Chaos In Medicine 1 [compatibility Mode]

CHAOS IN MEDICINE

Col K D Menon Medical Officer Defence Research & Development Organisation Instruments Research & Development Establishment Dehradun

CHAOS • Greek Mythology “The primeval emptiness of the universe before things came into being.” • Websters “A state of things in which chance is supreme.” • Traditional Notion of Chaos A condition of utter confusion, lacking in organisation, randomness etc.

CHAOS STUDY OF HOW SOMETHING CHANGES OVER TIME

• Easiest way is to make a graph. • e.g. - A baby’s weight changes over time. Wt = x y + c

Wt

b

a t0

t+1

Time

t+2

HYPOTHETICAL HOW WHEAT PRICES CHANGE OVER TIME

• Today’s price depends on yesterdays, tomorrow on today’s and so on. • A mathematical model using a hypothetical equation is used: x(t+1) = 1.9 – xt2 Where xt is the value of x at time t, x (t+1) value of x at time t+1 and 1.9 is an arbitrarily chosen constant. Let the value of x be 1. Then, 1 x(t+1) = 1.9 – xt2 = 1.9-1.0 = 0.9 2 x(t+1) = 1.9 – xt2 = 1.9 – (0.9)2 = 1.09

Input Value (xt)

New Value [x (t+1)]

1.0 0.9 1.09 0.712

0.9 1.09 0.712 1.393

Price

Time

THE GRAPH SHOWS 1. Complex unsystematic behavior including large sudden qualitative changes rather than a simple curve. 2. The indiscriminate pattern did not come from a haphazard process, but from a simple equation. It follows a rule : it is therefore Deterministic. 3. The equation that generated the chaotic behavior is simple. 4. Chaotic behavior came with just one variable. 5. The pattern is self generated. 6. No direct influence of sampling or measurement error is present.

Disorganized and complex looking behavior can come from an elementary Deterministic equation. Or In other words from a simple underlying Cause.

HISTORICAL DEVELOPMENT James Maxwell Clerk (1873) “ When an infinitely small change in the present state may bring about a finite difference in the state of the system in a finite time, the condition of the system is said to be unstable and renders impossible the prediction of future events, if our knowledge of the present state is only approximate and not accurate.” Henri Poincare (1909) Observed chaotic behavior while plotting planetary motions and trying to predict weather conditions.

EDWARD LORENZ (1917 –16 Apr 2008)

A Meteorologist with a PhD from Harvard’s while working on a weather model on a computer unintentionally fed in an input that had an error to the tune on 1 in1000. To his astonishment he found that the result was radically different from the one he had computed earlier. He concluded that small changes in the initial conditions can lead to highly diverse outcomes. This was the beginning of what is now known as

“The Chaos Theory”.

By the 1970’s with faster and more affordable computers these concepts started clearing up.

It is now realised that Chaos deals with something that evolves over time.

It was also realised that Chaos occurs only in Deterministic, Non-Linear, Dynamical Systems.

Deterministic

Because it follows a rule. It is bounded within well defined parameters.

System

Assemblage of interacting parts. A group or sequence of elements especially in the form of chronologically ordered set of data. e.g. Weather, Sensex,Wheat prices

Dynamical Systems Dynamics implies force, energy, motion or change. A Dynamical System is anything that moves, changes or evolves over time.

Change over time may be Discrete. ie separate or distinct eg earthquakes,volcanic eruptions Or Continuously eg air temperature, humidity Application of these concepts in mathematical modeling however needs special equations known as difference or discrete equations. Equations for discrete time changes are Difference Equations. Equations for continuous time changes are Differential Equations.

A Difference / Differential Equation is solved by a method known as Iteration. Iteration is a mathematical way of simulating discrete changes. To Iterate means to repeat an operation over and over. In Chaos

To apply the same equation repeatedly often with the outcome of one solution fed back as the input of the next.

This is the mathematical equivalent of Feedback. Positive Feedback amplifies the system. Negative Feedback dampens or inhibits the system.

NON-LINEARITY • Non linear means that that output is not directly proportional to input. • A change in one variable does not produce a proportional change or reaction in the related variables. • A system’s value at one time is not proportional to the values at an earlier time.

DIFFERENCES BETWEEN LINEARITY AND NON LINEARITY Linear

Non Linear

Behavior over time

Smooth and regular

Regular at first but often changes to erratic looking

Response to small changes in stimuli

Smooth and proportional

Usually greater than stimuli

Persistence of local pulses or signals

Decay or die over time

Highly coherent and persist for a long time

Coming back to Lorenz : “Imagine a rectangular slice of air heated from below and cooled from above by edges kept at constant temperature”. This is a simple model of the earth’s atmosphere. Lorenz described three time evolving variables: x y z

= = =

the convection flow the horizontal temperature distribution the vertical temperature distribution

and three parameters describing the character of the model: σ (sigma) p (rho)

= =

ß (beta)

=

ratio of viscosity to thermal conductivity temperature difference between top and bottom width to height ratio

And three differential equations describing the appropriate law of fluid dynamics: dx/dt = dy/dt = dz/dt =

σ (y-x) px – y – xz xy - ß z

Lorenz put this problem into the computer and in his own words went out for a cup of coffee and other things. The computer was instructed to repeat the equation again and again, using the output as the next input. It was also instructed to round off decimal points to the third place i.e 0.506127 as 0.506. This error of 1 part in 1000 should not have been significant. BUT IT WAS !

After enough time had elapsed this tiny error became an error as large as the range of possible solutions to the system.

Lorenz called this the “BUTTERFLY EFFECT”. “CAN THE FLAP OF A BUTTERFLY IN BRAZIL STIR UP A TORNADO IN TEXAS” Important because the flap of a butterfly wing is insignificant compared to the extreme weather condition of a tornado. Conclusion

In a Chaotic System the results depend very much on the initial conditions.

THIS IS KNOWN AS SENSITIVE DEPENDENCE ON INITIAL CONDITIONS

The next logical step was to do a multidimensional mapping using these numerical values. The results were like this

Starting from the initial conditions the calculations will look like the pathways in the pictures. Magnification of these lines will reveal that it is not one line but many. They are similar but not identical. The two voids in the picture represent the starting conditions. The lines near these remain close to this but the lines away from them are going further away. They map out an area of three dimensional space, but do not fill it. Yet they take up more than two dimensions. They have a dimension greater than 2 but less than 3. This is known as a fraction dimension.

A Fraction Dimension Shape is called a FRACTAL. Looking at the picture again we find that the initial conditions are stable. They are points in the system towards which neighboring points approach or are attracted and are called ATTRACTORS. An attractor is a set of states (points in the phase space), invariant under the dynamics, towards which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution. An attractor is defined as the smallest unit which cannot be itself decomposed into two or more attractors with distinct basins of attraction.

An Attractor with a fractal shape is called a: STRANGE ATTRACTOR Strange Attractors are hidden islands of stability or a subtle pattern of order at the heart of Chaos.

Lorenz’s Strange Attractor

What this means is – In a Non-Linear Dynamic System, a signal or pulse produces enormous change over a period of time which persist for quite some time even after the initial signal has decayed. But once the initial pulse or signal has been removed, the System slowly comes back towards the Strange Attractor.

NEW TERMS CHAOS DYNAMIC SYSTEMS NON LINEARITY DETERMINISTIC SENSITIVE DEPENDENCE ON INITIAL CONDITIONS • FRACTALS • STRANGE ATTRACTORS • • • • •

BENEFITS OF ANALYZING CHAOS Helps indicate whether haphazard looking fluctuations actually represent an orderly system in disguise. Greater accuracy in short term predictions. Reveal time limits of reliable predictions and identify conditions where long term predictions are largely meaningless. Recognizing Chaos makes modeling easier.

CONTROLLING CHAOS Can reveal circumstances which we might want to avoid. Guide a system out of it. Design a product or system to lead into or against it or control it or encourage it or enhance it or even exploit it. Fields where such work is going on Physiology, Study of Diseases, Neurological Disorders Weather Systems Encoding Electronic Data

Immune System

The immune system is an extraordinarily complex network of interacting cells and molecules (cytokines) that work together to ensure an appropriate response against a pathogen or toxin. It interacts with the environment and has memory. It can distinguish Self from Non-Self. Antigens Cytokines at the site of infection Migrating APC (Dendritic Cell/B Cell/Macrophage) MHC/Peptide (Epitope) complexes Transcription Factors T Cell Presentation T Cell Activation B Cell Activation T – B Cell Collaboration

Interaction

Knowledge about individual components vast, but how they interact is essential to understand functioning of the Immune System. Certain Findings that are Difficult to Explain Exponential increase in T cell response to IL 2. Unpredictable outcomes of cytokine gene knockout experiments such as a functional immune system in IL 2 -/- mice. Repeated rise and fall of TNF α levels in aqueous humor of eye following allogenic corneal transplantation.

Many scientists realised that mathematical modeling could give a better understanding.

Haemopoetic Stem Cell

T Cell

Thymus

Large Population of T Cells CD4- CD8-

CD4-CD8+ POSITIVE SELECTION

CD4+CD8+ CD4+CD8-

Adequate Affinity – Survive

CD4+CD8+ CD4+CD8+

SA/MHC II

SA/MHC

CD 4+

CD4+CD8+

CD 8+

SA/MHC I

NEGATIVE SELECTION

Presented with Self Antigens + MHC Complexes SA/MHC Low Affinity

Die by Apoptosis

Strong Interaction SA/MHC On APC Dendritic/ Macrophage

Low Affinity

Mature Naïve T Cell

Presented

SOME PERIPHERAL SELF PROTEIN ARE NEVER PRESENTED IN THE THYMUS (EXACT FRACTION UNKNOWN) SO T CELLS MAY BE STILL AUTO REACTIVE

Regulatory T Cell

SELF/ NON-SELF

STAT 4 T-bet

Activation

Th 1

CMI

I L 12

APC

I L 27 IL2

CD 80/CD 86

CD 28

MHC II

IL6 TGF ß

RORγt Activation STAT 3

Th 17 Other

Pathogens ? autoimmunity

TCR

Naïve CD4+T Cell

IL4

STAT 6 GATA 3

Th 2

Humoral

Activation

I L 10 / TGF ß

Th 3

? Adaptive Regulatory

Bone Marrow Tolerized

Mature B Cell

B Cell Signal 1

Pathogen Signal 2

Activated B Cell

Co stimulation T h 2 Cell

Plasma / Memory Cell

Antibody

Mathematical Modeling TCR Activation

Th 1 – Th 2 Differentiation

The inflammatory response seen in Autoimmunity is the final stage of a series of events that begins with TCR Activation. Usual explanation is high Affinity Bonding between an Antigen (peptide) that Cross-Reacts with a Self Antigen (ie cross-reacting Ag+MHC Complex). But persistence and prolongation of autoimmune states even after the cross reacting Antigen has gone has been noted. This has been difficult to explain.

T CELL RECEPTOR SIGNALING Present biological paradigm for TCR activation is a single TCR binding to a Peptide expressed on MHC I/II on an APC. But T Cells express many receptors of same specificity, while APC an array of different peptide-MHC. Estimated that CD 4 T Cells can respond to as few as 100-200 foreign complexes, while CD 8 T Cells may be able to respond to a single complex.

Important because TCR binds with all the different peptide-MHC complexes with some finite, but in some cases with low affinity. Yet T Cell can differentiate between the few high affinity interactions from majority of low ones. Since APC’s may present upto 105- 106 Complexes,each individual TCR must have a False Positive rate of less than 1 in 103 – 105.

This Specificity is seen in ability of T Cells to distinguish between altered peptide ligands differing by only one amino acid. What is significant however is the length of time the TCR is bound and not the affinity per se. Small changes in binding time of about 30 % can result in 1000 fold difference in the potency of the ligand.

MATHEMATICAL MODELING OF TCR ACTIVATION Uses the Kinetic Proof Reading Hypothesis. (Mechanism of Error Correction in Biological Process. First proposed by John Hopfield in 1974). Not Immediate Peptide/MHC

+

TCR

TCR Activation

Intermediate Steps ? Receptor Dimerization, ? Phoshporylation of Tyrosine residues, ? Docking of proteins containing src 2 homology region

Signal induction & TCR Activation

Time Delay Peptide/MHC

Signal Induction

TCR

Short Period Peptide/MHC

No Activation

TCR No Signal

Peptide/MHC

T Cell Activation

TCR

Longer Period Signal +

There maybe also nonlinear feedback between Kinase and Phosphatase molecules resulting in a Threshold for TCR Activation. Positive Feedback of protein phosphorylation events activated by TCR cells can result in Hysteresis enabling the TCR to act as a Bistable Switch. Hysteresis: Systems with Hysteresis can be summarised as a system that may be in a number of states, independent of the inputs.The prediction of the output is not possible without looking at the history of the input.

Mathematical Model As the TCR signal strength increases from left to right, at point †2 theTCR activation suddenly increases and the cell becomes activated (A)

In the reverse direction the T cell remains activated even when the signal has decreased below the threshold point at A, until the lower threshold point B is reached †1, when the T cell returns to its resting state. At a particular point of stimulus, the system has made a sudden jump. When the stimulus is decreased the jump back does not occur till a much lower value is reached. In between these regions, the system acts as an Bistable Switch.

Inferences

Low affinity binding may be sufficient to keep TCR in an activated state after an initial high affinity activation signal. Suggests that TCR interaction with low affinity self peptide MHC complexes may contribute to TCR activation and enhance T cell signaling. Auto Antigen activation could prolong an auto immune response after an initial high affinity signal to an antigen that cross reacts with a self antigen has gone. The model shows that TCR activation follows non-linearity with sensitive dependence on initial conditions, (ie the Chaos theory).

Th1 and Th2 Differentiation It has been noticed that in certain chronic conditions like chronic infectious states or chronic auto reactive states a marked depletion of Th1 cytokines and an increase in Th2 dominance is present. On other hand Th 1 dominance and Th 2 depletion is seen in certain other conditions. Th 1 Dominance

Th 2 Dominance

HIV & AIDS Lepromatous Leprosy TB, Leishmaniasis Prostate CA, Myelomas Lymphomas, Melanoma Rheumatoid Arthritis Multiple Sclerosis Diabetes Mellitus Type 1

Allergy Asthma SLE

Th1 and Th2 Differentiation Cytokines Cell Surface Signaling Molecules Intracellular Transcription Factors

All play a part

However overall process is still not understood clearly.

I L 4 binding to receptor on Thp cells

Signal Transducer & Activator of Transcription 6 (STAT 6)

+

Translocates to the nucleus Rapid induction of expression of GATA 3

_

Induction of Transcription Factor c-MAF Potent I L 4 Gene Specific Activator

Increased I L 4 Decreased T-bet production

Th 2 Differentiation

Similarly IL 12/IFN γ

STAT 1

+

GATA 3

T-bet

Th 1 Activation

Th 2 Activation Negative Feedback loop

Th 2 Activation

Th 1 Activation Negative Feedback loop

The fate of Th p cell differentiation depends on the dynamics of GATA 3 and T-bet expression within individual cells and the effect of Cytokines IFN γ & IL 4.

Mathematical Model of Single cell response to a Th2 signal by expressing GATA 3. x-axis = strength of external stimuli

Recently activated cells start with low levels of T-bet & GATA3.(A) Increasing stimuli till a threshold †2 increases GATA 3 steadily. At this level the cell rapidly reaches a state of high GATA 3 expression. (C) Auto activation of GATA 3 occurs at a maximum rate. At this level GATA 3 levels are relatively insensitive to external stimuli. In the region between †1 & †2 (B) even though below threshold level, the external stimuli is sufficient to sustain high levels of GATA 3.

The following points are highly significant: 1 Dynamical behavior of this model shows that T-bet – GATA 3 interaction acts as a bistable switch. 2 Th0/Thp

Th1/Th2 but not both Th1 signal induction

Th1 Th2 signal induction

Th2 3 Once a cell has been induced to produce and express T-bet/GATA3 it will continue to do so even when the extrinsic stimulus remains above the lower threshold but below the higher threshold.

Why This is Significant 1

The behavior of Th1 / Th2 system is non-linear, dynamic and deterministic. i.e. it follows the rules of the Chaos Theory.

2

The behavior of the system is sensitive on the initial values of cytokines and cells. i.e. Sensitive Dependence on Initial Conditions

3

It is proposed that the balance between Th1 /Th2 represent a stable state i.e. Strange Attractor.

4

Probably correcting the imbalance or moving the attractors can lead to improvement in the clinical picture of the disease. eg BCG / Interferon / I L 11 / Cellular Vaccination as non specific agents in melanoma, bladder carcinoma Others – Melatonin,Zinc,Selenium,Probiotic Bacteria Phytochemicals

Newtonian thought and medical science. There is a common view that biological sciences and physical sciences particularly mathematics and statistics are fundamentally different. Murray (1991) said, “If a mathematical model for any biological phenomenon is linear, it is almost certainly irrelevant from a biological viewpoint.” Descriptive Biology

Systems Biology

Theory of Relativity, The Quantum Theory, The Chaos Theory

“For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail.”

References 1. Robin E Callard, Andrew J Yates,Immunology and mathematics:crossing the divide, Immunology ,2005, 115, 21-23. 2. A Dalgleish, The relevance of non-linear mathematics (chaos theory) to the treatment of cancer, the role of the immune response and the potential for vaccines,Q J Med, 1999,92,347-359. 3. Gleick J, Chaos, London, Abacus 1987. 4. Personal communication from Dr Bindu M Krishna, Young Scientist Fellow (DST), Sophisticated Test & Instrumentation Centre, Cochin University of Science & Technology, Cochin, Kerala