OCS Study 

•

MMS 2001-043 

Sublethal Effects of Toxicants on 

Organisms: A Modeling Approach 

with Dynamic Energy Budgets 

Final Technical Summary 

Final Study Report 

U.S. Department of the Interior 

Minerals Management Service

Pacific OCS Region 

Sublethal Effects of Toxicants on 

Organisms: A Modeling Approach 

with Dynamic Energy Budgets 

Final Technical Summary 

Final Technical Report 

Authors 

Roger Nisbet 

Erik B. Muller 

Principal Investigators 

Prepared under MMS Cooperative 

Agreement Nos. 14-35-0001-30761 and 14-35-0001-30758 

by 

Southern California Educational Intiative and 

Coastal Marine Institute 

Marine Science Institute 

University of California 

Santa Barbara, CA 93106 

U.S. Department of the Interior 

Minerals Management Service Camarillo

Pacific OCS Region March 2001
Disclaimer 

This report has been reviewed by the Pacific Outer Continental Shelf Region, Minerals Management 

Service, U.S. Department of the Interior and approved for publication. The opinions, findings, 

conclusions, or recommendations in this report are those of the author, and do not necessarily reflect 

the views and policies of the Minerals Management Service. Mention of trade names or commercial 

products does not constitute an endorsement or recommendation for use. This report has not been 

edited for conformity with Minerals Management Service editorial standards. 

Availability of Report

Extra copies of the report may be obtained from: 

U.S. Dept. of the Interior 

Minerals Management Service 

Pacific OCS Region 

770 Paseo Camarillo 

Camarillo, CA 93010 

phone: 805-389-7621 

A PDF file of this report is available at: 

http://www.coastalresearchcenter.ucsb.edu/CMI/ 

Suggested Citation 

The suggested citation for this report is: 

Roger Nisbet and Erik Muller. Sublethal effects of toxicants on organisms: a modeling approach with 

dynamic energy budgets. MMS OCS Study 2001-043. Coastal Research Center, Marine Science 

Institute, University of California, Santa Barbara, California. MMS Cooperative Agreement Number 

14-35-0001-30761 and 14-35-0001-30758. 164 pages. 
Table of Contents 

FINAL TECHNICAL SUMMARY ....................................................................... 1 

FINAL TECHNICAL REPORT............................................................................ 5 

Introduction ............................................................................................................... 5 

Growth Models.......................................................................................................... 5 

Toxicity Models......................................................................................................... 6 

Modeling the Effect of Toxicants on the Parameters of Dynamic Energy 

 Budget Models..................................................................................................... 8 

Models Relating Individual and Population Response to Contaminants .................. 19 

From Molecules to Ecosystms Through Dynamic Energy Budget Models............... 24 

A Dynamic Energy Budget Model Based on Partitioning of Net Production ........... 39 

Survival and Production Variable Resource Environments ...................................... 65 

Stoichiometric Food Quality and Herbivore Dynamics ............................................ 92 

Sublethal Effects of Toxic Compounds on Dynamic Energy Budget Models: 

 Theory and Applications ..................................................................................... 114 

Predictive Power of Simple Population Models Based on Dynamic Energy 

 Budget Models..................................................................................................... 160 

A Dynamic Energy Budget Model Based on Partitioning of Net Production ........... 161 

Living with Variable Food; Survival and Production in a Dynamic Energy 

 Budget (DEB) Model for Individuals.................................................................. 162 

Modeling the Effects of Herbivore Stoichiometry on the Stability of 

 Plant-Herbivore Systems ..................................................................................... 163 

Final Technical Summary – Nisbet, Muller 

1 

FINAL TECHNICAL SUMMARY 

STUDY TITLES: 

Study I. Effects of Metal Toxicants on the Energy Budgets of Marine Organisms: A 

Modeling Approach 

Study II. Assessing Toxic Effects on Population Dynamics Using Individual-Based Energy 

Budget Models 

REPORT TITLE: Sublethal effects of toxicants on organisms: a modeling approach with 

dynamic energy budgets 

CONTRACT NUMBER: Study I: 14-35-0001-30761; Study II: 14-35-0001-30758 

SPONSORING OCS REGION: Pacific

APPLICABLE PLANNING AREA: Southern California

FISCAL YEAR(S) OF PROJECT FUNDING: Study I: FY95, FY96, FY97-FY98 no cost 

extension; Study II: FY 98, FY 99, FY 2000 no cost extension 

COMPLETION DATE OF THE REPORT: March 2001 

COST(S): Study I: FY 95 - $72,556, FY 96 - $75,030; Study II: FY 26,876, FY 22,824 

CUMULATIVE PROJECT COST: Study I: $147,586; Study II: $49,700 

PROJECT MANAGER: Russell J. Schmitt

AFFILIATION: University of California, Santa Barbara

PRINCIPAL INVESTIGATORS: 1

Roger Nisbet, 2Erik Muller 

ADDRESS: 1

Department of Ecology, Evolution and Marine Biology, University of 

California, Santa Barbara, CA 93106, 2Marine Science Institute, University of California, 

Santa Barbara, CA 93106

KEYWORDS: toxicity model; growth model; dynamic energy budgets; food quality; 

starvation; herbivore stoichiometry; produced water; synthesizing unit; Droop model; Monod 

model; mutliple limitations; 

BACKGROUND: This project and its follow-up were motivated by observations of 

Osenberg et al. (1992, Spatial scale of ecological effects associated with an open coast 

discharge of produced water. Pages 387-402. in J. P. Ray and F. R. Engelhardts, editors. 

Produced water: technological/environmental issues and solutions. Plenum Press, New 

York.) that the marine mussel Mytilus californianus grows and reproduces less in the vicinity 

of oil production platforms in the Santa Barbara Channel. Although they could not find 

increased environmental levels of contaminants common in produced water, they established 

that the mussels had accumulated barium in their shell to a level related to their distance to the 

nearest platform. Barium is a component of produced water, and the accumulation of this 

compound in mussels is a measure for the exposure of mussels to this and other compounds in 

produced water. Similar observations were made with other marine invertebrates. Their results 

show that impacts on biological variables were detectable over much greater distances than 
Sublethal Effects of Toxicants on Organisms: A Modeling Approach with Dynamic Energy Budgets

2 

physical variables. This implies that organisms integrate small changes in physiological 

properties over moderate time intervals and thereby produce significant effect. The results are 

important for two reasons. First, biological effects are necessary to detect very low levels of 

contamination. Second, population dynamics can be very sensitive to small changes in the 

physiological performance of individuals. This second point is particularly significant as 

direct ecological impacts at the level of populations and communities are commonly 

unmeasurable, especially for organisms with some dispersing life stage. 

OBJECTIVES: To develop models that describe the sublethal effect of toxicants on the vital 

rates of marine organisms. 

DESCRIPTION: Models for subletal effects of toxicants requires two components: a model 

describing the growth and reproduction of marine organisms in the absence of toxicants; and a 

model describing sublethal toxic effects for implementation in growth models. Both 

modeling components should be mechanistically justifiable, experimentally testable, and 

maximally general. The last point is especially important in light of the enormous number of 

possible combinations of species of organism and toxicant; moreover, it is rare to find a single 

pollutant in the environment, as usually a cocktail of toxicants is present. It is thus highly 

desirable to have one or a few models that fit many organism toxicant combinations. For the 

first modeling component, we developed and analyzed several dynamic energy budget (DEB) 

models. A DEB model describes the rates at which organisms assimilate and utilize energy 

from food for maintenance, growth, reproduction and development. These rates depend on 

the state of the organism (age, size, sex, nutritional status, etc.) and the state of its 

environment (food density, temperature, etc.). In DEB models, toxic effects show up as 

changes in parameter values. Although toxicants can have many different biochemical effects, 

it is feasible to summarize different modes of toxicant action through a few toxicity functions 

that describe alterations in the rates of feeding, respiration, growth and reproduction due to 

toxicant action, since the DEB modeling framework is sparse in parameters. 

SIGNIFICANT CONCLUSIONS: The most important finding from this study is that it is 

possible to describe many sublethal effects with a few simple toxicity functions. In these 

functions, toxic compounds affect the parameters determining the acquisition of energy and 

maintenance of viable physiological functions. Those parameters are affected simultaneously 

and the effect functions share the scaling (toxicity) parameter; thus, there is no need to 

distinguish effects on separate physiological parameters. Marine organisms included in the 

study are Mytilus californianus, M. edulis and Crassostrea gigas, toxicants included toluene, 

pentachlorophenol, tributyltin, mercury, copper, cadmium, various polyaromatic 

hydrocarbons, and produced water, and response variables included feeding rates, respiration 

rates, growth and reproduction. 

A second important conclusion is that care should be taken to interpolate observations about 

toxic effects on the level of individuals to predictions on populations and communities. 

Several compensatory mechanisms are in effect with the result that effects on the population 

level may appear at multiple trophic levels. 
Final Technical Summary – Nisbet, Muller 

3 

STUDY RESULTS: The study advanced along two lines: the development of DEB models 

and the development of toxicity models. There are two classes of DEB model: net assimilation 

and net-production models, the first class being at a more advanced state of development at 

the start of the project. To bring the net-production class of models on par with the netassimilation models, we developed a net-production model that describes the growth of all life 

stages of an organism (Lika and Nisbet, 2000). Furthermore, we explored the generality and 

testability of the most advanced net-assimilation model, the kappa-rule model, and its 

potential for use in ecotoxicology (Nisbet et al., 2000). Because food limitation and starvation 

phenomena are likely to aggrevate toxicological effects, we studied the dynamics of the 

kappa-rule model in a variable food environment (Muller and Nisbet, 2000). The model 

predicts that marine mussels have enhanced growth and reduced survival in strongly variable 

resource environments, and that mussels favoring growth over reproduction are more 

susceptible to stress (e.g., introduction of toxic compounds, food stress) than their 

conspecifics with a relatively high reproductive output. We also found that the quality of food 

(with regard to nutrient content) can have a profound effect on populations of herbivores and 

detritivores (Muller et al., in review). 

Although DEB models are sparse in parameters, the number of ways of including sublethal 

effects is still undesirably large, especially since information about which parameter may be 

affected by a certain toxicant is generally lacking. In an early contribution (Muller and Nisbet, 

1997), we explored the possibility of simplifying and generalizing toxic effects on parameters 

that determine the rates of energy allocation in an organism. In a subsequent manuscript 

(Muller and Nisbet, in preparation), we tested this generalization with a wide variety of 

combinations of organism and toxicant and show that it offers a good description of how 

toxicants change the rates of feeding, respiration and growth. Marine organisms included in 

the study are Mytilus californianus, M. edulis and Crassostrea gigas, toxicants included 

toluene, pentachlorophenol, tributyltin, mercury, copper, cadmium, various polyaromatic 

hydrocarbons, and produced water. We also found that the generalization gives a good 

description of changes in reproduction rates. These findings have implications for population 

dynamics, and we found that although the introduction of a toxicant always leads to an initial 

decline of production, the long term effect may be positive or negative, depending on 

compensatory mechanisms (Nisbet et al., 1997). To further investigate the potential effects of 

compensatory mechanisms, we performed a sensitivity analysis of the effects of contaminants 

on equilibrium densities in simple food chains (Nisbet et al., in prep). This work recognized 

that contaminants can affect rate processes at all trophic levels as well as influencing input 

and recycling of nutrients. 

STUDY PRODUCTS: 

Muller, E.B. and Nisbet, R.M. (1997) Modeling the Effect of Toxicants on the Parameters of 

Dynamic Energy Budget Models, In Environmental Toxicology and Risk Assessment: 

Modeling and Risk Assessment (Sixth Volume), p 71-81, F.James Dwyer, Thomas R. 

Doane and Mark L. Hinman Eds., American Society for Testing and Materials. 

Nisbet, R.M., Muller, E.B., Brooks, A.J. and Hosseini, P. (1997) Models relating individual 

and population response to contaminants. Environmental Modeling and Assessment 2, 7-

12. 
Sublethal Effects of Toxicants on Organisms: A Modeling Approach with Dynamic Energy Budgets

4 

Nisbet, R. M., Muller, E. B., Lika, K. and Kooijman, S. A. L. M. (2000) From molecules to 

ecosystems through dynamics energy budget models. Journal of Animal Ecology, 69:913-

926. 

Muller, E.B., Nisbet, R.M. (2000) Survival and Production in Variable Resource 

Environments. Bulletin of Mathematical Biology, 62:1163-1189. 

Lika, K, and Nisbet, R.M. (2000) A Dynamic Energy Budget Model Based on Partitioning of 

Net Production. Journal of Mathematical Biology, 41:361-386. 

Muller, E.B. and Nisbet, R.M. Sublethal effects of toxicants: a dynamic energy budget 

modeling approach compared to experimental results. In Preparation (manuscript 

included in full report). 

Muller, E.B., Nisbet, R.M., Kooijman, S.A.L.M., Elser, J.J. and McCauley, E. Stoichiometric 

Food Quality and Herbivore Dynamics. Under review with Ecology Letters. 
Final Study Report – Nisbet, Muller 

5 

FINAL STUDY REPORT 

INTRODUCTION 

This project and its follow-up were motivated by observations of Osenberg et al. (1992) that 

the marine mussel Mytilus californianus grows and reproduces less in the vicinity of oil 

production platforms. Although they could not find increased environmental levels of 

contaminants common in produced water, they established that the mussels had accumulated 

barium in their shell to a level related to their distance to the nearest platform. Barium is a 

component of produced water, and the accumulation of this compound in mussels is a 

measure for the exposure of mussels to this and other compounds in produced water. Thus, 

very low levels of pollutants in the environment may induce significant sublethal toxic effects. 

Our broad aim was to develop models that describe the sublethal effect of toxicants on the 

vital rates of marine organisms. 

We approached our problem along two distinct lines: 

• development of models that describe the growth and reproduction of marine organisms 

in absence of toxicants; 

• development of modules describing toxic effects that could be implemented in those 

growth models. 

Below we elaborate on the results of the two lines of investigation. In both lines we used three 

guiding principles: 

• the models should be mechanistically justifiable 

• the models should be experimentally testable; 

• the models should be maximally general. 

The first principle is required to maximize the predictive power of a model and to make its 

limitations explicit. The second point is obvious. The last point is very important in light of 

the enormous number of possible combinations of species of organism and toxicant. 

Moreover, it is rare to find a single pollutant in the environment; usually a cocktail of 

toxicants is present. It is thus highly desirable to have one or a few models that fit many 

organism toxicant combinations. The three guiding principles, however, may conflict one 

another. The first and second principle tend to make models more exclusive, while the last 

principle leads, by default, to inclusive models. The art of modeling is then to find the best 

compromise. 

GROWTH MODELS 

Organisms acquire energy from their environment and use it for growth and propagation. 

These and other expenditures are commonly modeled in terms of budgets. The simplest 

models assume a few fluxes that do not change over time, and use a mass or energy balance 

equation to analyze experimental results. A well-known example of this type of model in 

ecotoxicology is the “Scope For Growth” methodology. Although these models are useful in 

the analysis of toxicological data, they cannot be used to predict the effects of toxicants in a 

dynamic environment. More complex models use dynamic equations to describe the change of 

a potentially large number of many different budgets and fluxes. Models of both types abound 
Sublethal Effects of Toxicants on Organisms: A Modeling Approach with Dynamic Energy Budgets 

6 

in biology, and most are too specific as they decribe the particulars of a few genera only. Our 

interest is in simple dynamic models with a limited number of budgets, which we call 

dynamic energy budget (DEB) models. A DEB model describes the rates at which organisms 

assimilate and utilize energy from food for maintenance, growth, reproduction and 

development. These rates depend on the state of the organism (age, size, sex, nutritional 

status, etc.) and the state of its environment (food density, temperature, etc.). 

At the start of the project only one DEB model existed that met our guiding principle of 

maximum generality (see above), the kappa rule model, developed by Kooijman (1986) and 

subsequently generalized (Kooijman 2000). There are two classes of DEB model: net 

assimilation and net-production models. The kappa rule model is an example of the family of 

net assimilation models. To bring the net-production class of models on par with the netassimilation models, we developed a net-production model that describes the growth of all life 

stages of an organism (Lika and Nisbet, 2000; see below for copies of published papers, 

manuscripts and conference abstracts). Furthermore, we explored the generality and testability 

of the kappa-rule model and its potential for use in ecotoxicology (Nisbet et al., 2000). 

Although the kappa rule model has been used to describe the growth of many species, 

including many marine organisms, its implications for organisms in dynamic food 

environments remained largely obscure. Because food limitation and starvation phenomena 

are likely to aggrevate toxicological effects, we studied the dynamics of this model in a 

variable food environment (Muller and Nisbet, 2000). The model predicted enhanced growth 

and reduced survival in strongly variable environments. Food limitation can be experienced 

as a limitation of food quantity, which is the standard assumption in growth models, and as a 

limitation of food quality, especially for herbivores, such as mussels feeding on algae or 

diatoms. We explored the stoichiometric requirements of food for growth (Muller et al., in 

review). 

TOXICITY MODELS 

At the start of this project, toxicity models in a DEB modeling context dealt mainly with the 

accumulation of pollutants in organisms, i.e., without physiological toxic effects, or with the 

lethal effects of toxicants. We aimed at developing toxic effect functions describing sublethal

effects of toxicants. Toxicants can have many different biochemical effects, and it thus may 

seem difficult or impossible to fit many different toxic compounds in one modeling 

framework. However, DEB models provide such a framework, since those models are 

relatively sparse in parameters. In DEB models, toxic effects show up as changes in 

parameter values, and since those parameters are few in number, it is feasible to summarize 

different modes of action on the biochemical level through a few toxicity functions that 

describe alterations in the rates of feeding, respiration, growth and reproduction due to 

toxicant action. 

Although DEB models are sparse in parameters, the number of ways of including sublethal 

effects is still undesirably large, especially since information about which parameter may be 

affected by a certain toxicant is generally lacking. In an early contribution (Muller and Nisbet, 

1997), we explored the possibility of simplifying and generalizing toxic effects on parameters 

that determine the rates of energy allocation in an organism. In a subsequent manuscript 

(Muller and Nisbet, in preparation), we test this generalization with a wide variety of 
Final Study Report – Nisbet, Muller 

7 

combinations of organism and toxicant and show that it offers a good description of how 

toxicants changes the rates of feeding, respiration and growth. We also found that the 

generalization gives a good description of changes in reproduction rates. These findings have 

implications for population dynamics, and we found that although the introduction of a 

toxicant always leads to an initial decline of production, the long term effect may be positive 

or negative, depending on compensatory mechanisms (Nisbet et al., 1997). To further 

investigate the potential effects of compensatory mechanisms, we performed a sensitivity 

analysis of the effects of contaminants on equilibrium densities in simple food chains (Nisbet 

et al., in prep). This work recognized that contaminants can affect rate processes at all trophic 

levels as well as influencing input and recycling of nutrients. Model predictions are 

qualitatively consistent with data published by Carman et al. (2000), who found both high 

mortality of crustacean grazers, and elevated ammonium flux in diesel-contaminated benthic 

communities. 

REFERENCES 

Carman, K.R., Bianchi TS, and Kloep F (2000). Influence of grazing and nitrogen on benthic 

algal blooms in diesel fuel-contaminated saltmarsh sediments. Environ. Sci. Technol. 34: 

107-111. 

Lika, K., Nisbet, R. M. (2000). "A dynamic energy budget model based on partitioning of net 

production". Journal of Mathematical Biology, 41: 361-386. 

Muller, E. B., Nisbet, R. M. (2000). "Survival and production in variable environments". Bull. 

of Math. Biol., 62: 1163-1189. 

Nisbet, R.M., Muller, E.B., Brooks, A.J., Hosseini, P. (1997). “ Models relating individual 

and population response to contaminants”. Environmental Modeling and Assessment, 2: 

7-12. 

Nisbet, R.M., Muller, E.B., Lika, K., Kooijman, S.A.L.M. (2000). “From molecules to 

ecosystems through dynamic energy budget models". J. Anim. Ecol.,69: 913-926. 

Osenberg, C. W., Schmitt, R. J., Holbrook, S. J. and Canestro, D. 1992. Spatial scale of 

ecological effects associated with an open coast discharge of produced water. Pages 387-

402. in J. P. Ray and F. R. Engelhardts, editors. Produced water: 

technological/environmental issues and solutions. Plenum Press, New York. 
Sublethal Effects of Toxicants on Organisms: A Modeling Approach with Dynamic Energy Budgets 

8 

MODELING THE EFFECT OF TOXICANTS ON THE PARAMETERS OF 

DYNAMIC ENERGY BUDGET MODELS 

Erik B. Muller and Roger M. Nisbet 

REFERENCE: Muller, E.B. and Nisbet, R.M., “Modeling the effect of toxicants on the 

parameters of dynamic energy budget models,” 6th Symposium On Environmental 

Toxicology And Risk Assessment: Modeling And Risk Assessment, ASTM STP 622, J. 

Dwyer, T. Doane and M. Hinman Eds., American Society for Testing and Materials, 

Philadelphia, 1996. 

ABSTRACT: Toxicants negatively affect the rates of growth and reproduction of organisms. 

Dynamic energy budget models offer a convenient mathematical framework to describe 

growth and reproduction by individuals. Since these models take into account the lipid content 

of an animal, the accumulation of toxicants is easily incorporated. This paper deals with the 

subsequent effect of toxicants on growth and reproduction. We argue that the concept of noncompetitive inhibition is appropriate to describe the increased maintenance demands and 

reduced assimilation due to toxicant action. In this way, energy investment in growth and 

reproduction are indirectly reduced. 

KEYWORDS: toxicity model, energy budget, DEB, non-competitive inhibition, 

reproduction, toxicant. 

 Several research groups are currently engaged in developing mathematical models 

relating the effects of toxic compounds on individuals and populations of aquatic organisms 

[1, 2, 3]. The importance of this research is that many management decisions require insight 

into potential long-term effects of toxicants on populations, but much experimental 

information relates to measured, short-term effects on individuals. The research has two main 

parts: (i) modeling the effects on individual growth, reproduction and mortality of toxicantinduced changes in the rates of acquisition and utilization of energy by individual organisms, 

and (ii) determining the implications for population dynamics of these changes. 

 Toxicants affect organisms by changing some component(s) of the energy budget, as 

has been demonstrated by Scope For Growth studies [4, 5]. Dynamic energy budget (DEB) 

models describe the rules by which individual organisms assimilate and utilize energy from 

food [6, and references therein]. They incorporate feeding and assimilation rates dependent on 

the state of the individual and the environment, together with rules for energy allocation to 

maintenance, growth and reproduction (including priorities for energy allocation when food is 

scarce). Thus they provide a convenient mathematical framework within which to model the 

mechanisms whereby vital biological rates (growth, reproduction, respiration) are influenced 

by exposure to contaminants. 

 A recent theoretical study [7] has shown that the demography of a population of foodlimited organisms at equilibrium is very sensitive to details of the assumptions on their energy 

allocation priorities. The sensitivity arises because energy allocation priorities determine 

whether the primary effect of environmental stress is to change fecundity, juvenile 
Final Study Report – Nisbet, Muller 

9 

development time, juvenile mortality or adult mortality. That work was motivated by work on 

individual-based models of Daphnia populations [8, 9, 10], but the conclusions have wide 

applicability. In particular, there is potential for population dynamics to be affected, not only 

by the priorities for energy allocation, but also by the ways in which energy flows are 

influenced by toxicants. Thus we need to understand how to incorporate the effects of 

toxicants into DEB models. 

For any particular situation, we require three models: 

• a DEB model 

• a bioaccumulation model that describes the exchange of toxicants with the 

environment and their fate within an organism on the basis of physical properties 

(e.g. lipophilicity and affinity for ligands) 

• a model that describes the effects of toxicants on the parameters of the dynamic 

energy budget model. 

The first two models are easily combined, as has been shown previously [6, 11]. This is 

because the state variables of the first model (stored energy density and a measure of size) 

relate to the lipid and aqueous fractions, which are called compartments in a bioaccumulation 

model. We are developing the third model with three requirements: it should be 

mechanistically underpinned, it should be applicable to a wide range of toxicants and it should 

be mathematically tractable. This paper is a preliminary report of progress towards this 

model. From experimental literature at different levels of biological organization, we argue 

that the concept of non-competitive inhibition is useful to represent the sub-lethal action of 

many toxicants, and show how the parameters of a DEB model will be affected. We illustrate 

our approach with some calculations on the effects of "produced water" (discharge from 

marine petroleum production) on the growth of mussels. 

DYNAMIC ENERGY BUDGET MODELS 

 There are two types of dynamic energy budget models, designated kappa rule model 

and net production model. It is beyond the scope of this paper to present them in detail; indepth expositions can be found elsewhere [2, 6]. Instead, some basic notions are given here 

for the case when an adult grows under no-starvation conditions. Figure 1 outlines the energy 

flows in the models. 

 Three types of biomass are distinguished: energy reserves, structural or core biomass 

and gonads. The state variables are stored energy density, e, and structural biovolume, V. Food 

is taken up and converted into energy at a rate that is a function of the food density in the 

environment and of the size of an animal. Energy is spent on maintenance (at a rate 

proportional to core biomass), growth (proportional to size increase) and reproduction. The 

principal differences lie in the rules for energy allocation. In the net production model, 

maintenance demands are first debited from assimilated energy; what is left is partitioned 

between growth and energy reserves, depending how much energy is stored in the reserves. 

When the energy density exceeds a threshold value etr, energy is allocated from the reserves to 

reproduction at a rate σ proportional to the excess stored energy density. In the kappa rule 

model, assimilated energy is first added to the energy reserves. A fixed fraction κ of the 

energy flowing out of the reserves is spent on maintenance and growth, whereby maintenance 

demands take priority. The remainder is then used for reproduction. 
Sublethal Effects of Toxicants on Organisms: A Modeling Approach with Dynamic Energy Budgets 

10

FOOD STORAGE

STRUCTURAL

BIOMASS

GONADS

uptake & assimilation

growth

maintenance

RESPIRATION reproduction

RESPIRATION

a

FOOD

STORAGE GONADS

reproduction

STRUCTURAL

BIOMASS uptake & assimilation

growth

maintenance RESPIRATION

b

RESPIRATION

FIG. 1 Overview of energy flows (italics) and sources or sinks (bold) in an adult at non-starvation conditions 

as they are considered by the two dynamic energy budget models: (a) kappa rule model; (b) net 

production model. The dashed lines designate the overhead costs involved in assimilation, growth and 

reproduction. 

TABLE Dynamics in a growing and reproducing adult. 

KAPPA RULE MODEL NET PRODUCTION MODEL 

 

scaled stored energy density: 

de

dt = υV−1/3( ) f − e de

dt = 1 − e

3 1+

e

γ





 



 



 



 υf

V1/3 − mγ    

   − σ( ) e − etr

growth: 

dV

dt = eυV2/3 − mgV

e + g

dV

dt = e

3 υf

γ

V2/3 − mV 



 



 

scaled energy to reproduction: 

derm

dt = ( ) 1 −κVm

ge

υV2/3 ( ) − mV

( ) g + e − mgVp

 

 

 

 

 

 

derm

dt = σVVm

e − etr ( )
Final Study Report – Nisbet, Muller 

11

 The resulting dynamics of growth, scaled reproduction (erm) and stored energy density 

are summarized in the Table. They are given to show how the dynamics are affected when 

toxicants are present (see below). The symbols not yet introduced, m, g, γ, υ and Vm, are 

compound parameters reflecting maintenance costs, growth costs in the kappa rule model, 

growth costs in the net production model, the maximal assimilation rate and the maximal 

volume an organism will reach at unlimited food conditions, respectively. Finally, f is the 

scaled food density and Vp is the volume of an individual reaching adulthood. 

MODELING THE EFFECTS OF TOXICANTS 

 To model the effects of toxicants on growth and reproduction, a mechanistic concept is 

needed that can be used for many toxicants. This section focuses on what toxicants have in 

common from an energetic point of view, and how their action might be modeled in a 

dynamic energy budget framework. Toxicants that are primarily mutagens and other 

irreversible damaging agents are not dealt with here. It should be noted, though, that they 

affect mortality and the production of viable offspring and may therefore play an important 

role on the population level. 

 Many toxicants of ecotoxicological interest tend to interact with macromolecules. 

Cadmium, mercury and other heavy metals, either in ionic form or bound to an organic 

compound, bind to proteins [12]. This is because they have a high affinity for ligands, such as 

sulphydryl groups in amino acids, and may consequently disrupt the sulfur bridges in proteins. 

As a result, heavy metals change the tertiary conformation of enzymes, compete with the 

proper cofactor during synthesis of metallo-enzymes, and thereby inhibit many enzymatic 

reactions. 

 Another important group of toxicants is the class of lipophilic compounds, which 

includes aromatic and aliphatic compounds but also the organo-metals mentioned before. 

They have a high affinity for the apolar fractions in an organism and, therefore, readily 

dissolve in biomembranes, where they are believed to practice their main toxic activities [13]. 

This is in line with what has been known from quantitative structure activity relationship 

(QSAR) studies, which is that an effect of a toxicant is inversely related to its octanol-water 

partitioning coefficient [4]. Since this coefficient is proportional to the partitioning coefficient 

between cell membrane and buffer [14, 15], the octanol-water partitioning coefficient is often 

successfully applied in toxicity studies [3, 4, 11]. 

 Lipophilic compounds are toxic because they affect the membrane structure and the 

enzymes embedded therein [13]. Membranes containing these toxicants show an increased 

permeability to ions [15], so their function as a barrier is affected. This leads to the dissipation 

of energy and a reduction in metabolic rates [16, 17]. This is enhanced when the toxicant is a 

weak acid or base, so that it can cross the membrane in its undissociated, lipophilic form and 

then dissociates. In this way, protons are carried over the membrane. The interaction of 

enzymes with the membrane is affected by lipophilic toxicants causing enzymatic reactions to 

be hampered. The mechanism is poorly understood, but resembles the inhibitory effects of 

heavy metals. 

 So, many pollutants are toxic as a result of chemical affinities. This not only defines 

their partitioning behavior among body compartments, but also their effects on metabolism. In 
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the context of this paper, this means that the rates of energy fluxes and conversion efficiencies 

are reduced in the presence of toxicants. Consequently, we need assumptions to describe the 

effects of toxicants on the rate of energy fluxes and conversion efficiencies. 

 For the effect on energy fluxes, we propose to use the concept of the non-competitive 

inhibitor, which is a classic in enzyme kinetics [18]. This concept is based on MichaelisMenten kinetics but additionally assumes an inhibitor that reversibly binds to an enzyme, 

which completely inhibits the formation of a product. The binding of substrate and inhibitor 

are independent processes. If the chemical pseudo-equilibria are approached rapidly, the rate 

of the enzymatic reaction Vi is given by 

Vi = V0 1+

I

Ki





 



 

−1

 

 

where V0 is the rate in absence of inhibitors, I the inhibitor concentration and Ki the 

saturation constant. When more, independently acting inhibitors are present, the net rate is 

obtained by adding the respective fractions of concentrations and saturation constants to the 

denominator. 

40

60

80

100

120

140

0 5 10 15 20 25

ATP production rate in nmol O 2. g prot -1 . h -1

[Cd2+ ]

free in μM

FIG. 2 The rate of ATP production by mitochondria is inhibited by cadmium. The rate is appropriately 

described by non-competitive inhibition kinetics with Ki = 15.3 (±1.1) μM (see equation above). Data 

from Kesseler and Brand [19]. 
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 This relationship operates on the molecular level. Dynamic energy budget models, 

however, are focused on individual organisms. Two arguments support our proposal to use the 

concept of the non-competitive inhibitor in dynamic energy budget modeling. First, noncompetitive inhibition kinetics can be satisfactorily applied to systems that are more complex 

than single enzyme mixtures. Figure 2 illustrates that the inhibition of mitochondria by 

cadmium is appropriately described in a non-competitive way. Similarly, the respiration rate 

of cell suspensions with Nitrobacter winogradskyi is non-competitively inhibited by nitrous 

acid [20]. The second argument is that dynamic energy budget models treat the conductance 

of energy in a way that is conceptually similar to Michaelis-Menten kinetics. The link to noncompetitive inhibition kinetics is thus a natural one. 

 We also need to consider the effects of toxicants on energy conversion efficiencies. 

The efficiency of oxidative phosphorylation in potato tuber mitochondria is almost unchanged 

at low cadmium concentrations (see Figure 3), while the rate of ATP production drops 

immediately (see Figure 2). At higher cadmium concentrations, the efficiency is reduced, 

although not as drastically as the reduction in the ATP formation rate. This suggests that at 

relatively low toxicant concentrations the primary toxic effect is on rates. Because this effect 

is quite large, the toxicant concentration at which an animal cannot survive is relatively low 

(see below). Thus it is consistent with these data to assume that the toxic effect on conversion 

efficiencies is negligible at physiologically relevant toxicant concentrations, though of course 

other toxicants may have a more pronounced effect on conversion efficiencies. 

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25

Relative effective P/O ratio

[Cd2+ ]

free in μM

FIG. 3 Cadmium reduces the efficiency of oxidative phosphorylation, which is expressed as the fraction of 

ATP formation and oxygen consumption relative to the theoretical maximum stoichiometry. Data from 

Kesseler and Brand [19]. 
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 Having concluded that toxic effects are most likely to affect the rate parameters of the 

DEB models, we note that there are two parameters of that kind in both models: the energy 

conductance rate υ, which includes the assimilation rate, and the maintenance rate, m. The 

energy flux to growth is derived on the basis of balance equations and thus does not contain 

an independent rate parameter. This is also the case for energy used for reproduction in the 

kappa model formulation. The net production model, however, contains an independent 

reproduction rate parameter, σ. For the sake of simplicity, we assume that toxicants act 

equally adversely on all energy fluxes, thereby considering that the biochemistry involved has 

presumably a lot in common. When data show otherwise, there is, of course, no objection to 

differentiate between toxic effects on the rate parameters. Then, υ and σ are here simply 

multiplied by the same function of the toxicant concentration as were enzymatic rates before. 

The implementation of toxicants in the maintenance rate is more subtle. Maintenance 

demands are defined as the amount of energy required to keep an animal in a viable state for a 

period of time. This means that an animal cannot cut down on maintenance. To get sufficient 

energy to processes involved in maintenance, the maintenance rate should be multiplied by the 

inverse of the toxic effects function. The expressions for the rate parameters then become: 

υc = υ0

1+ c

mc = m0(1 + c)

σc = σ0

1 + c 

where c is the toxicant concentration scaled to the saturation constant. Just as for noncompetitive inhibitors, adding more toxicants is a matter of adding their scaled concentrations 

to the denominator. In an environment that is polluted with various toxicants, the final 

response will be dominated by the few compounds that are readily taken up and are present at 

the highest concentrations and are the most toxic. 

APPLYING THE TOXIC EFFECT FUNCTIONS

 To utilize the functions describing the effect of toxicants on rate parameters, we need 

to define the toxicant concentration c. Toxicants that are dissolved in adipose tissue are 

unlikely to exert direct effects. On the other hand, those dissolved in the lipid bilayer of 

membranes have a large impact. The simplest solution is to consider the target sites of 

toxicants as a part of structural biomass and assume that the water content of energy reserves 

and reproductive matter is negligible. The idea is that energy reserves and reproductive matter 

consist mainly of lipids and other insoluble storage materials [6]. The consequence is that 

energy reserves and reproductive matter are biochemically relatively inactive and thus hardly 

subject to toxic action. There are quite a few subtleties, though, involved in the mapping of 

biochemical composition to model variables [6], but these are beyond the scope of this paper. 

Then in accordance with others [1], the toxicant concentration in the aqueous fraction is the 

concentration in the toxic effect functions. This argument also covers lipophilic toxicants, 
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since the concentration in biomembranes is proportional to the aqueous concentration when 

equilibria are rapidly settled. 

 The toxicant concentration in the aqueous fraction can be described with a 

bioaccumulation model [11]. The key notions are that toxicant exchange with the environment 

is via the aqueous fraction, and that toxicants partition over body compartments, which 

depends on their affinity for the biochemical composition of the compartments. Since the 

compartment sizes are determined by the dynamic energy budget model, our effect functions 

fit in naturally under the restriction that the toxicant is not metabolized. 

 Some compounds have no effect when present at low concentrations. The maximum 

concentration at which no effect is observed is the no effect concentration. Some compounds 

are essential for growth but become toxic at higher concentrations. Among them are copper, 

zinc and nickel that are cofactors in metallo-enzymes. The concentration of such compounds 

in ionic form is controlled by metallothioneins that have a high affinity for such ions. Such 

proteins have also a high affinity for heavy metals without a known physiological function, 

such as cadmium and mercury. As a result, an organism may contain small concentrations of a 

heavy metal and yet show no physiological response. Whatever mechanism is behind the no 

effect concentration, it can be accounted for by correcting the concentration expected on the 

basis of partitioning behavior. The no effect concentration is then an additional parameter, 

which can be estimated to assess the risks of a toxicant [1]. 

3

3.2

3.4

3.6

3.8

4

0

0.0005

0.001

0.0015

0.002

0.0025

0 0.2 0.4 0.6 0.8 1

length in cm

scaled reproductive output

scaled toxicant concentration

FIG. 4 The final length and cumulative reproductive output of a mussel is indirectly reduced by toxic effects 

on the assimilation and maintenance rate. The results were calculated with simulations by assuming an 

initial 3 cm mussel in a constant environment during 120 days. 
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 To illustrate the impacts of toxicants, we predicted the growth and reproduction of the 

bay mussel Mytilus edulis with numerical methods. The purpose is to study how healthy, 3 cm 

long mussels that are outplanted near oil production platforms would be affected by toxicants 

in produced water [21]. For this we used the kappa rule model, which has been parameterized 

for bay mussels [22]. The environmental conditions and aqueous toxicant concentrations were 

supposed to be constant. As a result, the toxicant concentrations are scaled versions of each 

other and can be treated as a single variable. Furthermore, 10 % of the energy flowing out of 

the reserves is assumed to be used for reproduction and the scaled food density selected was 

80% of its maximum value. Then, the predicted length and reproductive output after 120 days 

is shown in Figure 4. It is obvious that growth and reproduction are strongly affected by 

toxicants, although indirectly via a decreased assimilation rate and increased maintenance 

requirements. A 3 cm long mussel cannot grow at scaled toxicant concentrations above 0.9. 

Then, the reproductive output drastically declines. This is because mussels are starving and 

need to cut down on reproduction in order to meet maintenance. At a scaled toxicant 

concentration slightly below 1, reproduction stops as well. From that concentration onwards, 

where the maintenance rate is doubled and the energy conductance rate halved, a 3 cm long 

mussel will die because of starvation. 

 

DISCUSSION 

 We started this paper by noting that a model of the effects of toxicants on growth and 

reproduction of organisms has 3 components: a DEB model, a bioaccumulation model, and a 

model of the effects of toxicants on the parameters of the DEB model. Previous work has 

addressed problems relating to the formulation of DEB models appropriate for application to 

toxicology [1, 2], and the formulation of bioaccumulation models [6, 11]. There is a much 

smaller literature on the third component, and the work reported here represents some 

preliminary ideas on that problem. 

 Our main conclusion is that the action of many toxicants are properly characterized by 

non-competitive inhibition. For this, we have analyzed data from experiments on the effects of 

toxicants at the level of single enzymes, organelles and organisms. The mechanism provides 

an underpinning for functional forms assumed by Kooijman and co-workers in recent research 

[1]. For example, they assumed that the first order Maclaurin expansion of an unknown 

toxicity function gives a satisfactory approximation under physiological relevant conditions. 

In other words, effects are linear in the toxicant concentration. This gives similar results to 

ours for the effect on maintenance rate, but for other rates gives a linear decrease which can 

only be valid at small concentrations; otherwise energy fluxes would become negative at high 

toxicant concentrations, which means that energy reserves are leaking into the environment. 

Here, ingestion and assimilation are irreversible processes even at relatively high toxicant 

concentrations, as they should be. 

 An increase in realism yields a model that can be applied to situations in which 

organisms are seriously affected by toxicants. Kooijman and co-workers have principally 

centered on no effect levels and included effects on conversion efficiencies. They analyzed 

data on growth and reproduction by assuming that toxic effects are exerted through one rate 

process or conversion efficiency at a time. For the estimation of the no effect concentration, 
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they have shown that it hardly matters which process is supposed to be affected. This is 

because the no effect concentration is to depend on when a mechanism starts to take effect 

rather than the mechanism itself. Hence, this approach has yielded a powerful tool to analyze 

standard toxicity tests. Our approach should be applicable to study the effects of toxicants on 

the growth and reproduction of individuals. 

 The next logical step is to test the model with experimental data on growth and 

reproduction. For this we will use data from mussels outplanted near an oil production 

platform [21]. Finally, we recall that this work was motivated by questions in population and 

ecosystem dynamics. It is possible to incorporate an assumption of non-competitive 

inhibition by a toxicant into a very simple biomass-based model of a plant-herbivore system. 

The result is a model that is capable of exhibiting a wide range of types of population 

dynamics (Nisbet and Hosseini, unpublished research). Of particular interest is the existence 

of a set of parameter values for which the fate of the system depends on initial conditions. 

This will also be a subject of future research. 

Acknowledgments We thank Andy Brooks, Bas Kooijman, Erik Noonburg, Craig Osenberg, 

Russ Schmitt and Will Wilson for discussions and advice. This research was supported by the 

Office of Naval Research (Grant No N00014-93-1-0952), and by the Minerals Management 

Service, U.S. Department of the Interior, under MMS Agreement No. 14-35-0001-30761. The 

views and conclusions contained in this document are those of the authors and should not be 

interperted as necessarily representing the official policies, either express or implied, of the 

U.S. Government. 

REFERENCES

[1] Kooijman, S.A.L.M. and Bedaux, J.J.M., “Analysis of toxicity test on Daphnia survival and 

reproduction”, Water Research in press. 

[2] Nisbet, R.M., Ross, A.H. and Brooks, A.J., “Emperically-based dynamic energy budget models: theory 

and an application to ecotoxicology”, Nonlinear World 3:85-106, 1996. 

[3] Lassiter, R.R. and Hallam, T.G., “Survival of the fattest: implications for accute effects of lipophilic 

chemicals on aquatic populations”, Environmental Toxicology and Chemistry 9:585-595, 1990. 

[4] Donkin, P., Widdows, J., Evans, S.V., Worral, C.M. and Carr, M., “Quantitative structure-activity 

relationships for the effect of hydrophobic organic chemicals on rate of feeding by mussels (Mytilus 

edulis), Aquatic Toxicology 15:277-294, 1989. 

[5] Widdows, J., Donkin, P., Brinsley, M.D., Evans, S.V., Salkeld, P.N., Franklin, A., Law, R.J. and 

Waldock, M.J., “Scope for growth and contaminant levels in North Sea mussels Mytilus edulis”, 

Marine Ecology Progress Series 127:131-148. 

[6] Kooijman, S.A.L.M., “Dynamic energy budgets in biological systems”, Cambridge University Press, 

UK, 1993. 

[7] Gurney, W.S.C., Middleton, D.A.J., Nisbet, R.M., McCauley, E., Murdoch, W.W. and De Roos, 

A.M., “Individual energetics and the equilibrium demography of structured populations”, Theoretical 

Population Biology in press. 

[8] Gurney, W.S.C., McCauley, E., Nisbet, R.M. and Murcdoch, W.W., “The physiological ecology of 

Daphnia: formulation and tests of a dynamic model of growth and reproduction”, Ecology 71:716-

732, 1990. 

[9] Nisbet, R.M., Gurney, W.S.C., Murdoch, W.W. and McCauley, E., “Structured population models: a 

toool for linking effects at individual and population level”, Biological Journal of The Linnean Society

37:79-99, 1989. 
Sublethal Effects of Toxicants on Organisms: A Modeling Approach with Dynamic Energy Budgets 

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[10] McCauley, E., Nisbet, R.M., De Roos, A.M., Murdoch, W.W. and Gurney, W.S.C., “Structured 

population models of herbiverous zooplankton”, Ecology in press. 

[11] Kooijman, S.A.L.M. and Van Haren, R.J.F., “Animal energy budgets affect the kinetics of 

xenobiotics”, Chemosphere 21:681-694, 1990. 

[12] Byczkowski, J.Z., “Effects of metal compounds on mitochondrial function: a review”, The Science of 

the Total Environment 37:133-162, 1984. 

[13] Sikkema, J., De Bont, J.A.M. and Poolman, B., “Mechanisms of membrane toxicity of hydrocarbons”, 

Microbiological Reviews 59:201-222, 1995. 

[14] Machleidt, H., Roth, S. and Seeman, P., “The hydrophobic expansion of erythrocyte membranes by the 

phenol anesthetics”, Biochimica et Biophysica Acta 255:178-189, 1972. 

[15] Seeman, P., “The membrane actions of anesthetics and tranquilizers”, Pharmacological Reviews

24:583-655, 1972. 

[16] Leão, C. and Van Uden, N., “Effects of ethanol and other alcohols an passive proton influx in the yeast 

Saccharomyces cerevisiae”, Biochimica et Biophysica Acta 774:43-48, 1984. 

[17] Cartwright, C.P., Juroszek, J.-R., Beavan, M.J., Ruby, F.M.S., De Morais, S.M.F. and Rose, A.H., 

“Ethanol dissipates the proton-motive force across the plasma membrane of Saccharomyces 

cerevisiae”, Journal of General Microbiology 132:369-377, 1986. 

[18] Segel, I.H., “Enzyme kinetics: behavior and analysis of rapid equilibrium and steady-state enzyme 

systems”, John Wiley & Sons, New York, 1993. 

[19] Kesseler, A. and Brand, M.D., “Localisation of the sites of action of cadmium on oxidative 

phosphorylation in potato tuber mitochondria using top-down elasticity analysis”, European Journal of 

Biochemistry 225:897-906, 1994. 

[20] Boon, B. and Laudelot, H., “Kinetics of nitrite oxidation by Nitrobacter winogradskyi”, Biochemical 

Journal 85:440-447, 1962. 

[21] Osenberg, C.W., Schmitt, R.J., Holbrook, S.J. and Canestro, D., “Spatial scale of ecological effects 

associated with an open coast discharge of produced water”, In: Produced Water (eds. Ray, J.R., and 

Engelhard, F.R.), Environmental Science Research 46:387-402, 1992. 

[22] Van Haren, R.J.F. and Kooijman, S.A.L.M., “Application of a dynamic energy budget model to 

Mytilus edulis (L.)”, Netherlands Journal of Sea Research 31:119-133. 
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FIGURE 2a 
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FIGURE 2b 
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FIGURE 3a 
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FIGURE 3b 
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FIGURE 4a 
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FIGURE 4b 
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FIGURE 6a 
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FIGURE 6b 
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FIGURE 7a 
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FIGURE 7b 
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FIGURE 8a 
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FIGURE 8b 
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The Department of the Interior Mission 

As the Nation's principal conservation agency, the Department of the Interior has responsibility for most 

of our nationally owned public lands and natural resources. This includes fostering sound use of our 

land and water resources; protecting our fish, wildlife, and biological diversity; preserving the 

environmental and cultural values of our national parks and historical places; and providing for the 

enjoyment of life through outdoor recreation. The Department assesses our energy and mineral 

resources and works to ensure that their development is in the best interests of all our people by 

encouraging stewardship and citizen participation in their care. The Department also has a major 

responsibility for American Indian reservation communities and for people who live in island territories 

under U.S. administration. 

The Minerals Management Service Mission 

As a bureau of the Department of the Interior, the Minerals Management Service's (MMS) primary 

responsibilities are to manage the mineral resources located on the Nation's Outer Continental Shelf 

(OCS), collect revenue from the Federal OCS and onshore Federal and Indian lands, and distribute 

those revenues. 

Moreover, in working to meet its responsibilities, the Offshore Minerals Management Program

administers the OCS competitive leasing program and oversees the safe and environmentally sound 

exploration and production of our Nation's offshore natural gas, oil and other mineral resources. The 

MMS Royalty Management Program meets its responsibilities by ensuring the efficient, timely and 

accurate collection and disbursement of revenue from mineral leasing and production due to Indian 

tribes and allottees, States and the U.S. Treasury. 

The MMS strives to fulfill its responsibilities through the general guiding principles of: (1) being 

responsive to the public's concerns and interests by maintaining a dialogue with all potentially affected 

parties and (2) carrying out its programs with an emphasis on working to enhance the quality of life for 

all Americans by lending MMS assistance and expertise to economic development and environmental 

protection.