Investigating Biological Systems Using Modeling: Strategies and Software

By Meryl E. Wastney, Blossom H. Patterson, Oscar A. Linares and 2 more

Investigating Biological Systems Using Modeling describes how to apply software to analyze and interpret data from biological systems. It is written for students and investigators in lay person's terms, and will be a useful reference book and textbook on mathematical modeling in the design and interpretation of kinetic studies of biological systems. It describes the mathematical techniques of modeling and kinetic theory, and focuses on practical examples of analyzing data. The book also uses examples from the fields of physiology, biochemistry, nutrition, agriculture, pharmacology, and medicine.

• Contains practical descriptions of how to analyze kinetic data
• Provides examples of how to develop and use models
• Describes several software packages including SAAM/CONSAM
• Includes software with working models

• Language: English
• Publisher: Academic Press
• Release date: Dec 2, 2012
• ISBN: 9780080926452

Meryl E. Wastney

Meryl Wastney is an Associate Professor in the Department of Pediatrics and the Department of Biomathematics and Biostatistics at the Georgetown University Medical Center in Washington D.C. She received her Ph.D. in biochemistry from Lincoln College at the University of Canterbury in New Zealand. She was a Fogarty Fellow in the Laboratory of Mathematical Biology, NCI, NIH, Bethesda, Maryland for three years and joined the Department of Pediatrics at Georgetown University in 1983. She is the author of 30 articles and has presented over 40 invited lectures and workshops on modeling biological systems.

Book preview

modeling.

Preface

Ample evidence now exists that complex biological systems can only be progressively understood with the aid of mathematical models. Mathematical modeling requires an understanding of several disciplines: biology, computing, kinetics, mathematics, and statistics. While many texts cover these topics individually, previous textbooks do not integrate this information to enable an investigator without a strong mathematical background to apply modeling for hypothesis testing. This textbook is an attempt to meet this need. References are made to material covered in other textbooks.

The approach described in this textbook is often called the SAAM (simulation, analysis, and modeling) approach or Berman approach to mathematical modeling. Briefly, it involves the process of mathematical modeling on a computer as an aid to understanding biological system behavior [1]. This approach has guided the development of the WinSAAM (Windows SAAM) modeling software. The SAAM program and its conversational version, CONSAM, have been under development at the Laboratory of Experimental and Computational Biology, National Institutes of Health, since the late 1950s. WinSAAM evolved from the SAAM and CONSAM programs. It was designed for use by biologists. For example, compartmental models to describe experiments performed on biological systems can be set up and changed by simply specifying differential equation parameters and providing the initial conditions for the experiment. Models can also be solved by entering explicit equations. Parameter values can be changed during a solution to simulate variable experimental conditions. Data can be fitted by ordinary, generalized, or weighted least-squares regression techniques. Plotting and statistical measures of fit make it easy to compare and evaluate mathematical or compartmental models. WinSAAM incorporates Windows features for interapplication communication, making interacting through WinSAAM with other software as simple as Select|Copy|Paste.

Section I is a general introduction to the field of modeling biological systems. Section II describes modeling software and compares several packages. Section III explains the tools for modeling and illustrates them using examples. Section IV covers topics related to the design of experimental studies and the steps involved in modeling biological data. Finally, Section V describes how to evaluate and use published models.

The book is designed for students (all sections), experimentalists who have data to analyze (Sections II-IV) or who are planning kinetic studies (Sections IV and V), and for modelers as a resource (Section III). Models contain a plethora of information about a system, and the book is also designed for those who wish to access information inherent in models for designing experiments, for assessing the state of knowledge in a particular area, or for teaching complex biophysical principles (Section V). There is some repetition in the book as the description of topics focuses more on theoretical aspects in some sections and practical approaches in others.

The principles of developing models to interpret kinetic data apply across disciplines, and examples in the text are chosen from many fields, including chemistry, biochemistry, physiology, pharmacology, animal science, medicine, and agriculture. Modeling is part art and part science and is best understood by a hands-on approach. The CD supplied with this book includes the latest version of WinSAAM as well as the examples that have been used in the text. Readers are encouraged to use and adapt the models for their own areas of interest and to incorporate modeling as a routine tool for data analysis and in experimental design.

We acknowledge the contributions made by our families and colleagues during the writing of this book, and specifically Dr. Janet Novotny for proofing some of the chapters.

Without data models can’t exist …

Without models data can’t be used …

R. Boston

1998

References

1. Berman, M. The formulation and testing of models. Ann. N.Y. Acad. Sci.. 1963;108:182–194.

I

INTRODUCTION

1

WHAT IS MODELING?

This chapter will provide an introduction to modeling by defining terms and discussing modeling philosophy. Modeling straddles the fields of biology and mathematics. Therefore, it could be termed ‘Biomodeling’. However, it differs from, but is related to, the fields of Biomathematics, Biostatistics, and Bioengineering where mathematics, statistics, and engineering are applied to the study of biological systems. Biomodeling involves the use of these scientific tools, a knowledge of biology, intuition, imagination, and creativity to answer the question, what biological process could explain these data? Modeling is therefore part art and part science. The approaches, concepts, and tools can be taught to some extent, but the art of modeling can only be acquired through practice and experience.

I. Definitions

Models are simplified representations of systems. They can be physical such as scale models of airplanes, or abstract such as mathematical models. Mathematical models are used widely in the fields of engineering, physics, economics, business, and meteorology where models are routinely used in weather prediction. Models are useful for studying complex systems where many processes occur simultaneously. In the case of weather forecasting, such processes might include temperature changes, precipitation patterns and expected movement of high and low pressure systems. The processes and their interactions can be represented mathematically by a set of equations (or model). Then, by solving the equations simultaneously, the solution of the model will mimic the behavior of the system (e.g., the path of a hurricane). As evident from this example, models are developed because it may not be possible, or it may be too costly, to probe the real system. Mathematical models therefore predict the response or behavior of a system. They can be also be used to predict the response of a system prior to an experiment on the actual system.

Models are used in all fields of biology. Specific examples are: 1) to calculate nutrient intake for optimal growth, 2) to represent blood circulation, 3) to predict the pharmacological response to a drug, 4) to determine the rate of uptake of compounds by cells, and 5) to calculate enzyme kinetics.

Modeling is the process of developing a model or set of equations to simultaneously represent the structure and behavior of a system. Modeling biological systems differs fundamentally from modeling physical systems because the structure of physical systems is usually known, whereas, the structure of biological systems is generally not known. Models of biological systems are based on observations of the system (Fig 1.1). This process of determining the structure of a system based on its behavior is called the inverse problem. There are a number of limitations in modeling biological compared to physical systems; data are often incomplete due to the limitations on sampling sites; sampling times and number of studies that can be performed; data are imprecise; constraints related to the biology and the experimental techniques must be embedded in the analysis (3,4).

Fig 1.1 Biological modeling is the process of determining the structure of a system from its response.

Because the structure of a biological system is not known, a model developed to fit data from the system also represents a hypothesis of the system. Models therefore are not static, but evolve over time as studies provide new information that extend and refine the model. A good example is the development of models that describe lipoprotein kinetics (5).

Tip: The aim of modeling is to create a mathematical likeness’ of a system so that the model behaves in the same way as the system.

II. Approaches to modeling

There are three general approaches to modeling biological systems. They relate to whether the model is defined before or after a study, whether the model parameters are related to the structure (physiology or chemistry) of a system, and the complexity of the model. The approach chosen by an investigator depends on the purpose for modeling the system.

1) A priori versus post priori:

A priori (or theoretical) models are developed based on existing information about a system. For example, a model for glucose metabolism could be based on in vitro studies of the individual enzyme reactions. An example of this approach is described by Garfinkel et al (11). By contrast, post priori (or empirical) models are based on new observations. With this approach, a model for glucose metabolism would be developed by injecting labeled glucose in vivo, measuring its disappearance from blood, and proposing a model to fit the data. An example of this approach is described by Foster et al. (10).

2) Descriptive versus mechanistic

Descriptive models consist of mathematical functions or equations that fit the observed data. Parameters in the equations generally have no relationship to entities in the system. An example is an equation that is the sum of exponentials. Although descriptive models are usually simpler than mechanistic models (described below), they are based on assumptions that need to be verified. Some limitations of these models are that they may not use all the information that is available in the data and they are often limited to the time range of the data.

As stated by Murray, a description of data is not an explanation (13). Mechanistic models consist of functions where the parameters are related to entities of the system under study. These models are based on the mechanisms of the system and can be used to explore and understand the structure and properties of a system. Mechanistic models have the disadvantage that the system structure needs to be specified. However, there are advantages in using these models, e.g., all the information in the data can be used, experiments can be proposed based on the findings of the model, and the models can be used to analyze data obtained from two or more different states which may serve to identify where differences in the system are occurring (6). This is important for comparing disease versus healthy states, or the effects of treated versus untreated states. The results of using descriptive versus mechanistic models have been described for calcium metabolism (12).

3) Large versus reduced

The goal of model development is to find the simplest model (i.e., fewest number of parameters) to fit the observed data. Within this framework, a system could be modeled using a ‘minimal’ model with a small number of parameters, or by a ‘large’ model with more parameters to represent additional biological complexity of the system. There is generally a tradeoff between mathematical rigor of a model and its biological accuracy. The minimal or reduced approach will often provide parameter values that are well-determined with low errors, but each parameter may represent the combination of several processes. By contrast, parameters in a large model generally relate to individual processes, but some may not be well-determined. A large model can sometimes be reduced to a simpler model by combining parts of the system for calculating specific parameters of the system (5).

Tip: Terms are summarized in the glossary (Appendix 1).

III. Model Types

Models can be classified in several ways based on mathematical form (9). The form of a model chosen to investigate a system, its complexity, and how precisely it represents an actual system are all influenced by the purpose for which the model is developed. Some examples of different types of models applied to biological systems are found in Robson and Poppi (14).

1) ‘Model-independent’ versus model-dependent

Algebraic models are generally used to determine a specific parameter such as rate of clearance. They are often referred to as ‘model-independent’ because the value that is calculated does not depend on knowing the structure of the system. However, some underlying model structure is assumed in these calculations. For example, absorption of a nutrient is sometimes calculated algebraically as the difference between intake and excretion of a labeled form of the nutrient in feces. This model assumes however, that the nutrient is not absorbed and then excreted back into the intestine. ‘Model-dependent’ implies that calculations about a system are related to the type of model used.

2) Deterministic versus stochastic

Deterministic models are specified explicitly and do not permit any random variability (e.g., the law of gravity) while stochastic models include an element of randomness and are based on probability. Stochastic models are used for determining rates of production, disposal and residence time of a compound (15). However, these parameters can also be determined from compartmental models (5, 8), (See Chapter 14). Most biological systems include features of both. For example, the existence of a pathway in a model (such as absorption from the gut) is deterministic in that the pathway exists in all subjects, but the rate of absorption differs among subjects, and is therefore, variable.

3) Linear versus Non-linear

A system is linear if any combination of inputs yields the same combination of outputs (7), i.e., the amount of a substance that is moved from one location or state to another is proportional to the amount of substance present. Such a system is said to follow first-order kinetics. Most biological systems are non-linear. This means that the movement is not proportional to the amount of substance present (6). A gentle perturbation of a nonlinear system however behaves linearly. Radioactive tracers are useful for studying biological systems because they have negligible mass, do not perturb the system, and therefore have linear kinetics (4). (Stable isotope tracers contribute mass and their use may perturb a system. Tracers are described in more detail in Chapter 3.)

4) Kinetic versus Dynamic

A kinetic model characterizes a system for a particular state while dynamic models describe changes in a system as it moves from one state to another. Kinetic models are generally linear while dynamic models include non-linearities. For example, a kinetic model could be used to describe glucose metabolism in a subject under normal conditions but a dynamic model would be required to describe the changes in glucose metabolism after the ingestion of a glucose load. The dynamic model would describe the glucose-induced release of insulin and the effect of insulin on returning blood glucose to normal levels.

5) Compartmental versus non-compartmental

Compartmental models represent a system by a series of ordinary differential equations. Non-compartmental models include all other types (partial differential equations, algebraic, stochastic). Compartmental models are useful for describing biological systems because these systems are often visualized in terms of pools or compartments. For example, a compound may be bound or free, inside the cell or outside, in plasma or in an extravascular space, and each state could be represented in a model by a separate compartment.

IV. Compartmental Models

Compartmental models have been widely applied in the study of biological systems, particularly in relation to analysis of isotope data (1, 2). Compartmental models assume that the material of interest is distributed throughout the system in discrete entities, called compartments. A compartment is considered to contain material that is homogeneous, or kinetically indistinguishable. A compartment may be defined physically (e.g., a specific body pool), or conceptually (e.g., all particles that turn over at a particular rate). For example, if a drug is lost from blood monoexponentially (Fig 1.2), metabolism of the drug can be represented by a one compartment model (Fig 1.3).

Fig 1.2 Drug disappearance from blood: monoexponential loss. Note that the Y axis is log scale, so that the exponential is given by a straight line.

Fig 1.3 One compartment model for monoexponential loss. F(1) is compartment 1, IC(1) is the initial condition in compartment 1, and L(0,1) is the fractional rate of loss to the outside from compartment 1.

To show the relationship between a monoexponential model and compartmental model, differentiate the equation for the exponential loss. It can be seen that the rate of loss is a constant fraction (k) of the amount present;

(1.1)

Themodel (Eq. 1.1) can be expressed using WinSAAM notation (see Appendix 2);

(1.2)

where;

F(1,t) is material in compartment 1 at time t, and is often written as F(1), or Y in Eq. 1.1

L(0,1) is transfer of material from compartment 1 to the outside, or k in Eq 1.1.

Equation 1.2 is often written as,

(1.3)

andthe model is shown graphically in Fig 1.3.

If the loss from a system is biexponential (Fig 1.4) the system can be described by two compartments (Fig 1.5).

Fig 1.4 Drug disappearance from blood: biexponential loss.

Fig 1.5 A two-compartment model for biexponential loss.

The system is described by the following two equations;

(1.4)

In this case, the material is introduced into compartment 1 by bolus injection and is a compound that exchanges with compartment 2 (e.g., an IV drug input into plasma and taken up by the tissues). If we were interested in a compound that is synthesized by the system, such as glucose, the model would need to include this input, represented by U(i). The equations for the model become;

(1.5)

This book will focus on mechanistic compartmental models, i.e., data-based modeling. This emphasis is chosen because this approach can be used to investigate the structure and function of a system.

REFERENCES

1. Anderson, D.H. Lecture Notes in Biomathematics; Compartmental modeling and tracer kinetics. Berlin: Springer-Velag, 1983.

2. Atkins, G.I. Multicompartmental models in biological systems. London: Methuen, 1969.

3. Berman, M. Kinetic modeling in physiology. FEBS Letters. 1969;2:S56–S57.

4. Berman, M. 1971. Compartmental modeling. In Advances in Medical Physics. J. Laughlin and E. Webster., editors, Boston.

5. Berman, M. Kinetic analysis of turnover data. Progr. Biochem. Pharmacol.. 1979;15:67–108.

6. Berman, M. Kinetic analysis and modeling: Theory and applications to lipoproteins. In: Berman. M., Grundy. S.M., Howard. B.V., eds. Lipoprotein Kinetics and Modeling. NY: Academic Press; 1982:3–36.

7. Brownell, G.L., Berman, M., Robertson, J.S. Nomenclature for Tracer Kinetics. Int J Appl Radiat Isotop. 1968;19:249–262.

8. Covell, D.G., Bernam, M., DeLisi, C. Mean residence time Theoretical development, experimental determination, and practical use in tracer analysis. Math Biosci. 1984;72:213–244.

9. Finklestein, L., Carson, E.R. Mathematical Modelling of Dynamic Biological Systems. UK: John Wiley and Sons, 1985.

10. Foster, D.M., Hetenyi, G., Jr., Berman, M. A model for carbon kinetics among plasma alanine, lactate, and glucose. Am. J. Physiol.. 1980;239:E30–E38.

11. Garfinkel, D., Kulikowski, C.A., Soo, V.-W., Maclay, J., Achs, M.J. Modeling and artificial intelligence approaches to enzyme systems. Fed Proc. 1987;46:2481–2484.

12. Jung, A., Bartholdi, P., Mermillod, B., Reeve, J., Neer, R. Critical analysis of methods for analysing human calcium kinetics. J. Theor. Biol.. 1978;73:131–157.

13. Murray, J.D. Mathematical Biology. Berlin: Springer-Velag, 1993.

14. Robson, A.B., Poppi, D.B. Modeling Digestion and Metabolism in Farm Animals. Lincoln: Lincoln University,