$|\mathcal{N}_{\frac{1}{4}}|\leq 9^{s}$. The union bound tells us

$\mathbb{P}\left(\max_{x\in\mathcal{N}_{\frac{1}{4}}}\left|\frac{1}{n}\|W_{T}x\|_{2}^{2}-1\right|\geq\frac{t}{2}\right)\leq
9^{s}\cdot
2\exp\left(-cn\min\left\\{\frac{t^{2}}{\kappa^{4}},\frac{t}{\kappa^{2}}\right\\}\right).$
(90)

By [60, Lemma 5.4],

$\|\frac{1}{n}W_{T}^{\top}W_{T}-I_{s}\|\leq
2\max_{x\in\mathcal{N}_{\frac{1}{4}}}\left|\langle(\frac{1}{n}W_{T}^{\top}W_{T}-I_{s})x,x\rangle\right|=2\max_{x\in\mathcal{N}_{\frac{1}{4}}}\left|\frac{1}{n}\|W_{T}x\|_{2}^{2}-1\right|.$
(91)

Since
$\frac{2}{3}\leq\sigma_{\min}(\Sigma)\leq\sigma_{\max}(\Sigma)\leq\frac{3}{2}$,
we have
$\|\Sigma_{T,T}^{1/2}\|,\|\Sigma_{T,T}^{-1/2}\|\leq\sqrt{\frac{3}{2}}$.
Therefore,

$\begin{split}\|\frac{1}{n}X_{T}^{\top}X_{T}\Sigma_{T,T}^{-1}-I_{s}\|=&\|\Sigma_{T,T}^{1/2}(\frac{1}{n}W_{T}^{\top}W_{T}-I_{s})\Sigma_{T,T}^{-1/2}\|\\\
\leq&\|\Sigma_{T,T}^{1/2}\|\|\frac{1}{n}W_{T}^{\top}W_{T}-I_{s}\|\|\Sigma_{T,T}^{-1/2}\|\\\
\leq&\frac{3}{2}\|\frac{1}{n}W_{T}^{\top}W_{T}-I_{s}\|.\end{split}$ (92)

Combine (90), (91) and (92) together, we have arrived at the conclusion.

  2. 2.

Now we consider the proof for (89). Note that
$\|H_{\delta}(\alpha^{\top}X_{T}^{\top}X_{\Omega})\|_{2}\leq t$ implies there
exists $\Lambda\subset\Omega$ such that all entry of
$|\alpha^{\top}X_{T}^{\top}X_{\Lambda}|$ are greater than $\delta$, and
$\left\||\alpha^{\top}X_{T}^{\top}X_{\Lambda}|-\delta\right\|_{2}\geq t$.
Thus,

$\begin{split}&\mathbb{P}\left(\left\|H_{\delta}(\alpha^{\top}X_{T}^{\top}X_{\Omega}/n)\right\|_{2}\geq
t\right)\\\ \leq&\mathbb{P}\left(\exists\Lambda\subseteq\Omega,\text{ such
that all entries of
}\left|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right|\geq\delta,\text{and
}\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\|_{2}\geq t\right)\\\
\leq&\mathbb{P}\left(\exists\Lambda\subseteq\Omega,\sqrt{|\Lambda|}\delta\leq
t,\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\|_{2}\geq t\right)\\\
&+\mathbb{P}\left(\exists\Lambda\subseteq\Omega,\sqrt{|\Lambda|}\delta>t,\text{
all entries of
}\left|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right|\geq\delta\right)\\\
\leq&\sum_{\begin{subarray}{c}\Lambda\subseteq\Omega\\\
|\Lambda|=\lfloor(t/\delta)^{2}\rfloor\wedge
r\end{subarray}}\mathbb{P}\left(\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\|_{2}\geq
t\right)+\sum_{\begin{subarray}{c}\Lambda\subseteq\Omega\\\
|\Lambda|=\lceil(t/\delta)^{2}\rceil\end{subarray}}\mathbb{P}\left(\text{all
entries of
}\left|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right|\geq\delta\right)\\\
\leq&\sum_{\begin{subarray}{c}\Lambda\subseteq\Omega\\\
|\Lambda|=\lfloor(t/\delta)^{2}\rfloor\wedge
r\end{subarray}}\mathbb{P}\left(\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\|_{2}\geq
t\right)+\sum_{\begin{subarray}{c}\Lambda\subseteq\Omega\\\
|\Lambda|=\lceil(t/\delta)^{2}\rceil\end{subarray}}\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t\right).\end{split}$ (93)

Since $t\geq\delta\geq C\max_{i\in
T^{c}}\|\Sigma_{i,T}\Sigma_{T,T}^{-1}\|_{2}\|\alpha\|_{2}$,

$3\sqrt{\lceil(t/\delta)^{2}\rceil}(\max_{i\in
T^{c}}\|\Sigma_{i,T}\Sigma_{T,T}^{-1}\|_{2})\|\alpha\|_{2}\leq
3\sqrt{2}(t/\delta)(\max_{i\in
T^{c}}\|\Sigma_{i,T}\Sigma_{T,T}^{-1}\|_{2})\|\alpha\|_{2}\leq t.$

By Part 3 of Lemma 6, for any $\Lambda\subseteq\Omega$, $t\geq
3\sqrt{|\Lambda|}\max_{i\in
T^{c}}\|\Sigma_{i,T}\Sigma_{T,T}^{-1}\|_{2}\|\alpha\|_{2}$, we have

$\begin{split}&\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t\right)\\\
\leq&\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n-\mathbb{E}\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t-\left\|\mathbb{E}\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\right)\\\
\leq&\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n-\mathbb{E}\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t-\left\|\Sigma_{\Lambda,T}\alpha\right\|_{2}\right)\\\
=&\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n-\mathbb{E}\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t-\left(\sum_{i\in\Lambda}(\Sigma_{i,T}\alpha)^{2}\right)^{1/2}\right)\\\
\leq&\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n-\mathbb{E}\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t-\sqrt{|\Lambda|}\max_{i\in T^{c}}|\Sigma_{i,T}\alpha|\right)\\\
\leq&\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n-\mathbb{E}\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t-\sqrt{|\Lambda|}\max_{i\in
T^{c}}\|\Sigma_{i,T}\Sigma_{T,T}^{-1}\|_{2}\|\Sigma_{T,T}\|\|\alpha\|_{2}\right)\\\
\leq&\mathbb{P}\left(\left\|\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n-\mathbb{E}\alpha^{\top}X_{T}^{\top}X_{\Lambda}/n\right\|_{2}\geq
t/2\right)\\\
\leq&2\exp\left(C|\Lambda|-cn\min\left\\{\frac{t^{2}}{\kappa^{4}\|\alpha\|_{2}^{2}},\frac{t}{\kappa^{2}\|\alpha\|_{2}}\right\\}\right).\end{split}$

Combine (93) and the previous inequality, one obtains

$\begin{split}&\mathbb{P}\left(\left\|H_{\delta}(\alpha^{\top}X_{T}^{\top}X_{\Omega}/n)\right\|_{2}\geq
t\right)\\\ \leq&\binom{r}{\lfloor(t/\delta)^{2}\rfloor\wedge
r}\exp\left(C\lfloor(t/\delta)^{2}\rfloor\wedge
r-cn\min\left\\{\frac{t^{2}}{\kappa^{4}\|\alpha\|_{2}^{2}},\frac{t}{\kappa^{2}\|\alpha\|_{2}}\right\\}\right)\\\
&+\binom{r}{\lceil(t/\delta)^{2}\rceil}_{+}\exp\left(C\lceil(t/\delta)^{2}\rceil-
cn\min\left\\{\frac{t^{2}}{\kappa^{4}\|\alpha\|_{2}^{2}},\frac{t}{\kappa^{2}\|\alpha\|_{2}}\right\\}\right).\end{split}$

$\square$

######  Lemma 8 (Properties of Soft-thresholding)

  1. 1.

Suppose $a,b>0$, $x,y\in\mathbb{R}$, $H_{\cdot}(\cdot)$ is the soft-
thresholding operator satisfying $H_{a}(x)={\rm sgn}(x)\cdot(|x|-a)_{+}$. Then
the following triangular inequality holds,

$|H_{a+b}(x+y)|\leq|H_{a}(x)|+|H_{b}(y)|.$ (94)

  2. 2.

Suppose $a,b>0$, $x,y\in\mathbb{R}^{p}$, if $\|H_{a}(x)\|_{\infty,2}\leq b$,
then

$|\langle x,y\rangle|\leq a\|y\|_{1}+b\|y\|_{1,2}.$ (95)

Proof of Lemma 8.

  1. 1.

$\begin{split}|H_{a+b}(x+y)|=&(|x+y|-a-b)_{+}\leq(|x|-a+|y|-b)_{+}\leq(|x|-a)_{+}+(|y|-b)_{+}\\\
=&|H_{a}(x)|+|H_{b}(y)|.\end{split}$

  2. 2.

$\begin{split}|\langle x,y\rangle|\leq&|\langle H_{a}(x),y\rangle|+|\langle
x-H_{a}(x),y\rangle|=|\sum_{j=1}^{d}\langle[H_{a}(x)]_{(j)},y_{(j)}\rangle|+|\langle
x-H_{a}(x),y\rangle|\\\
\leq&\sum_{j=1}^{d}\|[H_{a}(x)]_{(j)}\|_{2}\|y_{(j)}\|_{2}+\|x-H_{a}(x)\|_{\infty}\|y\|_{1}\leq\|H_{a}(x)\|_{\infty,2}\|y\|_{1,2}+\|x-a\|y\|_{1}\\\
\leq&b\|y\|_{1,2}+a\|y\|_{1}.\end{split}$

$\square$

###### Lemma 9

Suppose $X=[X_{1}^{\top},\ldots,X_{n}^{\top}]^{\top}\in\mathbb{R}^{n\times p}$
is a random matrix with independent random sub-Gaussian rows satisfying
Assumption 1, $\varepsilon_{i}\stackrel{{\scriptstyle
i.i.d.}}{{\sim}}N(0,\sigma^{2})$. Suppose $T\subseteq\\{1,\dots,p\\}$ is with
cardinality $s$, $P\in\mathbb{R}^{n\times n}$ is a projection matrix and
independent of $X_{T}$. Then, for any $t\geq\log(es)$,

$\mathbb{P}\left(\|X_{T}^{\top}P\varepsilon\|_{\infty}\geq
C\kappa\sqrt{nt}\sigma^{2}\right)\leq e^{-n}+e^{-Ct}.$

Proof of Lemma 9. For fixed vector $w\in\mathbb{R}^{n}$, since Assumption 2 is
satisfied, for $i\in T$, $X_{1i},\dots,X_{ni}$ are independent sub-Gaussian
distributed such that

$\mathbb{E}\exp\left(tX_{ji}\right)=\mathbb{E}\exp\left(te_{i}^{\top}\Sigma^{1/2}\Sigma^{-1/2}X_{j\cdot}^{\top}\right)\leq\exp\left(\frac{\kappa^{2}\|\Sigma^{1/2}e_{i}\|_{2}^{2}t^{2}}{2}\right)\leq\exp\left(\frac{\kappa^{2}\Sigma_{i,i}t^{2}}{2}\right)\leq\exp\left(\frac{3\kappa^{2}t^{2}}{4}\right).$

By Hoeffding-type inequality,

$\mathbb{P}\left(|X_{\cdot i}^{\top}w|\geq t\|w\|_{2}\right)\leq
2\exp\left(-c\frac{t^{2}}{\kappa^{2}}\right).$ (96)

Moreover, by [61, Lemma 1], for any $x\geq 0$,

$\mathbb{P}\left(\sum_{i=1}^{n}\varepsilon_{i}^{2}\geq(n+2\sqrt{nx}+2x)\sigma^{2}\right)\leq
e^{-x}.$

Set $x=n$ in the last inequality, we have

$\mathbb{P}\left(\|\varepsilon\|_{2}\geq\sqrt{5n\sigma^{2}}\right)\leq
e^{-n}.$ (97)

Combine (96) and (97) together and notice that
$\|P\varepsilon\|_{2}\leq\|\varepsilon\|_{2}$, we have

$\begin{split}&\mathbb{P}\left(\|X_{T}^{\top}P\varepsilon\|_{\infty}\geq
C\kappa\sqrt{nt\sigma^{2}}\right)\leq\sum_{i\in T}\mathbb{P}\left(|X_{\cdot
i}^{\top}P\varepsilon|\geq C\kappa\sqrt{nt\sigma^{2}}\right)\\\
\leq&\mathbb{P}\left(\|P\varepsilon\|_{2}\geq\sqrt{5n\sigma^{2}}\right)+\sum_{i\in
T}\mathbb{P}\left(|X_{\cdot i}^{\top}P\varepsilon|\geq
C\kappa\sqrt{nt\sigma^{2}},\|P\varepsilon\|_{2}\leq\sqrt{5n\sigma^{2}}\right)\\\
\leq&\mathbb{P}\left(\|\varepsilon\|_{2}\geq\sqrt{5n\sigma^{2}}\right)+\sum_{i\in
T}\mathbb{P}\left(|X_{\cdot i}^{\top}P\varepsilon|\geq
C\kappa\sqrt{nt\sigma^{2}}\bigg{|}\|P\varepsilon\|_{2}\leq\sqrt{5n\sigma^{2}}\right)\\\
\leq&e^{-n}+s\cdot 2\exp\left(-Ct\right)\leq e^{-n}+e^{-Ct}.\end{split}$

$\square$

###### Lemma 10

With probability at least $1-Ce^{-cn/s}$, the approximate dual certificate
defined in (42) can be written as $u=X^{\top}w$, where $\|w\|_{2}\leq
C\sqrt{s/n}$.

Proof of Lemma 10. By (42), we have $u=X^{\top}w$, where
$w=(w_{1}^{\top},\dots,w_{l_{\max}}^{\top})^{\top}$ and
$w_{l}=\frac{1}{n_{l}}X_{I_{l},T}\Sigma_{T,T}^{-1}q_{l-1}$. Thus
$\|w\|_{2}^{2}=\sum_{l=1}^{l_{\max}}\|w_{l}\|_{2}^{2}$. Also note that

$\begin{split}\frac{1}{n_{l}}\|X_{I_{l},T}\Sigma_{T,T}^{-1}q_{l-1}\|_{2}^{2}=&\langle\frac{1}{n_{l}}X_{I_{l},T}^{\top}X_{I_{l},T}\Sigma_{T,T}^{-1}q_{l-1},\Sigma_{T,T}^{-1}q_{l-1}\rangle\\\
=&\langle(\frac{1}{n_{l}}X_{I_{l},T}^{\top}X_{I_{l},T}\Sigma_{T,T}^{-1}-I_{|T|})q_{l-1},\Sigma_{T,T}^{-1}q_{l-1}\rangle+\|\Sigma_{T,T}^{-1/2}q_{l-1}\|_{2}^{2}\\\
=&\langle-
q_{l},\Sigma_{T,T}^{-1}q_{l-1}\rangle+\|\Sigma_{T,T}^{-1/2}q_{l-1}\|_{2}^{2}\\\
\leq&\|q_{l}\|_{2}\|\Sigma_{T,T}^{-1}q_{l-1}\|_{2}+\|\Sigma_{T,T}^{-1/2}q_{l-1}\|_{2}^{2}\\\
\leq&\frac{3}{2}\|q_{l}\|_{2}\|q_{l-1}\|_{2}+\frac{3}{2}\|q_{l-1}\|_{2}^{2}\leq
3\|q_{l-1}\|_{2}^{2}.\end{split}$

By (50), with probability at least $1-C\exp\left(-cn/s\right)$,

$\begin{split}\|w\|_{2}^{2}\leq&\sum_{l=1}^{l_{\max}}\frac{3}{n_{l}}\|q_{l-1}\|_{2}^{2}\\\
\leq&\frac{C}{n}(2\sqrt{s})^{2}+\frac{C}{n}\left(2\sqrt{s/\log(es)}\right)^{2}+\frac{C\log(es)}{n}\sum_{l=3}^{l_{\max}}\left(2^{4-l}\sqrt{s}/\log(es)\right)^{2}\\\
\leq&C\frac{s}{n}.\end{split}$

$\square$

Proof of Lemma 4. For any $1\leq i\leq p$, $1\leq j\leq
d,,\Lambda\subseteq(j),|\Lambda|=k$, by Lemma 6 with

$v=\Sigma^{-1}e_{i},\quad U\in\mathbb{R}^{p\times
k},U_{[\Lambda,:]}=I,U_{[\Lambda^{c},:]}=0,$

we have

$\mathbb{P}\left(\left\|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}\right\|_{2}\geq
t\right)\leq 2\exp\left(Ck-
cn\min\left\\{\frac{t^{2}}{\kappa^{4}},\frac{t}{\kappa^{2}}\right\\}\right).$
(98)

By the same method in Lemma 5 Part 2,

$\begin{split}&\mathbb{P}\left(\left\|H_{\alpha}\left((e_{i})_{(j)}-\frac{1}{n}X_{(j)}^{\top}X\Sigma^{-1}e_{i}\right)\right\|_{2}\geq\gamma\right)\\\
\leq&\mathbb{P}\bigg{(}\exists\Lambda\subseteq(j),\text{ all entries of
}|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}|\geq\alpha\text{
and
}\|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}\|_{2}\geq\gamma\bigg{)}\\\
\leq&\mathbb{P}\left(\exists\Lambda\subseteq(j),\sqrt{|\Lambda|}\alpha\leq\gamma,\|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}\|_{2}\geq\gamma\right)\\\
&+\mathbb{P}\left(\exists\Lambda\subseteq(j),\sqrt{|\Lambda|}\alpha>\gamma,\text{
all entries of
}|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}|\geq\alpha\right)\\\
\leq&\sum_{\begin{subarray}{c}\Lambda\subseteq(j)\\\ |\Lambda|=\lfloor
s/s_{g}\rfloor\end{subarray}}\mathbb{P}\left(\|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}\|_{2}\geq\gamma\right)\\\
&+\sum_{\begin{subarray}{c}\Lambda\subseteq(j)\\\ |\Lambda|=\lceil
s/s_{g}\rceil\end{subarray}}\mathbb{P}\left(\text{ all entries of
}|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}|\geq\alpha\right)\\\
\leq&\sum_{\begin{subarray}{c}\Lambda\subseteq(j)\\\ |\Lambda|=\lfloor
s/s_{g}\rfloor\end{subarray}}\mathbb{P}\left(\|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}\|_{2}\geq\gamma\right)+\sum_{\begin{subarray}{c}\Lambda\subseteq(j)\\\
|\Lambda|=\lceil
s/s_{g}\rceil\end{subarray}}\mathbb{P}\left(\|\left(e_{i}\right)_{\Lambda}-\frac{1}{n}X_{\Lambda}^{\top}X\Sigma^{-1}e_{i}\|_{2}\geq\gamma\right).\end{split}$

Combine (98) and the previous inequality together, we have

$\begin{split}&\mathbb{P}\left(\left\|H_{\alpha}\left((e_{i})_{(j)}-\frac{1}{n}X_{(j)}^{\top}X\Sigma^{-1}e_{i}\right)\right\|_{2}\geq\gamma\right)\\\
\leq&\binom{b_{j}}{\lfloor s/s_{g}\rfloor}\cdot 2\exp\left(C\lfloor
s/s_{g}\rfloor-cn\cdot
C\frac{s\log(es_{g}b)+s_{g}\log(d/s_{g})}{s_{g}n}\right)\\\
&+\binom{b_{j}}{\lceil s/s_{g}\rceil}\cdot 2\exp\left(C\lceil s/s_{g}\rceil-
cn\cdot C\frac{s\log(es_{g}b)+s_{g}\log(d/s_{g})}{s_{g}n}\right)\\\
\leq&4\left(\frac{2es_{g}b}{s}\right)^{2s/s_{g}}\exp\left(Cs/s_{g}-C\frac{s\log(es_{g}b)+s_{g}\log(d/s_{g})}{s_{g}}\right)\\\
\leq&4\exp\left(\frac{2s}{s_{g}}\log\left(\frac{2es_{g}b}{s}\right)+Cs/s_{g}-C\frac{s\log(es_{g}b)+s_{g}\log(d/s_{g})}{s_{g}}\right).\end{split}$
(99)

By (99) and the union bound, we have

$\begin{split}&\mathbb{P}\left(\max_{1\leq i\leq
p}\|H_{\alpha}(e_{i}-\frac{1}{n}X^{\top}X\Sigma^{-1}e_{i})\|_{\infty,2}\leq\gamma\right)\\\
\leq&\sum_{i=1}^{p}\sum_{j=1}^{d}\mathbb{P}\left(\left\|H_{\alpha}\left((e_{i})_{(j)}-\frac{1}{n}X_{(j)}^{\top}X\Sigma^{-1}e_{i}\right)\right\|_{2}\geq\gamma\right)\\\
\leq&d^{2}b\cdot
4\exp\left(\frac{2s}{s_{g}}\log\left(\frac{2es_{g}b}{s}\right)+Cs/s_{g}-C\frac{s\log(es_{g}b)+s_{g}\log(d/s_{g})}{s_{g}}\right)\\\
\leq&4\exp\left(2\log(s_{g})+2\log(d/s_{g})+\frac{3s}{s_{g}}\log\left(2eb\right)+Cs/s_{g}-C\frac{s\log(es_{g}b)+s_{g}\log(d/s_{g})}{s_{g}}\right)\\\
\leq&4\exp\left(-C\frac{s\log(es_{g}b)+s_{g}\log(d/s_{g})}{s_{g}}\right).\end{split}$

$\square$


As a result, we can write our partition function in the form

$Z_{NS}={N_{NS}^{P}\over P!}{1\over
2^{P}}(e^{-\beta\varepsilon_{2}}(1+e^{-\beta(\varepsilon_{1}-\varepsilon_{2})}))^{P}.$
(64)

This should be compared with

$Z_{NS}={N_{NS}^{P}\over P!}e^{-\beta P\varepsilon_{NS}}$ (65)

which is the result for the partition function for the most simple model in
which the nonspecific background is assumed to be uniform.

We now want to see whether our expression given in eqn. 64 is equivalent to
the single reservoir model. By equating eqn. 64 and eqn. 65 and taking the log
of both sides we have

$\varepsilon_{NS}=k_{B}T~{}\mbox{ln}2+\varepsilon_{2}-k_{B}T~{}\mbox{ln}~{}(1+e^{-\beta(\varepsilon_{1}-\varepsilon_{2})})$
(66)

We can simplify this by noting that the term involving the logarithm can be
simplified as

$\mbox{ln}(1+e^{-\beta(\varepsilon_{1}-\varepsilon_{2})})=\mbox{ln}(1+e^{-2\beta\Delta})\approx\mbox{ln}(1+1-2\beta\Delta)\approx\mbox{ln}~{}2+\mbox{ln}~{}(1-\beta\Delta),$
(67)

where we have used the fact that $\varepsilon_{1}-\varepsilon_{2}=2\Delta$.
Given that $\beta\Delta<<1$ (i.e. the energy difference between the two states
is small), we can use the Taylor series $\mbox{ln}~{}(1-x)\approx-x$ with the
result that

$\varepsilon_{NS}=\bar{\varepsilon}$ (68)

This result shows us that in the toy model of the nonspecific background of
Figure 26(B), the two nonspecific backgrounds are equivalent to a single
reservoir with an energy given by the mean of the energies of the two
reservoirs, establishing that in this pedagogically motivated model we can use
a single energy to describe the nonspecific background. Now let’s move to the
case of realistic distribution of nonspecific energies.

Figure 17 shows the distribution of nonspecific binding energies obtained by
taking the energy matrix describing the binding of LacI and applying it to all
sites across the E. coli genome (also see Figure 26(C) for a comparison with
the other models considered thus far). Other examples of the distribution of
nonspecific binding energies have been considered as well with similar outcome
[97, 98]. As a result, we can write the number of binding sites with energy
between $E$ and $E+dE$ as

$n(E)={N_{NS}\over\sqrt{2\pi\sigma^{2}}}e^{-(E-\bar{\varepsilon})^{2}/2\sigma^{2}},$
(69)

where $\bar{\varepsilon}$ is the mean of the distribution of nonspecific
binding energies and $\sigma$ provides a measure of the width of that
distribution.

To compute the partition function for the binding of a polymerase, for
example, to this nonuniform genomic background, we need to sum over all the
microscopic states available to the polymerase. Symbolically, the quantity we
need to evaluate is

$Z_{NS}=\sum_{E}n(E)e^{-\beta E}.$ (70)

In fact, since we are assuming a continuous distribution of energies, this
really is an integral of the form

$Z_{NS}=\int_{\infty}^{\infty}e^{-\beta
E}{N_{NS}\over\sqrt{2\pi\sigma^{2}}}e^{-(E-\bar{\varepsilon})^{2}/2\sigma^{2}}dE.$
(71)

This result can be rewritten as

$Z_{NS}={N_{NS}\over\sqrt{2\pi\sigma^{2}}}e^{-\bar{\varepsilon}^{2}/2\sigma^{2}}\int_{\infty}^{\infty}e^{-{1\over
2\sigma^{2}}[E^{2}-2E\bar{\varepsilon}+2\sigma^{2}\beta E]}dE.$ (72)

By completing the square, this integral results in

$Z_{NS}=N_{NS}e^{-\beta\bar{\varepsilon}}e^{\beta^{2}\sigma^{2}/2}$ (73)

which should be compared with the result we would get if we assumed a
homogeneous nonspecific background with only a single binding energy
$\varepsilon_{NS}$ resulting in the form

$Z_{NS}=N_{NS}e^{-\beta\varepsilon_{NS}}$ (74)

By equating these two expressions, we find that we can treat the nonuniform
background as though it were a homogenous genomic background with effective
binding energy

$\varepsilon_{eff}=\bar{\varepsilon}-{\beta\sigma^{2}\over 2}.$ (75)

The result above considered a single polymerase or repressor molecule bound to
the nonuniform nonspecific background. What happens in the case where we have
$P$ polymerases bound nonspecifically? Because each of those polymerases binds
independently of the others (because the number of polymerases is of order
$10^{3}-10^{4}$ and the genome size is greater than $10^{6}$ we don’t need to
worry about polymerases interfering with each other), the total partition
function for all of these polymerases bound to the nonspecific background is
given by

$Z_{NS}(P,N_{NS})={(\int_{\infty}^{\infty}e^{-\beta
E}{N_{NS}\over\sqrt{2\pi\sigma^{2}}}e^{-(E-\bar{\varepsilon})^{2}/2\sigma^{2}}dE)^{P}\over
P!}={N_{NS}^{P}e^{-\beta P\varepsilon_{eff}}\over P!},$ (76)

where once again $\varepsilon_{eff}=\bar{\varepsilon}-{\beta\sigma^{2}\over
2}$ and this result shows that if the distribution of binding energies is
Gaussian, then we can treat the nonspecific background as being equivalent to
a uniform nonspecific background with energy $\varepsilon_{eff}$. The point of
all of this analysis was simply to examine the validity of the convenient
simplifying assumption of some thermodynamic models of treating the
nonspecific background as uniform. As shown elsewhere [97, 98], this
approximation is quite reasonable.

##  15 Accounting for the effect of nonspecific promoter occupancy

So far our statistical mechanical treatment of the simple repression
architecture has treated the RNA polymerase and LacI proteins as isolated from
the pool of other transcription factors that are also littered across the
genomic DNA. In Figure 6 we plot the abundance of DNA-binding proteins per
cell across a number of growth conditions using the proteomic study from
Schmidt et al. [29]. These values include nucleoid-associated proteins that
also bind the genomic DNA. For growth in M9 minimal media with 0.5% glucose,
we find that there are about $3\times 10^{5}$ DNA-binding proteins per cell
and we can use this to make a simple estimate of genomic occupancy by these
proteins. Let us assume that each transcription factor binds the DNA as a
dimer (this will vary with the transcription factor species) and occupies a
DNA length of 15 bp (this varies from 7 bp to 38 bp in E. coli for
transcription factors listed on RegulonDB; [8]). For growth in 0.5% glucose,
we find that about 2.3 Mbp or about half the genome is occupied (15
bp$\times$3$\times$105DNA-binding proteins$\times$1/2 dimers per protein).

Given the high occupancy of DNA-binding proteins on the genomic DNA estimated
above, there might be some expectation that, in contrast to our current model
of simple repression, the occupancy of the genome by these other DNA-binding
proteins cannot be ignored. Here we consider the effect of their occupancy by
adding an explicit set of states to represent the case where these additional
DNA-binding proteins can occupy the roughly 60 bp promoter region of our
simple repression architecture. For simplicity we assume that these proteins
only bind nonspecifically, ignoring any potential sequence-specific effects.
In Figure 27(A) we show the states and weights of the simple repression
promoter, where we have included this additional set of states. We could have
extended this further, either by treating each additional DNA-binding protein
species separately, or by being more careful about our specification of these
additional states. However, the point of this exercise is to see what effect
the pool of nonspecifically bound DNA-proteins might have on our model. We can
calculate $p_{\text{bound}}$ which, if we invoke the weak promoter
approximation $\left({P\over N_{NS}}e^{-\beta\Delta\varepsilon_{P}}\ll
1\right)$, is given by

$p_{\text{bound}}={{P\over N_{NS}}e^{-\beta\Delta\varepsilon_{P}}\over
1+L\cdot{C_{NS}\over N_{NS}}+{R\over N_{NS}}e^{-\beta\Delta\varepsilon_{R}}}.$
(77)

$L$ represents the number of ways other DNA-binding proteins may bind the
promoter nonspecifically, and for simplicity is taken as the length of the
promoter region ($L$ $\approx$ $60$ bp). $C_{NS}$ represents the copy number
of all other DNA-binding proteins bound to the genome that we noted earlier.
Fold-change, which is the ratio of $p_{\text{bound}}(R\geq 0)$ to
$p_{\text{bound}}(R=0)$, will then be given by

$\text{fold-change}=\frac{1+L\cdot{C_{NS}\over N_{NS}}}{{P\over
N_{NS}}e^{-\beta\Delta\varepsilon_{P}}}\cdot\frac{{P\over
N_{NS}}e^{-\beta\Delta\varepsilon_{P}}}{1+L\cdot{C_{NS}\over N_{NS}}+{R\over
N_{NS}}e^{-\beta\Delta\varepsilon_{R}}}.$ (78)

Here, the RNA polymerase components ${P\over
N_{NS}}e^{-\beta\Delta\varepsilon_{P}}$ cancel out and upon some
rearrangement, we find that

$\text{fold-change}=\frac{1}{1+{R\over
N_{NS}}e^{-\beta\Delta\varepsilon_{R}}(1+L\cdot{C_{NS}\over N_{NS}})^{-1}}.$
(79)

Using $C_{NS}\approx 1.5\times 10^{5}$, which is based on our estimate of the
total DNA-binding protein copy number found above for growth in glucose (bound
as dimers), we calculate a value of $L\cdot{C_{NS}\over N_{NS}}\approx 2$.
Importantly, we find that this additional term in our fold-change equation
does not depend on the key parameters of our simple repression architecture,
namely the repressor copy number or repressor binding energy, and we can
arrive back to our original form of fold-change by a defining
$N_{NS}^{\prime}=N_{NS}\times(1+L\cdot{C_{ns}\over N_{NS}})$.

Figure 27: A crowded chromosome. (A) States and Weights for simple repression
with a pool of nonspecific DNA binding proteins. RNA polymerase (light blue),
a repressor, and other nonspecific DNA binding proteins compete for binding to
a promoter. The $R$ repressors and $P$ RNA polymerase bind with energies
$\Delta\varepsilon_{R}$ and $\Delta\varepsilon_{P}$, respectively. In
addition, there are $C_{NS}$ DNA binding proteins per cell that can bind the
promoter of length $L\approx 60$ bp. These proteins bind nonspecifically and
therefore only contribute an entropic term. $N_{NS}$ represents the number of
nonspecific binding sites on the genome. (B) Measured protein copy numbers are
shown for DNA binding proteins in E. coli across 22 growth conditions. Protein
copy numbers per cell were determined by Schmidt et al. [29] with proteins
identified based on their annotation in EcoCyc. Error bars are propagated from
the reported standard deviations. Protein copy numbers per genome equivalent
were calculated by estimating the total genomic content as a function of
growth rate using Cooper and Helmstetter’s model of E. coli chromosomal
replication [120, 121, 122].

The estimates so far were based on assuming that cells grow in 0.5% glucose at
a particular doubling rate. In different media, the growth rate will change
leading also to a modulation in the total number of transcription factors:
faster growing cells have a larger protein complement than their slower-
growing counterparts. However, faster growing cells also have more copies of
the genome as a means to keep up with the fast replication pace. Figure 27(B)
shows that these two effects cancel each other out. Specifically, variations
in the number of transcription factors as a result of changes in growth rate
are counteracted by the corresponding change in the average genome copy number
per cell such that the number of nonspecific binding proteins per base pair
remains approximately constant throughout a wide range of growth conditions.
As a result, the small effect of considering all nonspecifically bound
transcription factors remains unaltered regardless of growth rate.

References

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space considerations.

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