uri alon-an introduction to systems biology_ design principles of biological circuits-chapman &...

8/10/2019 Uri Alon-An Introduction to Systems Biology_ Design Principles of Biological Circuits-Chapman & Hall_CRC (2007)

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.O: . ' t ' t 1 1 t \1 1 1 I I Jl A i

()()

'I

S,/K,

I I\ ,I I ~I .'\ .' haction of X bound InS, a ~ a function ofS, conccnratinn. 1-J.tlf of X is bound when S, ~ K,.

A.2 I:I, I)ICJC; lJ I .'\ lm l\1 SSCW 1'1\()ll_IN 10 ANINIJl ICII\: MIC I I N iS ,'v\FNTFN F(jlii\TICJN- - - - -

1hc reprcssor protein X is designed to bind a small molecule i nd u ce r \ . which can beconsidered as its input signal. ll1c two can collide to form a bound complex, [XSJ 'Thercpressor is therefore found 1 in either free form, X, or bound form, [XS,], and a conservation law states that total concentration of re pressor protein is XT:

X . ~X+ [XS,[ (A.2.l)

rHE I N I ' U r flJ,\iCTION OF 1\ > K.) is known as zero-order because [XSJ -S, 0 , and the linear regime (S, K ~ )is known as first-order since [XS.J- Sx1 .

Recall that in cases like Lacl, only X unbound t o \ , is active in the sense that it can bindthe promoter D to block transcription. B e c a u ~ cfree X is active, we denote it by X . Activerepressor, X"= XT - [XS,], decreases with incre;lsing inducer levels:

X"= ~1+5,/K, concentration o X not bound to S

1\ .. C:OOPERMIVITY 0 1 ~I N I ~ J lJCIR lliNfJ-iN_C;\ND THF HILl l C ~ l J A T I O N

(A.2.5)

Before returning to the input function, we comment on a more realistic description ofinducer binding. Most transcription factors are composed of several repeated protein sub


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X and \ co l l i de to for ll the complex [XS,] at a ralc k . u and the cumplcx [XSxl fallsapart (dissociates) at a rate k, 11 'Jhus, the mass-action kinetic equation is: 2

d [XSJ/dt ~ k,, X S ,- k," [XS,) (A.2.2)

At steady .state, d[XSJ/dt = 0, and we find the chemical equilibrium relation:

K, [XS,) ~ S, (A.2.3)

where K, is the dissociation constant (for the lac repressor, K, - 1 p lOOO indllcer(IPTG) molccules/cell). 3 Using the conservation oftot;ll reprcssor X (Equation A.2.1), wearrive at a useful equation that recurs throughout biology (this equation is known as theMichaclis-Mentcn equation in the context of enzyme kinetics; we use the same name inthe present context of inducer binding):

Michaelis Mente11 equati on (A.2.4)

1 \Ve assume that Sxcan bind X r c g a r d l e ~ sofwln:ther it is bound to D or not.

'Usually the number ofSxmoleculcs is much larger thnn the number of X molecules, nnd so we need not worryabout conserv;1lion of Sx, Sx,total = Sx + [XSxl- For example, in the ltlc system, tlw number of Lad repressors, eachmade of a telramcr of l.acl proteins, is Xr - 10 units/cell, which is negligible relative to Sx, which is at least 1000/nll for a delectable response.

In Ihe G l ~ cof the /or r e p r e ~ . ~ o rKx - I pM. lhing lh


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uri alon-an introduction to systems biology_ design principles of biological circuits-chapman &...

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