We start by enumerating the possible states of the system and their Boltzmann weights, and summing the weights to obtain the partition function. We will do this first in terms of numbers of dimers and tetramers (D andT, respectively), and then use the equilibrium constant for the dissociation reaction of tetramers into dimers, KDT, to write the partition function in terms of KDT and the
total concentration of repressor [R].
When dimers are present in solution, there are 5 new states that the system can be in, in addition to the 5 states of the simple model that does not include dimers. All 10 states are shown schematically in Fig. A.1, along with the statistical weights which we will now derive.
To construct the weight of each state, we first note that the energy of each state issol/2 times the number of tetramer heads in solution, plussol/2 times the number of dimers free in solution, plus id per dimer or tetramer bound to the Oidoperator, plus1per dimer or tetramer bound to the O1 operator, plus ∆Floopif a loop is formed. So, for example, if a tetramer is bound at Oidand a dimer at O1, as in state (viii), the argument of the exponential is−β[(T−1)sol+(D−1)sol/2+1+id+sol/2], whereβ is the reciprocal of the Boltzmann constant times the temperature.
The multiplicity of each state consists of three parts. First, as in the simple model of [115]
(and, implicitly, of Chapter 2), there are 8πδω2 rotational configurations per repressor in solution; so, for example, in the state where their are neither dimers nor tetramers bound to the DNA (state (i)), there areT tetramers andD dimers in solution, or
8π2 δω
T+D
total rotational configurations.
Second, since each tetramer head has 2 orientations in which it can bind to the DNA and each tetramer has 2 heads, a tetramer bound at an operator contributes a factor of 4 to the multiplicity (2 heads times 2 configurations per head). However a dimer bound at either operator contributes a multiplicative factor of 2, since each dimer has only one binding domain. Finally we account for the ways of arranging the tetramers and dimers in solution by using a lattice model to describe the solution, as in [115]. The lattice has Ω lattice sites, so for example in state (viii), with a tetramer bound at Oid and a dimer bound at O1, there are (T−1)!(D−1)!(Ω−T−D−2)!Ω! ways of arranging the remainingT−1 tetramers andD−1 dimers in solution.
We next apply some simplifications. If we assume a dilute solution, such that ΩT+D, then
Ω!
(Ω−(T+D))!≈ΩT+D (A.1)
and likewise for similar terms. The parts of the multiplicities of each state that correspond to the
ways of arranging the dimers and tetramers in solution then become:
(i) Ω!
T!D!(Ω−T −D)! → ΩT+D
T!D! (A.2)
(ii),(iii) Ω!
(T −1)!D!(Ω−T−D+ 1)! → ΩT+D−1
(T−1)!D! (A.3)
(iv) Ω!
(T −2)!D!(Ω−T−D+ 2)! → ΩT+D−2
(T−2)!D! (A.4)
(v) Ω!
(T −1)!D!(Ω−T−D+ 1)! → ΩT+D−1
(T−1)!D! (A.5)
(vi),(vii) Ω!
T!(D−1)!(Ω−T−D+ 1)! → ΩT+D−1
T!(D−1)! (A.6)
(viii),(ix) Ω!
(T −1)!(D−1)!(Ω−T−D+ 2)! → ΩT+D−2
(T−1)!(D−1)! (A.7)
(x) Ω!
T!(D−2)!(Ω−T−D+ 2)! → ΩT+D−2
T!(D−2)! (A.8)
where the Roman numerals correspond to the numbering of the states in Fig. A.1.
Next we divide each weight by the weight of state (i) so that the weights of the 10 states become:
(i)→1 (A.9)
(ii)→4 8π2
δω
−1 ΩT+D−1 (T −1)!D!
T!D!
ΩT+D
e−β[(T−1)sol+Dsol2 +1+sol2 −T sol−Dsol2 ] (A.10)
= 4 8π2
δω −1T
Ωe−β(1−sol2 ) (A.11)
(iii)→4 8π2
δω
−1 ΩT+D−1 (T −1)!D!
T!D!
ΩT+D
e−β[(T−1)sol+Dsol2 +id+sol2 −T sol−Dsol2 ] (A.12)
= 4 8π2
δω −1T
Ωe−β(id−sol2 ) (A.13)
(iv)→16 8π2
δω
−2T2
Ω2e−β[T sol−2sol+Dsol2 +2sol2 +1+id−T sol−Dsol2 ] (A.14)
= 16 8π2
δω
−2T2
Ω2e−β(1+id−2sol2 ) (A.15)
(v)→8 8π2
δω −1T
Ωe−β[T sol−sol+Dsol2 +1+id+∆Floop−T sol−Dsol2 ] (A.16)
= 8 8π2
δω −1T
Ωe−β(1+id+∆Floop−2sol2 ) (A.17)
(vi)→2 8π2
δω −1
ΩT+D−1 T!(D−1)!
T!D!
ΩT+D
e−β[T sol+Dsol2 −sol2 +1−T sol−Dsol2 ] (A.18)
= 2 8π2
δω −1D
Ωe−β(1−sol2 ) (A.19)
(vii)→2 8π2
δω −1
ΩT+D−1 T!(D−1)!
T!D!
ΩT+D
e−β[T sol+Dsol2 −sol2 +id−T sol−Dsol2 ] (A.20)
= 2 8π2
δω −1D
Ωe−β(id−sol2 ) (A.21)
(viii),(ix)→8 8π2
δω −2
ΩT+D−2 (T−1)!(D−1)!
T!D!
ΩT+D
e−β[T sol−sol+Dsol2 −sol2 +1+id+sol2 −T sol−Dsol2 ] (A.22)
= 8 8π2
δω
−2T D
Ω2e−β(1+id−2sol2 ) (A.23)
(x)→4 8π2
δω
−2 ΩT+D−2 T!(D−2)!
T!D!
ΩT+D
e−β[T sol+Dsol2 −2sol2 +1+id−T sol−Dsol2 ] (A.24)
= 4 8π2
δω
−2D2
Ω2e−β(1+id−2sol2 ). (A.25)
Finally we define ∆1≡1−sol2 and ∆id≡id−sol2 . (Note that this is the same convention as in [115]: there ∆≡b+t−sol, wheretis the energy of the unbound head in solution, and then it is assumed that there is no cooperativity to the binding of the second head, such thatt=sol/2, as we have assumed here.) Note that in the arguments of the exponentials there is always an sol2 to go with eachid and1, so all thesol’s disappear and all1, id become ∆1,∆id.
We can now write the total partition function as
Z = 1 + 4 8π2
δω −1T
Ωe−β∆1+ 4 8π2
δω −1T
Ωe−β∆id+ 16 8π2
δω
−2T2
Ω2e−β(∆1+∆id) (A.26) + 8
8π2 δω
−1T
Ωe−β(∆1+∆id+∆Floop)+ 2 8π2
δω −1D
Ωe−β∆1+ 2 8π2
δω −1D
Ωe−β∆id + 16
8π2 δω
−2T D
Ω2 e−β(∆1+∆id)+ 4 8π2
δω
−2D2
Ω2e−β(∆1+∆id)
where we have combined states (viii) and (ix) into one term since they are mathematically identical.
To convert from numbers of dimers and tetramers to concentrations, we start by defining the
total amount of repressor in the TPM sample, [R], as
[R] = [D]
2 + [T], (A.27)
as in Chapter 2. As in [115], we will define [T] and [D] in terms of the number of lattice sites as
[T]≡ T
Ωv and [D]≡ D
Ωv, (A.28)
wherev is the size of each lattice site, chosen such that
1 v
8π2
δω = 1M. (A.29)
Also in keeping with [115] we define the dissociation constants as
Kd≡ 1 4v
8π2
δωeβ∆, (A.30)
where ∆is either ∆id or ∆1. In units of concentration this becomes
Kd= 1 M
4 eβ∆. (A.31)
Finally we define the J-factor as
Jloop ≡1 v
8π2
δωe−β∆Floop, (A.32)
or in units of concentration,
Jloop = 1 Me−β∆Floop. (A.33)
We can now write the partition function in terms of [T], [D], Kid,K1, andJloop. Consider first the looped state (state (v)),
8 8π2
δω −1
T
Ωe−β(∆1+∆id+∆Floop). (A.34)
If we replaceT /Ω with [T]v and group terms that can be combined into J-factors and dissociation constants we can rewrite Eq. (A.34) as
8 8π2
δω −1T
Ωe−β(∆1+∆id+∆Floop)= [T] 8v 8π2
δω −1
e−β∆1
!
e−β∆id 4v
8π2 δω
! 8π2
δω
4v e−β∆Floop
! .
(A.35) Then replacing grouped terms withJloop or the appropriateKd yields
[T] 8v 8π2
δω −1
e−β∆1
!
e−β∆id 4v
8π2 δω
! 8π2
δω
4v e−β∆Floop
!
= [T] 2 K1
1 Kid
Jloop
4 , (A.36)
which is the same as in the original model, with [R] replaced by [T] (which makes sense since only tetramers can form loops). Similar manipulations can be applied to the rest of the states, yielding a new partition function of
Z= 1 +[T]
K1
+ [T] Kid
+ [T]2 K1Kid
+[T]Jloop
2K1Kid
+ [D]
2K1
+ [D]
2Kid
+ [T][D]
K1Kid
+ [D]2 4K1Kid
(A.37)
= 1 +2[T] + [D]
Kid +2[T] + [D]
K1 +[T]2+ [T][D]
K1Kid +[T]Jloop
2K1Kid + [D]2
4K1Kid. (A.38)
Since [D]/2 + [T] = [R] (Eq. (A.27)),
Z= 1 + [R]
Kid
+[R]
K1
+ [R]2 K1Kid
+[T]Jloop
2K1Kid
, (A.39)
which is the same as Eq. (2.16) in Chapter 2.
Finally we will use KDT, the equilibrium constant forT ⇔ 2D, to express [T] in terms of [R]
andKDT. We start with the definition ofKDT:
KDT =[D]2
[T] , (A.40)
or
[T] = [D]2 KDT
. (A.41)
Substituting this into Eq. (A.27) yields
[D]
2 + [D]2
KDT = [R], (A.42)
which we can solve for [D]:
[D] =1 4
−KDT + q
KDT2 + 16KDT[R])
(A.43)
where the positive root is chosen because [D] must be positive and the discriminant is always positive.
However we want [T], not [D]; so we again use Eq. (A.27) to say that
[T] = [R]−[D]
2 = [R]−1 8
−KDT+ q
KDT2 + 16KDT[R])
. (A.44)
By substituting this into Eq. (A.39) we obtain our final partition function of
Z= 1 + [R]
Kid
+[R]
K1
+ [R]2 K1Kid
+[R]Jloop 2K1Kid
1− 1
8[R]
−KDT + q
KDT2 + 16KDT[R])
. (A.45)
Since the looping probability is the weight of the looped state divided by the partition function, we have our final result that
ploop, dimers=
[R]Jloop
2K1Kid
1 + K8[R]DT −8[R]1 p
KDT2 + 16KDT[R]) 1 + K[R]
id+[R]K
1 +K[R]2
1Kid +[R]J2K loop
1Kid
1 +K8[R]DT −8[R]1 p
KDT2 + 16KDT[R]) .
(A.46)