Lectures Systems Biolgy

8/13/2019 Lectures Systems Biolgy

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six lectures on systems biology

jeremy gunawardenadepartment of systems biology

harvard medical school

lecture 4

7 april 2011

part 2 seminar room, department of genetics


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0. why mathematical models?

1. posttranslational modification of proteins

2. microscopic cybernetics

!. development and evolution

a rather provisional syllabus


3/30

functional purpose of molecular complexity?


4/30

functional purpose of technological complexity?

how would you find out?

suppose you "now all about electromagnetism and #a$well%s e&uations

but have no idea what a radio is for


5/30

radio de-construction for biologists

deletion transfection interference

biochemistry

'uri (a)ebni", Can a biologist fix a radio? Or what I learned while studying apoptosis,*ancer *ell 2+17-12 2002


6/30

radio de-construction for biologists

the conventional e$periment in signal transduction

step function atsaturating concentration

look to see what changes

even if we loo" at cells in an organism

the organism /"nows its internalenvironment but we do not

what is the spatial and temporal pattern of EG in a de!eloping embryo?


7/30

radio de-construction for systems biologists

the mole"ular "omplexity inside "ells refle"ts the "omplexity of the

en!ironments in whi"h those "ells e!ol!ed

to understand this molecular comple$ity we have to integrate theenvironment into our conceptual and e$perimental methods

how do we turn this slogan into science?

use the outside to learn about the inside

how can we study cells from an input/output perspective?


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the constancy of the milieu intrieure

#he fixity of the milieu supposes a perfe"tion of theorganism su"h that the external !ariations are at ea"hinstant "ompensated for and e$uilibrated%%%% &ll of the !italme"hanisms, howe!er !aried they may be, ha!e alwaysone goal, to maintain the uniformity of the "onditions oflife in the internal en!ironment %%%% The stability of the

internal environment is the condition for the freeand independent life% *

**laude ernard, from ectures on the !henomena "ommon to #nimals and!lants, 17. uoted in * 3ross, Claude 'ernard and the "onstan"y of the internalen!ironment, he 5euroscientist, $+!06 1

*laude ernard, %ntroduction to the &tudy of 'xperimental (edicine, 16


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homeostasis

#he highly de!eloped li!ing being is an open system ha!ingmany relations to its surroundings ( in the respiratory andalimentary tra"ts and through surfa"e re"eptors,neuromus"ular organs and bony le!ers% Changes in thesurroundings ex"ite rea"tions in this system, or affe"t itdire"tly, so that internal disturban"es of the system are

produ"ed%Such disturbances are normally kept withinnarrow limits, because automatic adjustments within

the system are brought into action, and thereby wideoscillations are prevented and the internal conditionsare held fairly constant% *

*8alter *annon,Organi)ation for physiologi"al homeostasis, 9hysiological :eviews,)+!4!1, 12.

8alter *annon,he +isdom of the ,ody,8 8 5orton ; *o, 1!2.

8alter *annon,#he body physiologi" and the body politi", pilogue to 8.


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meanwhile. back in the real world

+atts governor (etal (ike autopilot

engineers had already developed machines that could implement

homeostatic behaviour

@ #indell, ,etween 0uman and (achine1 eedback. "ontrol and "omputing before"ybernetics, Aohns Bop"ins Cniversity 9ress, 2002.


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and down the road at (%

the analogy between machines and organisms was made e$plicit

D = :osenblueth, 5 8iener, A igelow, 'eha!ior, purpose and teleology, 9hilosophy of


12/30

linear approximation

near a steady state, a nonlinear system may be approximated by a

linear one *the +artmanGrobman #heorem-

steady state

nonlinear linear

offset from thesteady state

3acobian matrix

provided that no eigenvalue of the Aacobian has real part )ero


13/30

stability of steady states

a steady state is stable if all the eigen!alues of the .a"obian, at the

steady state, ha!e negati!e real part

homeostasis re&uires stability of the steady state but stability is not

sufficient to ensure homeostasis

in this case, sufficiently smallperturbations away from the steady

state cause the system to return to the steady state

shallow basinof stability


14/30

linear systems

two e&uivalent ways of writing a system of linear differential e&uations

single !ariable, higher deri!ati!es

matrix form

order 4 2

5 components 4 2


15/30

the problem of homeostasis 6with 7 component8

linear system with a stable steady state atx / 0

to be controlled so that it maintains the steady statex / r

against perturbations

set point


16/30

proportional control

apply negative feedbac" that is proportional to the discrepancybetweenxand r

error

x

b / 1, r / 2

3p/ 4 3

p/ 10

time time

controller gain


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proportional control

suffers from steady state errorthat can be minimised by increasing

the controller gain

how "an steady state error be a!oided?

normalised

steady state of thecontrolled system


18/30

biased proportional control

add a bias, so that there is positive control whenx / r

bias

if " / br, then

biased proportional control wor"s but it is not a robust solution -

the parameters have to be fine tuned


19/30

integral control

suppose there is another variable in the system,y, whose rate of

changeis proportional to the discrepancy betweenxand r

then, in any steady state,

integral control variable


20/30

integral control

useyas the control variable

integral "ontrol pro!ides a robust implementation of homeostasis

at the price of increasing the number of components


21/30

r / 1, b / 1, ki= 1

r / 1, b / 1, ki=

r / 1, b / 1, ki= !"#

r / 1, b / 1, ki= $!

ringing or hunting

integral control 9 transient behaviour


22/30

!% and !%: control

r / 1, b / 1, 3i/ 40, kp= 1

transient behaviour can be improved by adding proportional control

and shaped further by adding derivative control F9E@ controlG


23/30

evolution anticipated engineering

'i, Buang, l


24/30

chemotaxis in % coli

B erg, ' coli in (otion,


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perfect adaptation in theory

=7u( =u( =>u( =;u(

-7u( -u( ->u( -;u(

tumblingfre?uency

5 ar"ai, < (eibler, 5obustness in simple bio"hemi"al networ3s, 5ature


26/30

perfect adaptation in fact

suares - unstimulated cells

circles 9 7m( aspartate at t 4 @

each data point averaged over 7@@-$@@ cells

E "oli:94!7

C =lon, # 3


27/30

perfect adaptation as integral control

total receptor methylation acts as an integral control variable

4 methylation sites

# 'i, ' Buang, # E


28/30

linear system with ncomponents

external stimulus

observed output variable


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integral control is necessary for perfect adaptation

/perfect adaptation is ta"en to mean that the steady state value ofthe observed output variable,y, is independent of the stimulus,u

provided the underlying system is stable, then perfect adaptation

implies the e$istence of a generalised internal variable

that implements integral control

# 'i, ' Buang, # E


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summing up

6% mole"ular "omplexity refle"ts the "omplexity of external en!ironments

1% we need to integrate "omplex en!ironments into our "on"eptual andexperimental approa"hes ( an input7output perspe"ti!e

2% the origins of su"h ideas go ba"3 to physiology and engineering

4% homeostasis "an be robustly implemented by integral feedba"3 "ontrol

8% homeostasis *perfe"t adaptation- re$uires integral "ontrol, at least from alinear perspe"ti!e

9% nonlinear systems "an be approximated by linear ones, in the !i"inityof a reasonable steady state *+artmanGrobman #heorem-

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Lectures Systems Biolgy