thus w h e n In o f survivors is plot ted against t ime there is a l inear decl ine whose slope will equal K the dea th rate.

First o rde r dea th kinetics can be represented as follows:

C h a n g e in viable cells = initial n u m b e r X death rate

- d X / d t = KX

where:

- d X / d t = reduct ion in num ber s with t ime

K = dea th cons tan t

X = viable popula t ion

On integrat ion this equa t ion becomes:

In (Xt/Xo) = - K t

116

o r

and

where:

R. N. Smi th

X, = Xo X e -k'

K = In (Xt/Xo) × 1/t

X0 = initial popula t ion

Xt = popula t ion surviving after t ime t

Alternat ively since S = propor t ion o f survivors in the popula t ion

S = X , /Xo

then:

In S = - K t

K = - l n S / t

S = e -Kt

S = 2"718 -/'t

S = 0"3678 Kt

D E C I M A L R E D U C T I O N T I M E

This example also demonst ra tes a term widely used in biocide science, the decimal reduct ion time D. This, as the name implies, is the time taken for a ten-fold reduct ion in the n u m b e r of survivors. In the example in Table 1 it is obvious that D -- 1 h.

The decimal reduct ion t ime may be calculated from the death rate K as follows:

W h e n time equals the decimal reduct ion time, t = D,

Since

and so

S = 0 . 1

- K = In ( S ) . l / t

- K = ln0 .1 /D

D = l n 0 - 1 / - K


o r

D = - 2 - 3 0 3 / - K

= 2.303/K

The D value gives a simple unam biguous me thod for de termining the t ime taken to achieve a total kill with a known degree of certainty.

In the example given in Table 1 there was only 0.1 of a cell surviving after 7 h or only a 1 in 10 chance of the sample conta in ing a living cell. Thus, extending the t reatment time by another 8 h will give a one in one billion chance of a cell surviving and the sample being contaminated.

N O N - L I N E A R In SURVIVAL PLOTS

In practice the plots In of survivors against t ime are often non-linear. There are two forms of deviation from the normal log-l inear relationship. The first is a convex curve which eventually becomes linear and the second a concave curve which again becomes l inear after some time. In both cases the later l inear part of the curve represents the death rate K. The following models have been proposed to account for these phenomena .

C o n c a v e survival curve - - mixed populat ion

In this case the popula t ion is considered to consist of two distinct componen t s with different death rate constants (K) e.g. a mixed popula t ion of two species or a popula t ion of spores and vegetative cells.

Susceptible /q,, Dead

Resistant /L~ Dead

The more susceptible cells die more rapidly at the faster rate K~ and so the first (concave) part of the plot consists of the combined survival curves of the two organisms. Once all the more susceptible strain has been e l iminated the rate of decline will represent only the resistant strain and so the plot o f l n survivors against t ime becomes l inear with a slope of -/(2. Thus a plot of survivors against t ime will give a concave curve with a steep decline which becomes progressively less steep and finally becomes linear. If the l inear part of such a curve is extrapolated to t ime

118 R. N. Smith

14

o >

ft.

5

I [ 0 5 10 15

T i m e (h )

Fig. 2. C o n c a v e curve p r o d u c e d by d e a t h of a mixed p o p u l a t i o n c o n t a i n i n g two o r g a n i s m s A a n d B in wh ich the ini t ia l p o p u l a t i o n o f A = 5 X 10 5 v iab le cells ml ~ a n d

the d e a t h rate K(A) = 4 h - I : whi l s t B = 5 X 10 5 v iab le cells ml -I a n d K(B) = 1 h -I.

zero the In of survivors at this point will equal the resistant population in the original sample.

An example of a mixed population decline is given in Fig. 2. In this example the sample contains two populations each of 5 X l0 s viable cells ml -~ but the death rate of one i sK = 4 h -~ and the other K = 1 h-k The first population where K = 4 h- ~ is eliminated after only three hours and the decline thereafter is linear and represents the second death rate, K = 1 h -~. This demonstrates that the more susceptible population is rapidly eliminated.

It is fortunate that in practice most organisms react to biocides at much the same rate and so very often even mixed populations will show the normal linear decline.

Convex survival curve

Multiple hit This is a special instance of a two stage reaction, described below, but is one which often occurs when biocides are applied to microorganisms which tend to be associated in clumps. The mathematical model for the multiple hit ph e nome non was originally developed to describe death from gamma irradiation (Fowler, 1964) and may be expanded as follows:

A viable count assumes that each colony arose from a viable unit composed of a single cell but if the cells are clumped or associated

Kinetics o f biocide kill 119

together each viable unit (VU) will consist of two or more target cells. The multiple hit phenomenon occurs because each cell making up the clump must be inactivated before the viable unit ceases to register in a viable count and a death is recorded.

When cells are clumped a plot of the In of survivors against time will give a curve which is convex, at first several of the target cells within a clump may be killed but as long as one survives the clump is still viable. Thus the viable population at first shows very little decline and then a progressive decline until the point is reached when all the surviving clumps contain only one viable cell. At this point the decline of the In survivors is linear with a slope equal to - K , the death rate.

Model for a multiple hit curve. Assuming the first order kinetics described above the proportion of targets which survives regardless of clumping will be:

S = e -Kt

Therefore the probability (PH) of a viable unit receiving at least one hit will be:

P H = 1 - e -Kt

If n = number of targets, or cells in a clump, which must each receive a hit, or undergo a lethal reaction if that clump is to be killed, then the probability of clump death, i.e. all n targets each receiving a hit (PHn) will be:

PHn = (1 -- e-1"t)"

Example." In a case where there were two targets in each viable unit and the probability, or chance, of any target receiving a hit is 0. 5 or 50%, the probability of both targets in a viable unit receiving a hit will be:

0 .5×0 .5or0 .52 = 0.25or25%

In this case only a quarter of the VUs will be killed even though half the targets have been eliminated.

Similarly the actual proportion of viable units surviving, including those with <n hits, will be:

S = 1 - (1 - e - K t ) n

In this example

S = 1 - 0 - 2 5

= 0.75 or 75%

120 R. N. Smith

If the l inear section of the decline is extrapolated back to zero, the value of In S at this point will be greater than 1, and equal to the mean n u m b e r of target cells in each clump. Alternatively if l n X is plotted against t ime the intercept at t ime zero will equal the total n u m b e r of targets of cell.

A mult iple hit curve is displayed in Fig. 3. In this example the death rate K = 1 h -~ and there were initially 10 6 viable units each containing three viable cells giving a total of 3 x 106 viable cells in the original sample. After two hours only 0.36782 or 13.5% cells have survived and of the c lumps 93.6% will now contain only one viable cell. Thus, as can be seen in Fig. 3, the durat ion of the shoulder caused by the mult iple hit p h e n o m e n o n only lasts about one decimal reduction time when there are three targets per c lump and then the normal log-l inear relat ionship is established with a slope equal to -K .

Two stage reaction This produces a curve which is similar to the mult iple hit curve since while the m e c h a n i s m produc ing this curve is quite different the mathemat ica l interpretat ion is the same. This model assumes that the cells pass from a resistant to a susceptible stage before they die and the rate of reaction from resistant to susceptible is greater than the death rate. An example would be where resistant spores die via a susceptible intermediate state which is itself viable.

14! "

O

7

5

I I 0 5 10 15

T i m e ( h )

Fig. 3. Convex curve (mult iple hit) p roduced by death of c lumped cells. Populat ion, I × 10 ~ c lumps and 3 cells per c lump, K = h -t,


Thus:

Resistant (R) /q, Susceptible (C) K2 Dead

where K~ > 1£2. In this case:

S = (KI/(KI - K2))e -~2' - (K2/(K, - K2))e -A''

This type of decline again gives a convex curve dur ing the initial period when reactions 1 and 2 are proceeding similar to the mult iple hit curve. Once the first reaction is complete and all the cells are in the susceptible state then there is a log- l inear decline at the rate K2. The curve produced by the two stage reaction is indis t inguishable from the mult iple hit curve.

Time taken f o r a biocide to diffuse into the cell In some cases the death reaction is also dependen t on the concentra t ion of biocide which diffuses into the cell. The concentra t ion of biocide within the cell will be de te rmined by the diffusion rate and the contact time, Thus time is having two effects, first on the concentra t ion ofbiocide within the cell and second on the death rate K. This will give a modif ied death equat ion in which t ime t occurs twice:

d X / d t = K X t

which on integration becomes:

In (Xt/Xo) = - K t 2 / 2

In this si tuation the In survivors will be l inear with t ime squared and so a plot o f l n survivors against t ime will be convex, becoming increasingly steep with time. This accelerating decline cont inues until the point is reached where the biocide within the cell is in equi l ibr ium with that outside the cell and at this point the decline in In survivors becomes linear with a slope equal to K the death rate.

The curve demons t ra t ing this p h e n o m e n o n is displayed in Fig. 4. This demonstra tes that the shoulder of accelerating decline can last much longer than with a mult iple hit effect and can result in a biocide being given a smaller death rate than it really possesses. As can be seen in Fig. 4 the period of slow decline lasts much longer and with some biocides based on large organic molecules it may take eight or more hours before the slope becomes l inear and equals -K .

122 R. N. Smith

14

o

L

5

0 5 10 15

Time (h)

Fig. 4. C o n v e x curve due to the t ime t aken for the b ioc ide to diffuse in to the cell. Ini t ia l popu l a t i on , 1 × 10 6 ml -I v iab le cells.

S E C O N D O R D E R REACTIONS AS P S E U D O FIRST O R D E R REACTIONS

The first order reaction, Viable , Dead, does not include the biocide but in many cases the complexity of the reaction between the biocide and the cell is such that at least second order kinetics are appropriate.

In such cases the reaction can be represented by the second order reaction.

Viable + Biocide x Dead

Thus the rate of the reaction will be:

v = - d X / d t = K B X

where:

v = velocity of the reaction

B = biocide concentration

X = concentration of viable target cells

K -- death rate constant

However, if in this second order reaction, one of the reagents, the biocide, is in vast excess by comparison with the target sites in the microbial cells, then the concentration of the biocide will for all practical purposes remain unchanged before and after killing of the


target cells. Thus B becomes a cons tan t and so the react ion takes on the characterist ics o f first o rder reaction:

v = dX/dt = KX

This is referred to as a pseudo first o rder reaction. In pseudo first order, as in first o rde r reactions, the dea th rate K is i n d e p e n d e n t o f the concen t ra t ion o f target sites.

SECOND, T H I R D A N D H I G H E R O R D E R R E A C T I O N S A N D T H E E F F E C T OF B I O C I D E C O N C E N T R A T I O N

As has been shown the n u m b e r of free biocide molecules are so m u c h in excess as to r emain cons tant th roughou t a biocide-cel l reaction. This does not imply that concen t ra t ion has no effect upon biocide activity. In pract ice Watson (1908) demons t r a t ed that doubl ing or halving the concen t ra t ion can have a more than propor t iona l effect on biocide activity.

Since we are deal ing with second or possibly h igher orders of react ion the re la t ionship between biocide activity and biocide is based upon second a n d h igher o rder react ions and can be s u m m a r i s e d as follows:

a n d since D = 2 • 303/K then

B"/K = Co

B"/K = C~

Thus if the death rates Kj and K2 are calcula ted for two biocide concent ra t ions , B t and B2:

BI"/K1 = B f / K 2

BI~/Bf = K I / K 2

(BI/B2)" = KI/K:

n = log (K1/K2)/log (B1/B2)

Similarly:

B, " X D~ = Bfl X D2

Bl"/Bfl = D2/DI

(BdB:y = D2/DI

n = log (D2/DO/log (B1/B2)

124 R. N. Smith

where:

DI and D2 =

K~ and K2 =

B1 a n d B : =

H

C o =

cA-=

Further:

and

decimal reduct ion times

death rates

biocide concentrat ions

concentra t ion coefficient or concentra t ion exponent

constant

another constant (CD × 2.303)

logK = l o g B . n - l o g C u

logD = logC ~ - l o g B . n

Thus if log K is plotted against biocide concentra t ion a linear relat ionship is obta ined whose slope is the concentra t ion exponent n.

If the log of D is plotted against the log of biocide concentrat ion a l inear relat ionship is obta ined whose slope will equal - n .

The concentra t ion exponent n is derived from the kinetics of second and higher orders of reaction and for such reactions is equal to the order of reaction minus one. It can be used to demonst ra te the effect of changes in concentra t ion on the rate o f a biocide reaction. Chemica l reactions of orders greater than three are unknown, but nevertheless concentra t ion exponents for biocides ranging from 0-5 to 19 regularly occur. Examples of concentra t ion exponents are as follows:

Formaldehyde 1 Quar ternary a m m o n i u m c o m p o u n d s 1-2.5 Phenols 4-7 Alcohol 1-19

The implicat ions of the concentra t ion exponent are of considerable practical importance. Chang ing the concentra t ion of a biocide will change the activity of the biocide by an a m o u n t equal to the proport ional change in concentra t ion raised to the power of the concentra t ion exponent . Thus a high concentra t ion exponent will magnify many times the effect of concentra t ion changes on biocide activity. For example if the concentra t ion exponent of a biocide is 6, halving the concentra t ion will increase the D value or reduce the death rate K by 26 which results in sixty-four fold change.


R E F E R E N C E S

Fowler, J. F. (1964). Differences in survival curves for formal multi-target and multi-hit models. Physics Medical Biology, 9, 177-89.

Meynell, G. G. & Meynell, E. (1970). Theory and Practice of Experimental Bacteriology. Cambridge University Press, Cambridge.

Russell, A. D., Hugo, W. B. & Ayliffe, G. A. J. (1982). Principles and Practice of Disinfection, Preservation and Sterilisation. Blackwell Scientific Publications, London.

Watson, H. E. (1908). A note on the variation of the rate of disinfection with change in concentration of the disinfectant. Journal of Hygiene, Cambridge, 8, 536-40.

kinetics of biocide kill

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