A model of phycotoxin release by cyanobacterial cells

Ecological Modelling 110 (1998) 105 – 117

A model of phycotoxin release by cyanobacterial cells A.P. Belov * Centre of En6ironmental Science and Technology, Uni6ersity of Glamorgan, Pontypridd, CF37 1DL, South Wales, UK Received 18 May 1997; accepted 24 December 1997

Abstract The paper presents a semi-empirical mathematical model for phycotoxin release into water during the development of the freshwater cyanobacterial population. The data obtained from field observation and laboratory experiments demonstrate correlation between the development of cell population and phycotoxin concentration in water (National Rivers Authority, 1990. Toxic Blue-Green Algae. A Report by the National Rivers Authority. Water Quality Series, September 1990. London, pp. 128). This information provides an approach to model phycotoxin release process. This probability for toxin production and release is used to derive an explicit formula for toxin concentration in water. This formula predicts the appearance of the peak toxicity in water between 3 and 6 months from the beginning of population growth. The peak toxin concentration in water and the time of its occurrence depend on environmental conditions and the toxin release function. The logistic equation with resources depletion is used to describe population development. The explicit model makes it possible to be calibrated, and assumptions about probability distributions being validated in light of laboratory experiments and field observations. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Cyanobacteria; Phycotoxin release; Mathematical modelling

1. Introduction Cyanobacteria (blue-green algae) are ancient phototrophic prokaryotes naturally occurring in most aquatic ecosystems, both freshwater and * Present address: Department of Mathematical Sciences, University of Montana, Missoula, Montana 59812-1032, USA. Tel.: + 1 406 2435311; fax: + 1 406 2432674; e-mail: [email protected]

marine. The cells have a number of specific abilities which allow them to adjust themselves to the environmental conditions they are living in. For example, they can regulate their buoyancy depending on light intensity and spectral composition (Reynolds and Walsby, 1975). Blooms of cyanobacteria are not a new phenomenon (Gerald of Wales, 1191), though there is a general perception that blooms result from water eutrophication. Blooms of cyanobacteria often follow the

0304-3800/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII S0304-3800(98)00013-1

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surface scums due to the buoyancy auto-regulation mechanism failure to follow environmental changes. Not all blooms are toxic but in many cases toxins are detected in water and there are reports of a number of events of mammal intoxication (National Rivers Authority, 1990). Buoyant cyanobacteria, including the genera Anabaena, Aphanizomenon, Microcystis and Oscillatoria are known to produce or are potentially able to produce phycotoxins of different types. The most common belong to two classes: neurotoxins, which include alkaloids and peptides, and hepatotoxins, which include cyclic peptides. A small proportion of the toxin pool can be excreted during normal cell growth, but large detectable quantities of these compounds are released into the water by dead and disintegrated cells at the end of cyanobacterial population development when the surface scum occurs. However, a substantial release of phycotoxins from growing cultures of Oscillatoria has been observed in laboratory experiments, (National Rivers Authority, 1990). Experiments demonstrate that laboratory cultures of P. Par6um produced the greatest amount of toxins during the late stage of logarithmic growth and continued into the stationary phase and also in the late stage of growth (Carmichael, 1986). In laboratory experiments for the batch culture of Microcystis Aeruginosa the toxin was essentially retained within cells during lag and growth phases. After maximal growth was attained much of the toxins is released into water as bloom occurs (Codd et al., 1989). The rapid change of physical and environmental parameters, as well as variations in solar radiation and the composition of dissolved chemicals, may also affect the production and secretion of phycotoxins by the healthy cells. Not much is known about mechanisms which cause and regulate toxin production and release into the water by both ‘non-toxic’ and toxic strains of cyanobacteria. As a result, the forecasting of the occurrence and the predictions of the ecological implications of phycotoxic bloom are also obscure. A phenomenological description of phycotoxin production and release could be helpful in the understanding of complex phenomena of the cyanobacterial blooms which nowadays are common in polluted lakes and shallow seas.

2. Model The well-known 1-D equation for the vertical distribution of the toxin concentration s(x,t) in the water column is as follows (Eq. (1)) (s(x,t) (s(x,t) + n(x,t) (t (x =





( (s(x,t) Ds + G(x,t;ji)− ms(s,t) (x (x

(1)

where n is the water vertical velocity component, G(x,t;j) is the toxin production function, Ds is the vertical diffusion coefficient of the toxic substance, m is the destruction constant of the toxic compound dissolved in water, and ji accounts for the stress factors which regulate phycotoxin production and its emission into water. There are reports that destruction of phycotoxins in water is a more complex process compared with linear first order kinetics. Reduction of toxin concentration in water may exhibit a lag-phase which may vary from 2 to 80 days (Jones et al., 1994). The rate of phycotoxin destruction in water can depend on water temperature, on presence of aquatic bacteria and dissolved chemicals such as copper compounds. For example, microcystin is biodegraded in natural waters from 2 to 4 weeks (Jones et al., 1994), which gives an estimate for microcystin decay m= 0.05 day − 1. The boundary and initial conditions for the Eq. (1) can be taken as (s(x,t) = 0, (x x = 0 s(x,t) t = 0 = 0

(s(x,t) = 0, (x x = − h (2)

which assume that destruction of the toxic compounds takes place in water, and that there is no flux of toxins through the lake bottom at x= − h, and the lake surface at x= 0. Toxins could be used by other lake inhabitants, such as aquatic bacteria, and also removed from water (e.g. due to adhesion onto suspended particles, gilvin and gelbstof). These processes are linear in respect of the toxin concentration, and may be accounted by the first order kinetics along with the toxin destruction process.

A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

The production function G(x,t;j) depends on the density function of the cell distribution f(t;x,t) in respect of the age of the population (Belov and Wiltshire, 1995): G(x,t;ji )=

&

t

g(t; ji)f(t;x,t)dt

(3)

tion from the beginning to the end of the community evolution are presented in Fig. 1. More accurately, the algal density n(x,t) may be assessed by integration over the biological life time interval of a cell, t + rather than the destruction time t  (Eq. (7)):

0

where the release function g(t;j) depends on the characteristic biologic life time t and the destruction time t  of the cyanobacterial cell. The age distribution of algae is related to the algal density distribution n(x,t). Therefore, the value of f(t;x,t)dt is the number of cells per unit of volume, which at the particular time t have an age between t and t +dt. Using the observational data about freshwater blue-green algal populations (Reynolds and Walsby, 1975; National Rivers Authority, 1990), it is possible to approximate the probability density distribution by the Beta-function: f(t;x,t)=N(x,t)t a(t  −t)b

(4)

where a =a(t) and b =b(t) are the parameters which depend on absolute time t which account for the diversions in the cell age distribution during the evolution of algal population. The beta-function probability density is thought to be a reasonable approximation to model probability of toxin release by the cyanobacterial cell because of the finite definition interval which correlates with a limited period of cell development, (National Rivers Authority, 1990). The norm-function N(x,t) is found applying the definition of the probability density distribution function (Eq. (4)): n(x,t)=

&

t

f(t;x,t)dt

(5)

0

The integration in (Eq. (5)) yields the relation between the density and the norm: N(x,t)=

n(x,t) B(a + 1,b +1)t a + b + 1

(6)

where B(x,y) is the Beta-function, and t  accounts for the mean time span between the birth and the disintegration of a cell. Characteristic demographic distributions for the algal popula-

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n(x,t)=

&

t+

f(t;x,t)dt

(7)

0

Consequently, in Eq. (6) for the norm-function N(x,t) the Beta-function B(a+1,b + 1) should be replaced by the incomplete Beta-function B1 + e (a+1,b + 1), (Abramovitz and Stegun, 1965): N(x,t)=

n(x,t) B1 + e (a+1,b + 1)t a + b + 1

(8)

where 1+ e= t + /t . In case of e 1, it is possible to use the Eq. (5) and Eq. (6). Otherwise, in the relevant expressions the Beta-function B(a+ 1,b +1) must be replaced by the incomplete Beta-function B1 + e (a+1,b +1), as in Eq. (8). The stress factors ji may depend on certain physical and environmental parameters such as (Eq. (9)): j1 = t +

   (n (x

m

,

j2 = t (T( ,G( ),

j3 = j3(cm ) (9)

where ((n/(x)m is the critical value of the density gradient, cm is the critical concentration of nutrient m, which may be an inorganic micronutrient such as Mo, Zn and Fe, and this variation can affect cell biochemistry. The change in concentration of macro-nutrients such as P, N, C can affect growth of cyanobacteria, and may also enhance production of toxin by fluctuation in time and space forming thin layers which contrast from the homogeneously dissolved nutrients in water. T( and G( are characteristic values of water temperature and solar radiation. Production of toxin by cyanobacteria is altered by temperature when it changes beyond upper and lower limits of temperature optimum. The effect of light intensity and spectral composition on toxicity is less obvious, and there are indications that the toxicity of M.

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Fig. 1.

A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

aeruginosa culture increases with illumination, although the decline in toxicity has also been recorded in the same conditions (van der Westhuizen and Eloff, 1985). The significant increase in toxicity is detected in phosphate starved cultures with more than normal concentration of toxin being found in water. The similar stimulation of toxicity is found as population growth decreases due to shortage of N (Carmichael, 1986). The density of bloom measured by biomass per unit of volume does not correlate with toxicity (ibid.). Sharp microgradients of density may cause fluctuations of light intensity which can influence photondriven biochemical processes in cell, and, therefore, to act as a stress factor. Generally, the decline of toxicity for cyanophyta takes place for cultures in nutrient rich medium at optimal temperature, illumination and pH, as cells are no longer under ‘stress’. Environmental conditions can have a considerable effect on toxin production (Carmichael, 1986). Using information about toxin release in water by the intact and disintegrated cells (National Rivers Authority, 1990), toxin release function may be modelled as follows, g(t;ji )=A(ji )t p(t  −t)q

(10)

where p=p(t + ,t ) and q =q(t + ,t ) are the parameters of the toxin release. The magnitude of p and q may also depend parametrically on the stress factors which can be evaluated from the experimental data. The dependency of p and q on stress factors may be evaluated from how these parameters can affect the probability of toxin release. Given the shortage of experimental data, it is possible to assume that concentration of micro-nutrients and physical factors such as T( and G( , and, implicitly, ((n/(x), influence the magnitude of p and q resulting in change of toxin release by the cell. Variation of these parameters, Figs. 3 and 4, indicate that for the fixed p and for q=8}16, release of toxin into water reaches a maximum for the old slowly growing intact cells, though for qB 6 most of toxin is released by the

109

old decaying cells. For the fixed q at the smaller magnitude of p, pB 2, there is almost uniform release of toxin during the cell’s life-span. Whereas for p\ 3, the maximum of emission shifts towards old decaying cells, Fig. 2. Because of complex nature of toxin production which involves metabolic and genetic mechanisms it can be assumed that microquantities of trace elements such as Zn, Hg and particularly heavy metals, and their complexes, may also affect toxin production rate. Generally, synergesis of various chemicals found in polluted water can act as a stress factor. Assessment of the factor A(j3) follows from the assumption that a ‘typical cell’ bears the characteristic properties of the population. The amount of phycotoxin which is produced and released during the life cycle of cyanobacterial cell is M(j3)=

&

t

g(t;j3)dt

(11)

0

and this output depends on the maturity of the population. From Eq. (10) and Eq. (11), it follows that A(j3)=

M(j3) B(p+ 1,q + 1)t p + q + 1

(12)

The average toxin amount M(j3) is specific to the kind of cyanobacteria and can depend on environmental conditions and the chemical composition of the lake water during the development of the population. The characteristic course of the toxin release into water which is based on the data presented in the NRA Report, (National Rivers Authority, 1990), Fig. 2, is shown in Fig. 3. Using Eq. (3), Eq. (4), Eq. (10), Eq. (11) and Eq. (12), we find the explicit form for the toxin production function G(x,t;ji), (Eq. (13)): G(x,t;ji)= n(x,t)M(j3) B(a(t)+ p+1,b(t)+q+ 1) B(a(t)+ 1,b(t)+ 1B(p +1,q + 1)t  (13)

Fig. 1. Example of the time-evolution of the cyanobacterial population demographic distribution. The demographic distribution function is scaled to: (a) the initial carrying capacity; (b) to the varying population density. The characteristic time for the resources depletion is T = 200 days, the depletion parameter D= 18; t is the absolute time in days and t is the non-dimensional age of a cell assuming the life-time of the cell as unit.

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Fig. 2. The course of toxin concentration in water during the cyanobacterial population evolution, t is the non-dimensional age of cells. NRA data, (National Rivers Authority, 1990).

where the production of toxins is correlated with the ageing of the cyanobacterial population. The equation for the averaged concentration of the phycotoxin in lake water is obtained by averaging Eq. (1) over the water column depth dS(t) =G( (t;ji )−mS(t) dt

S(t)=

where the phycotoxin concentration in water during the development of the population is (Eq. (15)):

&

0

s(x,t)dx

(15)

−h

and the average density of cells in water

&

1 0 n(x,t)dx (16) h −h while in the averaged production function G( (t;ji) the density n(x,t) must be replaced by n¯(t) from Eq. (16). Eq. (14) accounts for the boundary conditions (Eq. (2)), and the assumption that the hydrodynamic vertical velocity which is zero both n¯ (t)=

(14)

1 h

Fig. 3. (see right) Toxin emission into the water by a cyanobacterial cell. The non-dimensional time t for the cell’s life and it is supposed that biological life of cell covers 85% of the toxin release time. The non-dimensional toxin release function is shown as a function of the absolute time t in days; (a) for the parameter q, taking p= 12.4; (b) for the parameter p, taking q = 1.2.

A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

Fig. 3.

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A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

112

at the water surface and at the bottom of the water column, is applied. It is also assumed that, due to the continuity equation, the velocity divergence is nil.

3. Results The annual trend in toxin concentration S(t) depends on changes in the average density of algal cells n¯(t) in water and the parameters a(t) and b(t) as the functions of time. The density of the cells may be obtained from the dynamical model of the development of a cyanobacterial population in an eutrophic lake (Belov and Wiltshire, 1995; Belov and Giles, 1996). In order to evaluate water toxicity change, it is also possible to assume that biological growth of cells under conditions of limited available resources is provided by the classic Verhulst-Pearl equation, (Renshaw, 1993):



dn¯ (t) n¯ (t) =rn¯ (t) 1− dt n



(17)

where r is the intrinsic rate of natural increase for bacterial growth with unlimited resources. In the natural habitat, depletion of resources due to the exhaustion of nutrient pool and seasonal variations in environmental conditions can be accounted for by the decreasing carrying capacity n = n (t) of the population, which is assumed to be n (t) =

n0 1+ (ut)D

(18)

where n0 is the initial value of the carrying capacity, u$1/tD with tD the characteristic depletion time for the limiting level of available resources, and D is the power for the depletion rate. Properties of non-autonomous logistic equations are discussed by Hallam (1986). An example of the characteristic change of carrying capacity for cyanobacterial population in the lake is shown in Fig. 4. The variation of limiting resources in natural water is a complex process. The pool of nutrients is exploited not only by cyanobacteria but also by other phytoplankton and green algae. Cyclic development of con-

curring populations may have different cycles which result in the release of basic nutrients back into water within one season. There is a regular influx of nutrients into freshwater reservoirs due to human activity. Eq. (17) for deteriorating resources accounts for the exhaustion of available nutrients, micro-nutrients and seasonal weather change. Cyanobacterial population can end as a bloom, which may occur in the middle of summer or even in late spring. In this case catastrophic deterioration of habitat occurs due to self-development of cyanobacteria. Forecasting this event requires accounting for different environmental and hydrological factors including light-induced buoyancy regulation of cells, and is within the scope of the more complex model (Belov and Giles, 1996). Parameters u and D are the cumulative parameters which determine the variation of carrying capacity in equation for cyanobacterial population growth (Eq. (17)), when n (t) is taken in form of Cauchy’s distribution (Eq. (18)). In Eq. (18) parameters have explicit physical meaning as for u being the inverse resources depletion time and for D defining the rate of deterioration. For example, for D= 18 the carrying capacity falls from the maximal level to almost nil within 50 days, and is defined as D= ln((n /n )/ln(t/tD ), where dn = n0 − n . The depletion interval increases as D decreases, and the length of this interval depends on u. Parameters u and D may depend on transport, mixing and growth processes in water column. These processes have various characteristic times ranging from seconds to weeks. Parameters depend also on the population itself due to strong feedback because of the increase of population density which is affected by solar radiation and the availability of nutrients. To account explicitly for these processes the logistic Eq. (17) is too general, and much a more complex theory and model should be used. The parameters a(t) and b(t) are the power functions which represent the ageing of cyanobacterial populations. Some certainty about a and b dependence on time can be achieved assuming that the cell’s age probability distribution density follows the Eq. (4), and this determines the course of a(t) and b(t) to be

A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

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Fig. 4. Change of the carrying capacity n (t), where t in days is the absolute time, T is the resources depletion characteristic time in days, and the depletion parameter D= 18.

a(t) =





a

a −1 exp( − kt) a0 b2 b(t) =b1 − 1+b2exp( − kt) 1+

(19)

where a0 and a are the parameters of a(t) at the beginning and the end of population development; b1 and b2 are used to express the same values of b(t): b =b2 − b1

b2 b0 =b1 − 1 +b2

(20)

and k $1/td, where td is the characteristic time of the ’demographic’ trend in cyanobacterial population. The time td depends on the environmental conditions and the division rate of the cells. The accurate values of the parameters which are in a(t) and b(t) are to be adjusted in the light to

experimental data. As an example, the typical time courses of a(t) and b(t) are presented in Fig. 5. The solution of the modified Verhulst-Pearl Eq. (17) with the varying carrying capacity (Eq. (18)) is n¯ (t)= 1+ ED (t)+



n0

 n

n0 u − 1− r n0

D

D! exp(− kt) (21)

where for the integer D (Eq. (22)) ED (t)= D!

 u r

D D

(rt)D − k % (− 1)k (D−k)! k=0

(22)

If D is a non-integer, e.g. D= [D]+ e, where [D] is the integer part of D and 0BeB 1, the function ED(t) takes the form (Eq. (23))

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Fig. 5. Variation of the demographic parameters a and b during population development. The characteristic time trend in the age of cyanobacterial population is td, the absolute time t in days. (a) the parameter a(t;td), (b) the parameter b(t;td).

A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

115

Fig. 6. Variation of non-dimensional average cell population density in the water n¯ (t), where t is the absolute time in days, T is the resources depletion characteristic time in days, and D= 18, r =0.126.

ED (t)= G(D +1)

!

 u r

D

(rt)D − k % (−1)k G(D −k + 1) k=0

[D] − 1

+ (−1)[D]

"

(rt)1 + e M(1,2 + e,rt) G(2 + e)

(23)

where M(a,b,z) is the confluent hypergeometric function (Abramovitz and Stegun, 1965), and consequently, in the denominator of Eq. (21) the square brackets become simply [(n0 /n0)-1], without a factorial term. The expression for n¯(t) contains the five parameters n0 , n0, r, u and D where the first three represent the initial state of the algal population and the lake water, and the last two account for the diversity in habitat. All parameters are to be assessed from experimental data. Nevertheless, model calculations for the annual course of the cell’s density n¯(t) yield the scale for

the plausible values of these semi-empirical parameters. Fig. 6 gives an example of the annual development of the average cell density in water for a typical set of parameters. The solution of Eq. (14) for average toxin concentration in lake water yields: S(t) =

&

M(j3)exp(− mt) t n¯ (z) B(p+ 1,q + 1)t  0 B(a(z)+ p+ 1,b(t)+ q+ 1) dz B(a(z)+ 1,b((z)+ 1)

(24)

where functions a(t) and b(t) are given by Eq. (19), and function n¯(t) by (Eq. (20)). The result of the integration of Eq. (24) for the phycotoxin concentration in water, is shown in Fig. 7. As an example, for the characteristic time for depletion of resources TD $ 200 days the maximum values of water toxicity are expected in August/Septem-

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A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

Fig. 7. Variation of average non-dimensional toxin concentration S(t) in a lake water during the development of a cyanobacterial population, where t in days, is the absolute time, T is the resources depletion characteristic time in days, and D =14, r= 0.126, td = 40 days m − 1 =1.67 day.

ber if the algal population starts to grow in February. The growth of water toxicity follows the degradation of the algal population with a shift in time of about 15 – 30 days after the number of cells begins to decrease. This effect is a result of algal bloom which usually follows the surface scum. Commonly, this phenomenon takes place in the middle latitude lakes at the end of summer (National Rivers Authority, 1990). Cyanobacterial blooms and scum formation are characteristics of buoyant cyanobacterial cells (blue-green algae). The description of the population development is one objective for the dynamical model for evolution of cyanobacterial cells in nutrient saturated lake water (Belov and Wiltshire, 1995).

4. Conclusions The phenomenological model which is discussed above strongly depends on experimental data, both for the assessment of model parameters and the verification of the functional forms. These parameters should be obtained from the qualitative analysis of ecological data and the appropriate laboratory experiments. On the other hand, the toxin trend in water depends on the algal density n¯(t) which, in turn, should be obtained from the general dynamical model for the cyanobacterial population development (Belov and Wiltshire, 1995; Belov and Giles, 1996), or from the computer simulation models (Howard et al., 1991; Patterson et al., 1994; Sherman and Webster, 1994; Howard et al., 1996).

A.P. Belo6 / Ecological Modelling 110 (1998) 105–117

The explicit solution that is found in this paper provides an opportunity for the assessment of characteristic values of the parameters u, r, k, D, m, a, b, p and q. Values of M, n0 and t , t + remain uncertain, but, nevertheless, may be evaluated from the laboratory experiments with a particular specimen of cyanobacteria. The characteristic values of the first set of parameters that are used in the pilot calculations demonstrate the plausible course of the average density of cells n¯(t) and the water toxicity S(t). These functions are presented in the relative units and within the accuracy of the unknown scaling factors, as is shown in Figs. 6 and 7. Calibration of these graphs may be provided using appropriate ecological data. The other problem which should be studied in association with phycotoxin emission is the reliance of model parameters on the external environmental factors such as intensity and spectral composition of solar radiation, the spectrum of nutrient concentration in water, pH and the temperature of water. There is another source of the indirect dependence of parameters on periodic variations such as diurnal and seasonal cycles. Investigation of all these characteristics is an objective of a general model of the development of cyanobacterial population and further experiments. Nevertheless, the model provides a framework which may be used to prepare laboratory experiments to verify the assumptions made about probability distributions and the functional dependencies of model parameters.

Acknowledgements I am grateful to Professors R.J. Wiltshire and G.A. Codd for valuable and stimulating discussions.

.

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