Bioinformatics An Introduction 4th Edition (Jeremy Ramsden)

324

Regulatory Networks

of B, and the binding ratio r equals c divided by b 0r = c/b0 (number of bound ligands per biopolymer)

becomes

r equals StartFraction n upper K 0 a Over 1 plus upper K 0 a EndFraction periodr =

nK0a

1 + K0a ^.

(23.14)

Suppose now that the sites are not independent but that the addition of a second (and

subsequent) ligand next to a previously bound one (characterized by an equilibrium

constant upper K 1K1) is easier than the addition of the ﬁrst ligand. In the case of a linear

receptor B, the problem is formally equivalent to the one-dimensional Ising model

of ferromagnetism, and neglecting end effects, one has

r equals StartFraction n Over 2 EndFraction left parenthesis 1 minus StartFraction 1 minus upper K 0 a Over left bracket left parenthesis 1 minus upper K 0 a right parenthesis squared plus 4 upper K 0 a divided by q right bracket Superscript 1 divided by 2 Baseline EndFraction right parenthesis commar = ⁿ

(

1 −

1 −K0a

[(1 −K0a)²+ 4K0a/q]¹^/²

)

(23.15)

where the degree of coöperativity qq is determined by the ratio of the equilibrium

constants,q equals upper K 1 divided by upper K 0q = K1/K0. Forq greater than 1q > 1 this yields a sigmoidal binding isotherm. Ifq less than 1q < 1,

then binding is anticoöperative, as, for example, when an electrically charged particle

adsorbs at an initially neutral surface; the accumulated charge repels subsequent

arrivals and makes their incorporation more difﬁcult.

Sustained Activation

Effective stimulation in the immune system often depends on a sustained surface

reaction. When a ligand (antigen) present at the surface of an antigen-presenting cell

(APC) is bound by a T-lymphocyte (TL) (see Sect. 14.6), binding triggers a confor-

mational change in the receptor protein to which the antigen is ﬁxed, which initiates

further processes within the APC, resulting in the synthesis of more receptors, and

so on. This sustained activation can be accomplished with a few, or even only one

TL, provided that the afﬁnity is not too high: The TL binds, triggers one receptor,

then dissociates and binds anew to a nearby untriggered receptor (successive binding

attempts in solution are highly correlated). This “serial triggering” can formally be

described by

normal upper L plus normal upper R right arrow normal upper R Subscript normal upper L Superscript asteriskL + R →R^∗

(23.16)

(with rate coefﬁcient k Subscript normal aka), where the starred R denotes an activated receptor and

normal upper R Subscript normal upper L Superscript asterisk Baseline right harpoon over left harpoon normal upper R Superscript asterisk Baseline plus normal upper LR^∗

L ^⇌^R^∗⁺^L

(23.17)

with rate coefﬁcientk Subscript normal dkd for dissociation of the ligand L from the activated receptor, and

the same rate coefﬁcient k Subscript normal aka for reassociation of the ligand with an already activated

receptor. The rate of activation (triggering) is minus d r slash d t equals minus k Subscript normal a Baseline r l−dr/dt = −karl, solvable by noting

that d l slash d t equals minus k Subscript normal a Baseline left parenthesis r plus r Superscript asterisk Baseline right parenthesis plus k Subscript normal d Baseline r Subscript normal upper L Superscript asteriskdl/dt = −ka(r + r^∗) + kdr^∗

L^{. One obtains}

l left parenthesis t right parenthesis equals StartFraction k Subscript normal a Baseline tau Over 1 minus upper Y e Superscript negative t divided by tau Baseline EndFraction plus StartFraction k Subscript normal a Baseline left parenthesis l 0 minus r 0 right parenthesis minus k Subscript normal d Baseline minus 1 divided by tau Over 2 k Subscript normal a Baseline EndFraction commal(t) =

kaτ

1 −Ye⁻^t^/^τ⁺^k^a⁽^l⁰⁻^r⁰⁾⁻^k^d⁻¹^/^τ

2ka

(23.18)