564
13 Modeling Biological Systems
13.3 Optimality Conditions Applied to Organized Systems
13.3.1 Specifying Optimality Conditions
In dynamical systems undergoing ordering, those subsystems that optimize the
efficient utilization of available energy and that optimize the production of critical
structures often have tested a wide variety of competing dynamical pathways. Those
pathways which optimize conditions for faster and more robust reactions will be the
most successful, i.e. they will become dominant. In addition to the level of activity,
the conditions of quasi-stability, the degree of adaptability, the extent of viability,
and the ease of reproducibility come into the optimization process of life systems.
A variety of approximation schemes have been applied to find the state of
relevant variables which minimize functions, such as power consumption, cost, or
the so-called ‘fitness function’. Finding the extremes of functions and of data is an
important challenge in many fields of study, including mathematical biology, biochemical pathway studies, network analysis, ecology, neuronal connection studies,
statistics, and complex systems. (See Appendix I.)
Optimization conditions can often be expressed as the minimization of a ‘cost’
function, f (x), under a set of n constraints g l (x) = 0. Here, the {x} ≡
{x 1 , x 2 , · · · , x N } is a set of N independent variables which affect the cost, with
1 ≤ l ≤ n < N. Analytic techniques exist to minimize known functions under
constraints. One of the most useful is the ‘Lagrange Multiplier Method’ (see
Appendix Sect. I.1).
If the function has a large number of variables with many minima, finding those
with function values below a certain level can be a daunting task. For example,
a long protein chain of amino acids will have many possible flexation and rotation
loci along the protein chain, making an astronomically large number of distinct final
folds, all with a particular folded energy. Finding what nature picks, or what might
be a useful medicinal protein, requires optimization strategies. Some methods used
are described in Appendix Sect. I.2.
To show the utility of optimization methods for relatively simple examples, we
will examine the questions: Are there optimal arterial radii and bifurcation angles in
animals? and: Can one explain the known scaling relationship between an animal’s
metabolism and its mass?
13.3.2 Optimized Arterial Radius and Bifurcation Angles
The purpose of an artery is to act as a conduit of blood for the delivery of oxygen,
nutrients, hormones and other compounds to cells, as well as to transport immune
cells and to carry away cell waste material. The heart must pump blood with a flow
sufficient to supply each cell of the animal with the needed ingredients to maintain
its life and functions. To be efficient, the body has evolved to minimize its power
13 Modeling Biological Systems
13.3 Optimality Conditions Applied to Organized Systems
13.3.1 Specifying Optimality Conditions
In dynamical systems undergoing ordering, those subsystems that optimize the
efficient utilization of available energy and that optimize the production of critical
structures often have tested a wide variety of competing dynamical pathways. Those
pathways which optimize conditions for faster and more robust reactions will be the
most successful, i.e. they will become dominant. In addition to the level of activity,
the conditions of quasi-stability, the degree of adaptability, the extent of viability,
and the ease of reproducibility come into the optimization process of life systems.
A variety of approximation schemes have been applied to find the state of
relevant variables which minimize functions, such as power consumption, cost, or
the so-called ‘fitness function’. Finding the extremes of functions and of data is an
important challenge in many fields of study, including mathematical biology, biochemical pathway studies, network analysis, ecology, neuronal connection studies,
statistics, and complex systems. (See Appendix I.)
Optimization conditions can often be expressed as the minimization of a ‘cost’
function, f (x), under a set of n constraints g l (x) = 0. Here, the {x} ≡
{x 1 , x 2 , · · · , x N } is a set of N independent variables which affect the cost, with
1 ≤ l ≤ n < N. Analytic techniques exist to minimize known functions under
constraints. One of the most useful is the ‘Lagrange Multiplier Method’ (see
Appendix Sect. I.1).
If the function has a large number of variables with many minima, finding those
with function values below a certain level can be a daunting task. For example,
a long protein chain of amino acids will have many possible flexation and rotation
loci along the protein chain, making an astronomically large number of distinct final
folds, all with a particular folded energy. Finding what nature picks, or what might
be a useful medicinal protein, requires optimization strategies. Some methods used
are described in Appendix Sect. I.2.
To show the utility of optimization methods for relatively simple examples, we
will examine the questions: Are there optimal arterial radii and bifurcation angles in
animals? and: Can one explain the known scaling relationship between an animal’s
metabolism and its mass?
13.3.2 Optimized Arterial Radius and Bifurcation Angles
The purpose of an artery is to act as a conduit of blood for the delivery of oxygen,
nutrients, hormones and other compounds to cells, as well as to transport immune
cells and to carry away cell waste material. The heart must pump blood with a flow
sufficient to supply each cell of the animal with the needed ingredients to maintain
its life and functions. To be efficient, the body has evolved to minimize its power