Molecular Cell Physiology - MCDB 3280
3280 Home

Unit 16 - Self-Organizing Behavior,
Networks and Physiology

Systems and Networks are Central to Biology

“The greatest challenge today, not just in cell biology and ecology but in all of science, is the accurate and complete description of complex systems."

  • "Scientists have broken down many kinds of systems. They think they know most of the elements and forces."
  • "The next task is to reassemble them, at least in mathematical models that capture the key properties of the entire ensembles.” (E. O. Wilson, Consilience p.85. Knopf, New York, 1998)

"The thousands of components of a living cell are dynamically interconnected, so that the cell’s functional properties are ultimately encoded into a complex intracellular web [network] of molecular interactions."

  • "This is perhaps most evident with cellular metabolism, a fully connected biochemical network in which hundreds of metabolic substrates are densely integrated through biochemical reactions." (Ravasz E, et al.)

Significance of Network Theory

Many types of complex networks have similar, perhaps generic, properties when modeled mathematically.

Two points about the importantance of understanding network theory:

  1. The mathematics used to characterize types of networks provide a means to model and understand extremely complex biological interactions that might otherwise be too difficult to meaningfully approach.
     
  2. When a network displays organization it is likely there is no overall blueprint that sets up this organization. Rather, the demonstration that an organized network exists suggests trying to understand factors (rules) that govern the behavior of individuals in the network that lead to the overall structure.

Non-Random Networks are Ubiquitous

Many networks display similar, non-random, properties independent of the exact nature of their elements.

  • This allows generic approaches to be used across interdisciplinary boundries in circumstances as different as:
    • connections between web pages on the Internet
    • the behavior of a highly interconnected electric power grid, such as this section of the NY state power grid:


 

    • ecological ineractions, such as this food web:


 

    • the network of social interactions between individuals in a society (think of "six-degrees of separation")
    • the biochemical interactions that comprise a cell's metabolism
    • the interactions between genes in an organism. For example, here is part of the network that controls the cell cycle:

    • neuronal behavior in the highly interconnected neurons of the brain

Perhaps even more important, it can be often be shown shown that a small number of organizing principles lead to a network's organized structure.

  • That is, simple rules applied iteratively to the elements of a network can lead to a non-random, highly organized structure.
    • There is no rule that gives the structure itself, only rules that govern have individual elements behave.
    • Such self-organizing behavior may be able to explain--and help understand--otherwise almost mysterious, apparently chaotic complexity.

Understanding how a network evolves and behaves will be a very important step in the development of the fields of genomics, proteomics and systems physiology.

Systems Tend to Optimize

It appears that often the overall system represented by a network has been nearly optimized by natural selection (or human ingenuity). So, studying systems by examining them as networks networks may yield important general principles.

  • A striking example of such system selection is the optimal cruising speed of things that fly, as shown here:

  • Note the scaling over about 12 orders of magnitude.
  • "Shorter wings for speed and maneuverability (triangles) yield higher cruise speeds than those optimized for soaring (diamonds). Most systems (circles) are compromises." (Csete and Doyle)

Self-Organizing Behavior

Networks are a complex example of self-organizing behavior. Before examing them in detail, a few simpler examples of self-organizing behavior are described to develop an understanding of this line of theoretical modeling.

An example of self-organizing behavior:
An inorganic reaction that cycles

Inorganic chemical reactions: "The Belousov-Zhabotinsky reaction is a chemically active medium which maintains self-oscillations and [spiral] propagating waves. For an observer this is seen as a periodic change of color from red to blue, or a propagating blue wave on the red background." (source)

  • Above is one example of the B-Z reaction. Note the spital nature of the reaction cycle.

  • Here are snapshots of other B-Z reactions:

  • This type of behavior results from a oscillating chemical reaction such as this:

Ferroin is an indicator analogous to those
used to show an acid-base change.

It is both interesting and important to realize that such reactions can be modeled mathematically as elements in a regular grid that interact with each other according to simple rules.

The rules are stated abstractly as follows.

  • At any given step each element has a state say a,b,c...,z
  • In a 2D simulation, each element has six neighbors (except those at the edge)
  • The state of an element in step (X+1) is some function of the combined state of its neighbors in step X.

For example below is a 100 by 100 element grid at step 240 that follows comples rules. Note its similarity to the BZ reaction. Color indicates state of an element--element is also called a "cell". (source)

  • Here are the rules that generated this example:
    • "A cell in state 0 is said to be "healthy". A cell in state n is said to be "ill". All other cells, those in states 1, 2, ..., n-1, (n=250) are said to be "infected".
    • "If the cell is healthy then its new state is [a/k1] + [b/k2], where a is the number of infected cells among its neighbours, b is the number of ill cells among its neighbours, and k1 and k2 are constants.
      • [ ] means the integer part of the number enclosed, so that, for example, [7/3] = [2+1/3] = 2.
    • "If the cell is ill (i.e., in state n) then it miraculously becomes healthy (i.e. its state becomes 0)
    • "If the cell is infected then its new state is [s/(a+b+1)] + g, where a and b are as above, s is the sum of the states of the cell and of its neighbours and g is a constant. s/(a+b+1) is the average state of the cell and its neighbours, so
      • In other words, this rule states that the new state of a cell is the average of its state and its neighbours states plus a constant which may be thought of as the tendency of the infection to spread."

An example of self-organizing behavior:
The Game Of Life

Complex self-organization is explored in a simulation called the "Game of Life" (also called Cellular Automaton Theory) that has numerous Web sites associated with it (that are readily accessed by entering the term "Game of Life" into your favorite browser search engine).

In the simplest type of game here are the rules:

  • A dead cell is represented by a blank cell and a live cell is represented by a filled, colored cell
  • Natural Birth: A cell is born when it has exactly three live neighbors.
  • Natural Survival: A cell stays alive when it has two or three live neighbors.
  • Loneliness: A cell dies when it has less than two live neighbors.
  • Overcrowding: A cell dies when in has more than three live neighbors.
  • Here is an animated gif of a simple oscillating pattern that follows these rules:

  • Here is an oscillating patttern that's a bit more complicated:

An Example of Self-organizing Behavior:
Schooling of Fish

A school of fish seems to behave almost as a single organism as it twists and turns in response to the pressure of water currents and to the perception of predators.

It is almost certain that in most cases the school has no leader and no "intention" of going anywhere in particular.

Mathematical models based on simple rules can easily explain schooling patterns.

The rules are basically a varient of identical, genetically programmed behaviors of each individual fish such as:

  • Maintain an average distance from near neighbors
  • Swim in the average direction of near neighbors
  • Respond to intense and rapidly changing signals from perceptive organs such as the lateral-line-system

An actual example of a computer simulation based on such rules (shown to give you a feeling for how this is modeled) yields the following sequences of school behaviors in response to a predator:

  • This modeling is based on the following. Any other individual member of the school (i) within the region (r) causes repuilsion (move away from) to occur, in field (p) causes parallel swimming, and in region (a) causes attraction (movement towards).

  • The net movement is based on summation of all of the vectors R of all of the (n) individuals within any of these fields. (Further the angle w of the vector is taken into account systematically.)
  • Speed can also be factored in, and equations devised to represent all of these interactions. Varying the constants in the equations gives various types of schooling behavior.
    • For example the probability (p) of turning angle ø is:

is related by various equations to the summation of all of the vectors described above.

Mathematical Modeling of Networks

Some networks, such as this one are randomly connected:

Some networks are very regular, such as these where near neighbors are connected or all points connect to all others:

However, many large networks have a structure between these extremes because their elements are connected according to self-assembly rules of the types discussed above.

Mathematical modeling provides tools for understanding these non-random yet non-regular networks.

Small-World Networks

Some networks are of the small-world type.

  • In such a network there are numerous clusters of richly interconnected elements and a small number of connections between the clusters.
  • This type of network has been offered as a model of the "six-degrees-of-separation" concept.
  • A small world network falls between a regular and random network in its properties, as depicted in this figure:

  • p is the probability that a randomly chosed connection will be randomly redirected elsewhere (i.e., p=0 means nothing is changed, leaving the network regular; p=1 means every connection is changed and randomly reconnected, yielding complete randomness).
  • A small-world network can be generated from a regular one by randomly disconnecting a few points and randomly reconnecting them elsewhere.
  • Another way to think of a small world network is that some so-called 'shortcut' links are added to a regular network as shown here:

  • The added links are called shortcuts because they allow travel from node (a) to node (b), to occur in only 3 steps, instead of 5 in this example.
    • Another way to think of this is as follows. Phenomena such as "six-degrees" don't occur because everyone is connected to lots and lots of other people. Rather individuals are closely interconnected in local subwebs, and a small number of powerfully connective links between these subwebs allow rapid contact between them.
    • Here is a very diagrammatic depiction that makes the point of a few densely connected local nodes and shortcut links in a small world network this resembles the spoke-and-hub system of modern airline connections.

  • A network or order (0<p<1 as in the figure above) can be characterized by the average shortest length L(p) between any two points, and a clustering coefficient C(p) that measures the cliquishness of a typical neighbourhood (a local property).
    • These can be calculated from mathematical simulations and yield the following behavior (Watts and Strogatz):

    • Such modeling allowed testing against real-world phenomena by Watts and Strogatz.

    • In terms of L and C defined above, a small-world graph is much more highly clustered than an equally sparse random graph (C >> Crandom), and its characteristic path length L is close to the theoretical minimum shown by a random graph (L ~ Lrandom).
      • The reason a graph can have small L despite being highly clustered is that a few nodes connecting distant clusters are sufficient to lower L.
      • Because C changes little as small-worldnesss develops, it follows that small-worldness is a global graph property that cannot be found by studying local graph properties.

Metabolic Application

The following example (Wagner and Fell) of a complex metabolic network shows an application of small-world theory.

  • They modeled the known reactions of 287 substrates that represent the central routes of energy metabolism and small-molecule building block synthesis in E. coli. This included metabolic subpathways such as:
    • glycolysis
    • pentose phosphate and Entner-Doudoro pathways
    • glycogen metabolism
    • acetate production
    • glyoxalate and anaplerotic reactions
    • tricarboxylic acid cycle
    • oxidative phosphorylation
    • amino acid and polyamine biosynthesis
    • nucleotide and nucleoside biosynthesis
    • folate synthesis and 1-carbon metabolism
    • glycerol 3-phosphate and membrane lipids
    • riboflavin
    • coenzyme A
    • NAD(P)
    • porphyrins, haem and sirohaem
    • lipopolysaccharides and murein
    • pyrophosphate metabolism
    • transport reactions
    • glycerol 3-phosphateproduction
    • isoprenoid biosynthesis and quinone biosynthesis
  • These form a network because some compounds are part of more than one pathway and because most of them include common componants such as ATP and NADP.
  • These graphs show that considering either reactants or substrates, the clustering coefficient C>>Crandom, and the length coefficient L is near that of Lrandom, characteristics of a small world system.

  • Further they were able to determine key componants in the system by seeing which nodes were most highly connected. The top three (not including ATP or NAD) were glutamate, pyruvate and CoA.
  • They were also able to detemine which local "small-world" cluster was most central in terms of direct connections to other clusters. The answer was the TCA cycle.
  • In this test case the results agree with many other studies of E. coli metabolism. The power of the method lies in how it could help examine networks where individual rate equations and even substrate concentrations are not known in advance.
  • There are various mathematical methods that can be used to automatically assign the sub-communities in a large Web.

Scale-Free Networks

Often small-world networks are also what are called scale-free. In a scall-free networks the characteristic clustering is maintained even as the networks themselves grow arbitrarily large.

  • Here is a picture of a part of such a scale-free network

The mathematical properties and methods of analysis of such scale-free networks allow broad types of analysis, modeling and simulation.

  • Again, remember the point is that such types of analysis can lead to understanding key factors about the underlying physiology of the system that would otherwise either not be evident or be essentially impossible to adequately study.

In any real network some nodes are more highly connected than others.

  • P(k) is the proportion of nodes that have k-links.
    • For large, random graphs only a few nodes have a very small k and only a few have a very large k, leading to a bell-shaped Poisson distribution, such as this one:

    • Scale-free networks fall off more slowly and are more highly skewed than random ones due to the combination of small-world local highly connected neighborhoods and more 'shortcuts' than would be expected by chance.
      • Such networks are governed by a power law of the form

      • Because of this power law relationship, a log-log plot of P(k) versus k gives a straight line of slope

  • Some networks, especially small-world networks of modest size do not follow a power law, but are exponential, a point that can be significant when trying to understand the rules that underlie the network.

Self-Similarity and Fractals

A way to characterize a scale-free network is to note that it is self-similar, no matter how greatly it is magnified.

A familiar example of a scale-free, sel-similar situation is the case of fractals. Recall that any sub-part of a fractal looks like its larger parent, independent of scale. Here are some examples

Fractal Tree

source

Temporal Fractal Dynamics

Many biological processes display a fractal nature when examined temporally, as shown for human heart rate in the next figure.

There is interesting speculation (Goldberger et al.) as to why such fractal dynamics might arise.

  • "A defining feature of healthy function is adaptability, the capacity to respond to unpredictable stimuli and stresses. Functional plasticity requires a broad range of integrated outputs."
  • "Fractal physiology, exemplified by long-range correlations in the human heartbeat...may be adaptive from at least two perspectives
    • "long-range correlations serve as a (self-)organizing mechanism for highly complex processes that generate fluctuations across a wide range of time scales"
    • "the absence of a characteristic scale inhibits the emergence of highly periodic behaviors (mode-locking), which would greatly narrow functional responsiveness.
      • "This latter conjecture is supported by findings from life-threatening conditions such as heart failure, where the breakdown of fractal correlations is often accompanied by the emergence of a dominant mode."
      • "Transitions to strongly periodic dynamics are observed in many other pathologies, including Parkinson's disease (tremor), obstructive sleep apnea, sudden cardiac death, epilepsy, and fetal distress syndromes.
      • The paradoxical appearance of highly ordered dynamics with pathologic states (often termed "disorders") exemplifies the concept of complexity loss in disease and aging. this can be seen in the next figure (curves offset) which shows changes is the fractal pattern of heartbests in aging and disease:

Go the next page:

Scale-Free Examples