RULE BOOK

Guidelines to a Mathematical Biology

1^st Edition

ENTERPRISE BIOLOGY SOFTWARE PROJECT, 2007

Medina, Washington USA

Copyright, 2007, By Robert P. Bolender

PREFACE

A mathematical biology depends on a rule-based approach; one that includes a well-crafted strategy for obtaining the best results. As we begin to define this new research landscape, rules can be very helpful in assembling discovery platforms for the life sciences by addressing standardization, validity, integration, prediction, and accountability. This book of rules suggests a pathway to a mathematical biology based largely on biological data, mathematics, and technology. Each rule is stated, explained with examples, discussed, and summarized with a list of advantages and caveats.

Biology consists of many interconnected and self-renewing parts arranged within a hierarchy of size. The arrangement is a curious one, wherein parts are contained within parts and where a given part may contain many similar or different parts likewise divisible into smaller parts. Here, the challenge becomes one of figuring out how to study these parts in such a setting with the goal of figuring out how everything works. There is, however, a small problem. As soon as we try to study these parts, their properties change in response to our intrusions and we end up increasing the complexity of an already complex task. Given the magnitude of this undertaking, one quickly discovers that mathematics and technology become our only options.

Biological parts arranged hierarchically behave in unusual ways that will not - at first - seem the least bit obvious. Did you know, for example, that to study a part one has to know at least three things about it - its volume, mean volume, and number. This means that each part comes with its own equation: Volume = meanVolume x Number. Why is it useful to know this? In a dynamic system, the parts are forever changing their volume, mean volume, and number, but at different times and in different directions. Imagine this. Put one part inside another part and now we have two interacting equations, one running as a subset of the other. By continuing this process up and down the hierarchy, we end up with a vast network of interacting parts and equations. Within this framework, a laboratory experiment becomes a portion of that network and when its results are added to the literature, they contribute a local piece to the global puzzle. In effect, a mathematical biology requires a comprehensive strategy for assembling and solving biological puzzles – big ones and small.

The good news is that we have reached a point in our story where we can introduce a set of standards and rules for studying these parts. The central challenge for the reader will be to discover the fundamental importance of biological organization, biochemistry, molecular biology, stereology, mathematics, and relational databases in building a mathematical biology – not as individual disciplines but as a unified hybrid science.

TABLE OF CONTENTS

Preface

1. Conceptual Framework

2. Sampling

3. Hierarchical Parts

4. Experiments as Equations

5. Optimal Data

6. Interpretations

7. Gold Standards

8. Connections

9. Change

10. Bias and Animal Variability

11. Counting Molecules

12. Complexity

13. Reverse and Forward Engineering

14. Integrating Data

15. Mathematical Phenotypes

16. Dimensional Consistency

17. Standardization

18. Universal Databases

19. Semiquantitative Data

Prologue

References

1. CONCEPTUAL FRAMEWORK

Rule 1. Treat biology as a mathematical entity, one subject to the laws of nature.

This is the pivotal rule, one that allows the remaining rules to fall logically into place. On closer inspection, however, the rule also gives us license to press the reset button and begin a new game. Consider, if you will, what really happens in our laboratories.

Whenever we run an experiment, two largely independent games are launched – one orchestrated by us the other by biology. Our research goal often consists of looking for changes in one or a few variables, wherein the measure of success often depends on an ability to detect such changes and to discuss their probable causes and implications. In contrast, biology plays a far more challenging game. It must recognize the mischief created by our experiment and then figure out how to design and implement an appropriate response. Whereas our goal may be a successful publication, biology’s goal may extend from making a few minor repairs to fighting for its very survival. Not surprisingly, these two games come with different rules and different outcomes. Biology plays according to the rules of nature, whereas most of us have been trained to play largely by man-made rules that often define a descriptive and semiquantitative biology. What does this tell us? As long as we continue to base our research on assumptions and semiquantitative methods, we will be playing the game but not playing to win.

Continuing the gaming analogy above, I would argue that we – as scientists - are playing the small game. In contrast, biology must always play the big one. However, things can change. A data-driven biology offers us the option of also playing in biology’s league, but only if we are willing to play by nature’s rules. Consider the difference. Instead of merely finding a change in one or a few variables, we can enjoy the privilege of watching biology craft ingenious solutions to an endless string of bewildering problems.

Here is the key question. When biology changes, what really changes? Everything. Guided by the steady hand of mathematics, our research methods can imitate real-world biology by allowing everything to become part of a common mathematical process. Quantitative methods can replace semiquantitative ones and our quantitative data can become connected and coherent with those of biology. Why should we be willing to settle for less?

But, can we explore biology as a mathematical science without being overwhelmed by a tangle of equations? Yes, of course we can. All we have to do is tap into the organizing principles of biology and let them work for us. This Rule Book outlines the strategy of such an approach, wherein the rules become the stepping-stones to discovery.

2. SAMPLING

Rule 2. Use unbiased sampling methods.

Unbiased sampling methods guarantee that every part (structure) under consideration has an equal chance of being sampled. This applies to parts of all sizes, ranging from macroscopic to microscopic – from organisms to molecules. A sample becomes representative only when it faithfully reflects all the parts of interest in the parent structure.

The point of unbiased sampling is to obtain a representative sample that can be extrapolated back to the original material. Such procedures provide dependable quantitative data. The method of sampling, however, determines what information can be recovered. Homogenization provides an unbiased sample, but the process forfeits most of the structural information and we are left largely with global (total) data. In contrast, sectioning intact structures forfeits one dimension of information that must be recovered with serial section reconstructions or stereological methods. In both cases, an adequate recovery depends on the validity of the experimental equation and on the data used to evaluate it.

Let’s look at some examples of homogenization and tissue sectioning.

A. Homogenization (Structural Order Minimized): The structural integrity of the original object containing the parts of interest is lost by the homogenizing process. Typically, this is the standard reductionist approach to studying molecules, wherein parts are separated and isolated.

A part – or a collection of parts – can be homogenized and an aliquot taken for analysis. Such an aliquot represents an unbiased sample, provided the data can be extrapolated back to the original material. In practice, the assay is repeated several times and the average taken. If the homogenate is fractionated, then the rules of analytical fractionation apply, wherein both recoveries and balance sheets will be needed to extrapolate the data (De Duve, 1974). Isolating parts and relating them to a mg of protein of the isolate cannot be expected to satisfy the unbiased sampling requirement.

Advantages

1. Access to molecules: Homogenizing biological material provides access to molecules originally sequestered in structural compartments.
2. Convenient counting of molecules: By extracting the molecules and suspending them in solution, their concentration can be determined using some type of an optical density (OD) measurement; absorbance, transmission, intensity, lumens, et cetera.
3. Automation: Analyzing the large data sets now being created by experiments in genetics, molecular biology, and biochemistry have been well-served by automating OD measurements.
4. Formation of data ratios: One of the best sources of order in biology can be found in relationship of one part to another. When two parts (e.g., molecules) are related to the same reference phenotype (a volume, surface, mg protein, gram of tissue, et cetera), the reference phenotype cancels, leaving a dimensionless ratio. Forming such a ratio also minimizes methodological bias and animal variability – provided the sampling is unbiased.

Caveats