# Absorption, scattering and shadow by a noncommutative black hole with global
monopole

M. A. Anacleto<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Campina Grande Caixa Postal 10071, 58429-900 Campina Grande,
Paraíba, Brazil  F. A. Brito111Corresponding author<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Campina Grande Caixa Postal
10071, 58429-900 Campina Grande, Paraíba, Brazil  Departamento de Física,
Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa,
Paraíba, Brazil  J. A. V. Campos<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-970 João
Pessoa, Paraíba, Brazil  E. Passos<EMAIL_ADDRESS>Departamento de
Física, Universidade Federal de Campina Grande Caixa Postal 10071, 58429-900
Campina Grande, Paraíba, Brazil

###### Abstract

In this paper, we investigate the process of massless scalar wave scattering
due to a noncommutative black hole with a global monopole through the partial
wave method. We computed the cross section of differential scattering and
absorption at the low frequency limit. We also calculated, at the high
frequency limit, the absorption and the shadow radius by the null geodesic
method. We showed that noncommutativity causes a reduction in the differential
scattering/absorption cross section and shadow radius, while the presence of
the global monopole has the effect of increasing the value of such quantities.
In the limit of the null mass parameter, we verify that the cross section of
differential scattering, absorption and shadow ray approach to a non-zero
value proportional to a minimum mass.

##  I Introduction

Hawking Hawking:1971vc  black hole radiation emission studies revealed a
physically relevant possibility for us to enter the universe of quantum
gravity. A very important point raised by Hawking is related to the final
stage of black hole evaporation, so that when the mass of the black hole
approaches the Planck mass, a quantum theory of gravity is needed. However,
stimulated by the need to study the structure of spacetime on the Planck
scale, many other approaches have been proposed considering noncommutative
geometry Padmanabhan:2003gd ; Szabo:2006wx ; Ansoldi:2006vg . As with position
and moment uncertainty in conventional quantum mechanics, we can find a
noncommutative relationship in Einstein theory of general relativity. Thus,
position measurements can fail to commute by suggesting a noncommutative
manifold Nicolini:2008aj . Studies as proposed by Nicolini Nicolini:2005vd
showed that noncommutativity can be implemented in general relativity by
modifying the matter source. This can be done by modifying the mass density by
replacing a Dirac delta function with a Gaussian distribution of minimum width
$\sqrt{\theta}$ given by
$\rho_{\theta}(r)=M(4\pi\theta)^{-3/2}\exp{(-r^{2}/4\theta)}$, or even
considering replacing by a Lorentzian distribution Nozari:2008rc  tanking the
following form
$\rho_{\theta}(r)=M\sqrt{\theta}\pi^{-3/2}(r^{2}+\pi\theta)^{-2}$, where
$\theta$ is the noncommutative parameter and $M$ is the total mass that due to
the uncertainty of noncommutativity is diffused throughout the linear region
of size $\sqrt{\theta}$. Besides the noncommutativity, the study of
topological defects has been used in different areas of physics as a way to
improve understanding the behavior of the early Universe.

It is well known in the literature that global monopoles are types of
topological defects that are formed by a spontaneous symmetry breaking of
global symmetry and that a large part of its energy is concentrated into a
point more specifically in the center Vilenkin:2000jqa  and, a charge of this
nature is characterized by a spontaneous symmetry breaking of original global
$O(3)$ symmetry to $U(1)$ Kibble:1976sj . In recent years many authors have
devoted themselves to studying black holes with global monopoles
BezerradeMello:1996si ; Pitelli:2009kd ; Sharif:2015kna ; Shaikh:2019fpu ;
Haroon:2019new , initiated by Barriola and Vilenkin Barriola:1989hx  which was
the first study considering a metric of black hole with global static
monopoles. In Anacleto:2017kmg , the differential scattering cross section and
the absorption of a black hole with global monopole in $f(R)$ gravity have
been computed by the partial wave method.

In the present work, we have the main purpose of exploring the effect of
quantum gravity corrections that contribute to the process of massless scalar
wave scattering by a noncommutative black hole with a global monopole. The
problem of scattering a massless scalar wave across a black hole has been
constantly studied by several authors Townsend:1997ku ; Futterman:1988ni ;
Li:2022wzi ; Xing:2022emg ; Bisnovatyi-Kogan:2022ujt ; Zeng:2021dlj ;
Mourad:2021qgo ; Jha:2021bue ; Li:2022jda ; Gogoi:2022wyv ; Karmakar:2022idu ;
Lobos:2022jsz ; Tsupko:2022yzg ; Zeng:2022fdm ; Heydari-Fard:2021qdc ;
Heydari-Fard:2021pjc ; Khodadi:2021gbc ; Fathi:2020sfw ; Chen:2022ngd , and
many works considering the low frequency limit ($GM\ll 1$) can be found in the
literature Matzner:1977dn ; Westervelt:1971pm ; Peters:1976jx ;
Sanchez:1976fcl ; Sanchez:1976xm ; Sanchez:1977si ; Sanchez:1977vz ;
DeLogi:1977dp ; Doran:2001ag ; Dolan:2007ut ; Crispino:2009ki ;
Starobinskil:1974nkd . The use of the partial wave method can also be found in
several different contexts Gibbons:1975kk ; Page:1976df ; Churilov1973 ;
Moura:2011rr ; Jung:2004yh ; Jung:2004yn ; Doran:2005vm ; Dolan:2006vj ;
Castineiras:2007ma ; Benone:2014qaa ; Marinho:2016ixt ; Das:1996we ;
Macedo:2016yyo ; deOliveira:2018kcq ; Hai:2013ara . Besides, this method has
also been applied to examine scalar wave scattering by acoustic black holes
Crispino:2007zz ; Dolan:2009zza ; Oliveira:2010zzb ; Dolan:2011zza ;
Anacleto:2012ba ; Anacleto:2012du ; Dolan:2012yc ; Anacleto:2015mta ;
Anacleto:2016ukc  and also to exploring the scattering of scalar waves by a
noncommutative BTZ black hole Anacleto:2014cga .

In previous studies Anacleto:2019tdj , we investigated the effect of
noncommutativity in black hole by considering a Lorentzian smeared mass
distribution on the scattering problem, where we find a differential
scattering cross section for small angles given approximately by
$d\sigma/d\Omega\approx 16G^{2}M^{2}/\vartheta^{2}[1+(4/M)\sqrt{\theta/\pi}]$.
In the present work, we will see that the presence of noncommutativity and the
global monopole changes the behavior of the differential scattering cross
section in a nontrivial way. In addition, we also saw in Anacleto:2019tdj
that for the low frequency limit the absorption cross section is always
proportional to the area of the event horizon of the noncommutative
Schwarzschild black hole $\sigma\approx
16G^{2}M^{2}/\vartheta^{2}[1+(2/M)\sqrt{\theta/\pi}]^{2}$. We will see that
the absorption cross section increases when we add the influence of the global
monopoles on the geometry. At the small mass limit $M\to 0$, we have a result
that differs from studies done for other geometries Anacleto:2018acl ;
Anacleto:2020lel ; Anacleto:2020zhp , and at this limit the absorption cross
section has a direct relationship with the mass $M$. Therefore, starting from
a Lorentzian distribution, analytical results are more easily examined than
using a Gaussian distribution where the analysis is done numerically. In
Anacleto:2020efy ; Anacleto:2020zfh , employing the Lorentzian distribution,
the authors have found logarithmic corrections for entropy as well as the
condition for the black hole remnant. In Campos:2021sff , we have investigated
the quasinormal modes and the shadow radius of a noncommutative Schwarzschild
black hole. We show that, in the zero-mass limit, the shadow ray does not
vanish, being proportional to a minimum mass for the finite noncommutative
parameter and the black hole becomes a black hole remnant. Therefore, in the
present work, we aim to explore the effect of quantum gravity corrections that
contribute to the process of massless scalar wave scattering by a
noncommutative black hole with a global monopole. We will also apply the null
geodesic method to calculate the shadow radius in order to verify the
influence of noncommutativity and the global monopole.

The paper is organized as follows. In Sec. II we consider the influence of
noncommutativity and the global monopole on the Schwarzschild black hole
metric to determine the differential cross section and absorption. In Sec. III
we apply the null geodesic method to compute the shadow of the noncommutative
black hole with a global monopole. In Sec. IV we make our final
considerations. Here we adopt the natural units $\hbar=c=k_{B}=G=1$.

##  II Noncommutative Black Hole with a global monopole

In this section, we introduce the noncommutative black hole with a global
monopole for the purpose of determining the differential scattering cross
section and absorption for this model. Thus, we adopt the partial wave method
to calculate the phase shift at the low energy limit. The spherically
symmetrical noncommutative black hole with global monopole is described by the
following line element

$\displaystyle
ds^{2}=F(r)dt^{2}-\frac{dr^{2}}{F(r)}-r^{2}\left(d\vartheta^{2}+\sin^{2}\vartheta
d\phi^{2}\right).$ (1)

The function $F(r)$ is given by

$\displaystyle F(r)$ $\displaystyle=$ $\displaystyle
1-8\pi\eta^{2}-\frac{2{\cal M}_{\theta}}{r},$ (2) $\displaystyle=$
$\displaystyle
1-8\pi\eta^{2}-\frac{2M}{r}+\frac{8M\sqrt{\theta}}{\sqrt{\pi}r^{2}}.$ (3)

Here the event horizon, $r_{\eta}$, and the Cauchy horizon, $r_{\theta}$, read

$\displaystyle r_{\eta}$ $\displaystyle=$
$\displaystyle\frac{2M}{(1-8\pi\eta^{2})}-4\sqrt{\frac{\theta}{\pi}}+\cdots=\frac{2M}{(1-8\pi\eta^{2})}\left[1-\frac{2(1-8\pi\eta^{2})}{M}\sqrt{\frac{\theta}{\pi}}\right],$
(4) $\displaystyle r_{\theta}$ $\displaystyle=$ $\displaystyle
4\sqrt{\frac{\theta}{\pi}}+\cdots.$ (5)

The mass $M$ can be written in terms of $r_{\eta}$ as follows

$\displaystyle
M=\left(1-8\pi\eta^{2}\right)\left(\frac{r_{\eta}}{2}+2\sqrt{\frac{\theta}{\pi}}\right).$
(6)

The Hawking temperature is given by

$\displaystyle T_{H}=\dfrac{\left(1-8\pi\eta^{2}\right)}{4\pi
r_{\eta}^{2}}\left(r_{\eta}-4\sqrt{\frac{\theta}{\pi}}\right).$ (7)

For $\eta=0$, we have the result obtained in Anacleto:2020zfh .

In order to obtain a minimum mass in the final stage of black hole
evaporation, we compute the specific heat capacity using the formula

$\displaystyle C=\left(\dfrac{\partial M}{\partial
r_{\eta}}\right)\left(\dfrac{\partial T_{H}}{\partial
r_{\eta}}\right)^{-1}=-2\pi\left(r_{\eta}-8\sqrt{\theta/\pi}\right)\left(r_{\eta}+16\sqrt{\theta/\pi}\right).$
(8)

Writing in terms of $r_{s}$ we have

$\displaystyle
C=\dfrac{-2\pi\left(r_{s}-12(1-8\pi\eta^{2})\sqrt{\theta/\pi}\right)\left(r_{s}+12(1-8\pi\eta^{2})\sqrt{\theta/\pi}\right)}{(1-8\pi\eta^{2})^{2}}.$
(9)

Note that for $r_{s}=r_{min}=12(1-8\pi\eta^{2})\sqrt{\theta/\pi}$, we have
$C=0$ and the black hole becomes a remnant. So we have $r_{min}=2M_{min}$ and
from there we find the following minimum mass
$M_{min}=6(1-8\pi\eta^{2})\sqrt{\theta/\pi}$.

Therefore, expressing equation (7) in terms of $r_{s}$, we have

$\displaystyle
T_{H}=\dfrac{\left(1-8\pi\eta^{2}\right)^{2}\left(r_{s}-8(1-8\pi\eta^{2})\sqrt{\theta/\pi}\right)}{4\pi\left(r_{s}-4(1-8\pi\eta^{2})\sqrt{\theta/\pi}\right)^{2}}.$
(10)

Now substituting the radius $r_{min}$ into (10) we find the following maximum
temperature of the black hole remnant

$\displaystyle T_{Hmax}=\dfrac{(1-8\pi\eta^{2})}{64\pi\sqrt{\theta/\pi}}.$
(11)

In Fig. 1 we have the graph of temperature as a function of $r_{s}$ for
$\Theta=0.06$ and $\eta=0.03$, where $\Theta=\sqrt{\theta}/(M\sqrt{\pi})$. For
a minimum radius $r_{min}=0.70371$ and a minimum mass $M_{min}=0.3519$, then
the value for the maximum temperature is $T_{Hmax}=0.08101$. The maximum
temperature value corresponds precisely to the peak of the graph.

Figure 1: We have the Hawking temperature for the non-commutative case with
monopoles compared to the usual Schwarzschild case.

###  II.1 Differential Scattering Cross Section and Absorption

Now, we start our analysis of the massless scalar field equation to examine
the scattered wave in the background (1), given by

$\displaystyle\dfrac{1}{\sqrt{-g}}\partial_{\mu}\Big{(}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi\Big{)}=0.$
(12)

Next, we use the following separation of variables in the Klein-Gordon
equation (12)

$\displaystyle\Psi_{\omega lm}({\bf r},t)=\frac{{\cal R}_{\omega
l}(r)}{r}Y_{lm}(\vartheta,\phi)e^{-i\omega t},$ (13)

being $Y_{lm}(\vartheta,\phi)$ the spherical harmonics and $\omega$ the
frequency.

Hence, we find the radial equation for ${\cal R}_{\omega l}(r)$ as follows

$\displaystyle F(r)\frac{d}{dr}\left(F(r)\dfrac{d{\cal R}_{\omega
l}(r)}{dr}\right)+\left[\omega^{2}-V_{eff}\right]{\cal R}_{\omega l}(r)=0,$
(14)

with

$\displaystyle
V_{eff}=\frac{F(r)}{r}\frac{dF(r)}{dr}+\frac{F(r)l(l+1)}{r^{2}},$ (15)

the effective potential.

Next, we introduce a new radial function, $\chi(r)=F^{1/2}(r){\cal R}(r)$,
such that

$\displaystyle\frac{d^{2}\chi(r)}{dr^{2}}+V(r)\chi(r)=0,$ (16)

being $V(r)$ the potential, given by

$\displaystyle
V(r)=\frac{[F^{\prime}(r)]^{2}}{4F^{2}(r)}-\dfrac{F^{\prime\prime}(r)}{2F(r)}+\dfrac{\omega^{2}}{F^{2}(r)}-\frac{V_{eff}}{F^{2}(r)}.$
(17)

At this point, by performing a power series expansion in $1/r$ in the above
equation, we find

$\displaystyle\frac{d^{2}\chi(r)}{dr^{2}}+\left[\frac{\omega^{2}}{(1-8\pi\eta^{2})^{2}}+{\cal
U}_{eff}(r)\right]\chi(r)=0,$ (18)

The effective potential is now given by

$\displaystyle{\cal
U}_{eff}(r)=\frac{4M{\omega}^{2}}{(1-8\pi\eta^{2})^{3}r}+\frac{12\ell^{2}}{r^{2}}+\frac{32M{\omega}^{2}}{(1-8\pi\eta^{2})^{4}r^{2}}\sqrt{\frac{\theta}{\pi}}+\cdots,$
(19)

Here, due to the modification of the term $1/r^{2}$ in the effective potential
Anacleto:2017kmg ; Anacleto:2019tdj , we define

$\displaystyle\ell^{2}$ $\displaystyle\equiv$
$\displaystyle-\frac{(l^{2}+l)}{12(1-8\pi\eta^{2})}+\frac{r_{\eta}^{2}{\omega}^{2}}{4(1-8\pi\eta^{2})^{2}}.$
(20)

Note that the suitable asymptotic behavior for the effective potential is
satisfied, i.e., ${\cal U}_{eff}(r)\rightarrow 0$ when $r\rightarrow\infty$.
Next, we can find the phase shift analytically, at the low frequency limit, by
means of the following Ansatz Anacleto:2017kmg ; Anacleto:2019tdj

$\displaystyle\delta_{l}\approx
2(l-\ell)=2\left(l-\sqrt{-\frac{(l^{2}+l)}{12(1-8\pi\eta^{2})}+\frac{r^{2}_{\eta}{\omega}^{2}}{4(1-8\pi\eta^{2})^{2}}}\right).$
(21)

Hence, the phase shift $\delta_{l}$ for $l\rightarrow 0$ becomes

$\displaystyle\delta_{l}=\delta_{0}+\delta_{l\geq
1},\quad\delta_{0}=-\frac{{\omega}r_{\eta}}{(1-8\pi\eta^{2})},\quad\delta_{l\geq
1}=0.$ (22)

Furthermore, with determined $\delta_{l}$ we can calculate the differential
scattering cross section using the formula Anacleto:2017kmg ; Yennie1954 ;
Cotaescu:2014jca

$\displaystyle\dfrac{d\sigma}{d\Omega}=\Big{|}\frac{1}{2i{{\omega}}}\sum_{l=0}^{\infty}a^{1}_{l}\frac{P_{l}(\cos\vartheta)}{1-\cos\vartheta}\Big{|}^{2},$
(23)

where $P_{l}(\cos\vartheta)$ are the Legendre polynomials, being

$\displaystyle
a^{1}_{l}=a^{0}_{l}-\frac{l+1}{2l+3}a^{0}_{l+1}-\frac{l}{2l-1}a^{0}_{l-1},$
(24)

and

$\displaystyle a^{0}_{l}=(2l+1)\left(e^{2i\delta_{l}}-1\right).$ (25)

Therefore, from expression (24) for $l=0$, we have

$\displaystyle a^{1}_{0}=a^{0}_{0}-\frac{1}{3}a^{0}_{1},$ (26)

and from relation (25), we have as a result:

$\displaystyle a^{0}_{0}=e^{2i\delta_{0}}-1,$ (27)

and

$\displaystyle a^{0}_{1}=3\left(e^{2i\delta_{1}}-1\right)=0,$ (28)

because $\delta_{l\geq 1}=0$. Then the equation for the differential
scattering cross section (23) can be now written as follows

$\displaystyle\dfrac{d\sigma}{d\Omega}=\Big{|}\frac{1}{2i{{\omega}}}\sum_{l=0}^{\infty}a^{0}_{l}\frac{P_{l}(\cos\vartheta)}{1-\cos\vartheta}\Big{|}^{2}=\Big{|}\frac{1}{2i{{\omega}}}\sum_{l=0}^{\infty}(2l+1)\left(e^{2i\delta_{l}}-1\right)\frac{P_{l}(\cos\vartheta)}{1-\cos\vartheta}\Big{|}^{2},$
(29)

and that at the small angle limit equation (29) takes the form

$\displaystyle\frac{d\sigma}{d\Omega}$ $\displaystyle=$
$\displaystyle\frac{4}{{\omega}^{2}\vartheta^{4}}\Big{|}\sum_{l=0}^{\infty}(2l+1)e^{i\delta_{l}}\sin(\delta_{l}){P_{l}(\cos\vartheta)}\Big{|}^{2},$
(30) $\displaystyle=$
$\displaystyle\frac{4}{{\omega}^{2}\vartheta^{4}}\Big{|}e^{i\delta_{0}}\sin(\delta_{0})P_{0}(\cos\vartheta)+\sum_{l=1}^{\infty}(2l+1)e^{i\delta_{l\geq
1}}\sin(\delta_{l\geq 1}){P_{l}(\cos\vartheta)}\Big{|}^{2}.$ (31)

Moreover, at the low frequency limit, by using the result of (22), we find the
following result for the differential scattering cross section

$\displaystyle\frac{d\sigma}{d\Omega}\Big{|}^{\mathrm{lf}}_{{\omega}\rightarrow
0}=\frac{4}{{\omega}^{2}\vartheta^{4}}\delta^{2}_{0}=\frac{4}{\vartheta^{4}}\frac{r_{\eta}^{2}}{(1-8\pi\eta^{2})^{2}}=\frac{16M^{2}}{(1-8\pi\eta^{2})^{4}\vartheta^{4}}\left(1-\frac{2(1-8\pi\eta^{2})}{M}\sqrt{\frac{\theta}{\pi}}\right)^{2}+\cdots.$
(32)

For $\eta=0$ and $\theta=0$ we obtain the result for the Schwarzschild black
hole case. Now, an interesting result for the differential scattering cross
section emerges when we take $M\rightarrow 0$ and thus find a non-zero result
given by

$\displaystyle\frac{d\sigma}{d\Omega}\Big{|}^{\mathrm{lf}}_{M\rightarrow
0}\approx\frac{64\theta}{\vartheta^{4}\pi(1-8\pi\eta^{2})^{2}}=\frac{16M^{2}_{min}}{9\vartheta^{4}(1-8\pi\eta^{2})^{4}},$
(33)

where $M_{min}=6(1-8\pi\eta^{2})\sqrt{\theta/\pi}$ is the minimal mass
Anacleto:2020zfh . Note that the differential scattering cross section is
increased by the monopole effect.

At this point, we will compute the absorption cross section for a
noncommutative black hole with a global monopole, at the low frequency limit,
which can be obtained as follows:

$\displaystyle\sigma_{abs}$ $\displaystyle=$
$\displaystyle\frac{\pi}{{\omega}^{2}}\sum_{l=0}^{\infty}(2l+1)\Big{(}1-\big{|}e^{2i\delta_{l}}\big{|}^{2}\Big{)}=\frac{4\pi}{{\omega}^{2}}\sum_{l=0}^{\infty}(2l+1)\sin^{2}(\delta_{l}),$
(34) $\displaystyle=$
$\displaystyle\frac{4\pi}{{\omega}^{2}}\left[\sin^{2}(\delta_{0})+\sum_{l=1}^{\infty}(2l+1)\sin^{2}(\delta_{l\geq
1})\right].$ (35)

Now, by taking the limit ${\omega}\rightarrow 0$ with $\delta_{l}$ given by
(22) the absorption reads

$\displaystyle\sigma_{abs}^{\mathrm{lf}}$ $\displaystyle=$
$\displaystyle\frac{4\pi\delta_{0}^{2}}{{\omega}^{2}}=\frac{4\pi
r^{2}_{\eta}}{(1-8\pi\eta^{2})^{2}}=\frac{16\pi
M^{2}}{(1-8\pi\eta^{2})^{4}}\left(1-\frac{2(1-8\pi\eta^{2})}{M}\sqrt{\frac{\theta}{\pi}}\right)^{2}+\cdots.$
(36)

We can recover the result for the absorption of the Schwarzschild black hole
in the absence of noncommutativity and global monopole by making
$\eta=\theta=0$. Furthermore, an interesting result for the absorption arises
when the mass parameter goes to zero and in this case we obtain a non-zero
result given by

$\displaystyle\sigma_{abs}^{\mathrm{lf}}$ $\displaystyle\approx$
$\displaystyle\frac{64\theta}{(1-8\pi\eta^{2})^{2}}=\frac{16\pi
M^{2}_{min}}{9(1-8\pi\eta^{2})^{4}}.$ (37)

In Fig. 2 we show the behavior of the absorbing cross section for $l=0$ when
varying $M$ from 1 to the minimum mass value. Note that by varying the $\eta$
parameter, the absorption has its value increased. Here, it is worth
mentioning that the results obtained in (33) and (37) are not present in the
usual case of the Schwarzschild black hole when the mass parameter approaches
zero.

Figure 2: Here we see the behavior of the absorption cross section for $l=0$
when varying $M$ from 1 to the minimum mass value, corresponding to the values
of the parameters $\Theta$ and $\eta$.

Besides, for absorption we can express the result in terms of the area of the
event horizon of the noncommutative black hole with global monopole, so we
have

$\displaystyle\sigma_{abs}^{\mathrm{lf}}=\frac{4\pi{r_{\eta}^{2}}}{(1-8\pi\eta^{2})^{2}}\approx{\cal
A}.$ (38)

We can also express Eq. (37) in terms of a minimum area as follows

$\displaystyle\sigma_{abs,\theta}^{\mathrm{lf}}=\frac{4\pi
r^{2}_{min}}{9(1-8\pi\eta^{2})^{4}}=\frac{A_{min}}{9(1-8\pi\eta^{2})^{4}},$
(39)

where $A_{min}=4\pi r^{2}_{min}$ is a minimal area. Therefore, we verified
that at the limit of $M$ going to zero, the black hole becomes a remnant of
black hole and for $\eta=0$ the absorption is equal to a minimal area due to
the effect of noncommutativity.

Now, we will calculate the absorption at the high frequency limit. For this we
rewrite the potential (19) as follows

$\displaystyle{\cal
U}_{eff}(r)=\frac{4M{\omega}^{2}}{(1-8\pi\eta^{2})^{3}r}+\frac{\bar{\ell}^{\,2}}{(1-8\pi\eta^{2})r^{2}}+\frac{48M{\omega}^{2}}{(1-8\pi\eta^{2})^{4}r^{2}}\sqrt{\frac{\theta}{\pi}}+\cdots,$
(40)

where

$\displaystyle\bar{\ell}^{\,2}\equiv$
$\displaystyle(l^{2}+l)-\dfrac{3\left(r_{\eta}^{2}+16M\sqrt{\theta/9\pi}\right){\omega}^{2}}{(1-8\pi\eta^{2})}.$
(41)

In this case, to determine the phase shift we apply the following Ansatz

$\displaystyle\delta_{l}\approx\dfrac{3}{4}\sqrt{l-\bar{\ell}\,}.$ (42)

For $l\rightarrow\infty$ the phase shift is given by

$\displaystyle\delta_{l}\approx\dfrac{3}{4}\sqrt{\dfrac{3}{2}\dfrac{\left(r_{\eta}^{2}+16M\sqrt{\theta/9\pi}\right){\omega}^{2}}{(1-8\pi\eta^{2})l}}.$
(43)

Therefore, at the limit of $l\rightarrow\infty$ , the absorption is given by

$\displaystyle\sigma_{abs}=\frac{4\pi}{{\omega}^{2}}\sum_{l=0}^{\infty}2l\delta_{l}^{2}=\frac{4\pi}{{\omega}^{2}}\sum_{l=0}^{\infty}2l\dfrac{27}{16}\dfrac{\left(r_{\eta}^{2}+16M\sqrt{\theta/9\pi}\right){\omega}^{2}}{2l(1-8\pi\eta^{2})},$
(44)

such that we obtain the following result for absorption at the high frequency
limit

$\displaystyle\sigma_{abs}^{hf}=\dfrac{27\pi}{4}\dfrac{\left(r_{\eta}^{2}+16M\sqrt{\theta/9\pi}\right)}{(1-8\pi\eta^{2})}=\frac{27\pi
M^{2}}{(1-8\pi\eta^{2})^{3}}\left(1-\frac{8(1-8\pi\eta^{2})}{3M}\sqrt{\frac{\theta}{\pi}}\right).$
(45)

Here we have successfully obtained absorption at the high frequency limit by
the partial wave method. Note that for $\eta=\theta=0$, we recover the
absorption result for the Schwarzschild black hole at the high frequency limit
obtained usually by the null geodesic method, $\sigma_{abs}^{hf}=27\pi M^{2}$.

For $\eta=0$ and $\theta\neq 0$, we find

$\displaystyle\sigma_{abs}^{hf}=27\pi M^{2}-72\pi
M\sqrt{\frac{\theta}{\pi}}+\cdots,$ (46)

showing that the absorption is reduced when we vary $\theta$.

For $\eta\neq 0$ and $\theta=0$, we obtain

$\displaystyle\sigma_{abs}^{hf}=27\pi M^{2}(1+24\pi\eta^{2}+\cdots).$ (47)

In this case the absorption is increased when we vary $\eta$, which coincides
with the results obtained by the null geodesic method.

###  II.2 Numerical analyses

In order to investigate the behavior of the radial equation (14) in the
asymptotic limits we define the tortoise coordinate $\varrho$ as follows:

$\displaystyle\frac{d}{d\varrho}=F(r)\frac{d}{dr},\quad\quad
F(r)=1-8\pi\eta^{2}-\frac{2M}{r}+\frac{8M\sqrt{\theta}}{\sqrt{\pi}r^{2}},$
(48)

such that

$\displaystyle\varrho=r+\frac{r^{2}_{\theta}}{r_{\theta}-r_{\eta}}\ln\Big{|}{r-r_{\theta}}\Big{|}-\frac{r^{2}_{\eta}}{r_{\theta}-r_{\eta}}\ln\Big{|}{r-r_{\eta}}\Big{|}.$
(49)

Thus, the radial equation (14) can be written as

$\displaystyle\dfrac{d^{2}{\cal R}_{\omega
l}(r)}{d\varrho^{2}}+\left[\omega^{2}-V_{eff}\right]{\cal R}_{\omega l}(r)=0.$
(50)

Note that, when applying the limits $r\rightarrow r_{\eta}$ and
$r\rightarrow\infty$, the effective potential defined in (15) tends to zero.
Consequently, we can examine the scattering problem by exploring the radial
solution at these asymptotic limits. In this way, we have the following
boundary conditions:

$\displaystyle R_{\omega
l}(r)\approx\Big{\\{}\begin{array}[]{cc}A_{in}e^{-iw\varrho}+A_{out}e^{iw\varrho},&\quad\varrho\rightarrow+\infty\quad(r\rightarrow\infty),\\\
A_{tr}e^{-iw\varrho},&\quad\varrho\rightarrow-\infty\quad(r\rightarrow
r_{\eta}),\end{array}$ (53)

with $|A_{in}|^{2}=|A_{out}|^{2}+|A_{tr}|^{2}$. Next, we determine the
absorption by applying equation (34) with the phase shift given by

$\displaystyle e^{2i\delta_{l}}=(-1)^{l+1}\frac{A_{out}}{A_{in}}.$ (54)

Here we present the numerical results that were obtained by numerically
solving the radial equation (14). For this purpose we have adopted the
numerical procedure performed in Dolan:2012yc .

In the table 1 we show the comparison between the analytical and numerical
results for $\omega\rightarrow 0$ and $l=0$.

In Fig. 3, we plot the partial absorption cross section for the $l=0$ mode for
different values of $\Theta=\sqrt{\theta}/(M\sqrt{\pi})$ and $\eta$. We can
see by comparing the curves (Fig. 3, Fig. 3, Fig. 3 and Fig. 3) for different
values of $\Theta=\sqrt{\theta}/(M\sqrt{\pi})$ and $\eta$ that the partial
absorption is increased in relation to the Schwarzschild black hole when we
vary $\eta$ with fixed $\Theta$. Now, in Fig. 3 and Fig. 3 when vary $\Theta$
with fixed $\eta$ the partial absorption is decreased due to the
noncommutative effect. For $\Theta=0$ and $\eta=0$ the graph shows the result
of the partial absorption for the Schwarzschild black hole.

In Fig. 4 we plot the partial absorption for $l=0$ mode by setting the values
of $\Theta=0.06$ and $\eta=0.03$. We can observe that when we reduce the mass
value, the absorption amplitude is not null as we can see from equation (37).
The partial absorption cross section graphs for $l=0,1,2,3,4$ modes are shown
in Fig. 5.

In Fig. 6 we present the graphs of the total absorption cross section. The
results for fixed $\Theta$ and varying $\eta$ values are shown in figures 6
and 6. Graphs for fixed $\eta$ and varying $\Theta$ values are shown in
figures 6 and 6. The lines in Fig. 6 represent the values for high frequency
absorption, by applying the null geodesic method presented in the next
section.

Finally in Fig. 7 we plot the result of the differential scattering cross
section for $M\omega=1.0$ and in Fig. 8 for $M\omega=2.0$. The effect of
noncommutativity and the global monopole on the differential scattering cross
section is more significant at large angles.

Table 1:  Analytical and numerical results for $\omega\rightarrow 0$ and $l=0$. | $\Theta=0.00$ | $\Theta=0.05$ | $\Theta=0.10$ | $\Theta=0.12$
---|---|---|---|---
$\eta$ | Eq. (37) | Numerical | Eq. (37) | Numerical | Eq. (37) | Numerical | Eq. (37) | Numerical
0.00 | 16.0000 | 15.9988 | 12.9600 | 12.9592 | 10.2400 | 10.2396 | 9.24160 | 9.24105
0.03 | 17.5334 | 17.5324 | 14.2735 | 14.2729 | 11.3486 | 11.3484 | 10.2725 | 10.2723
0.05 | 20.7420 | 20.7400 | 17.0364 | 17.0351 | 13.6952 | 13.6942 | 12.4607 | 12.4598
0.07 | 27.0657 | 27.0603 | 22.5273 | 22.5249 | 18.4051 | 18.4031 | 16.8727 | 16.8703
0.09 | 39.7687 | 39.7637 | 33.6864 | 33.6867 | 28.1086 | 28.1063 | 26.0187 | 26.0171
0.12 | 96.5153 | 96.5062 | 84.5912 | 84.5858 | 73.4531 | 73.4475 | 69.2179 | 69.2136

Figure 3: Absorption cross section for $l=0$.
Figure 4: Absorption cross section for small $M$.

Figure 5: Absorption cross section for $l=0,1,2,3,4$.

Figure 6: Total absorption cross section. The lines represent the values for
high frequency absorption, the values obtained by the null geodesics are: (a)
$27.0,28.9,30.5,32.8,35.9$, (b) $27.0,24.9,23.0,21.2,19.6$, (c)
$19.6,21.2,22.5,24.3,26.8$ and (d) $35.9,33.4,31.0,28.8,26.8$. Figure 7:
Differential scattering cross section for $M\omega=1.0$. Figure 8:
Differential scattering cross section for $M\omega=2.0$.

##  III Null Geodesics and Shadow

In this section, we will apply the null geodesic method to compute the
absorption at the high frequency limit and we will also determine the shadow
radius of the noncommutative black hole with global monopole. Therefore, at
this limit, we check the validity of our numerical results.

###  III.1 Null Geodesics

As a starting point we consider the following Lagrangian density:

$\displaystyle\mathcal{L}\equiv\dfrac{1}{2}g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu},$
(55)

being “.” the derivative with respect to an affine parameter. Hence, by
applying the line element (1), we have

$2\mathcal{L}=F(r)\dot{t}^{2}-\dfrac{\dot{r}^{2}}{F(r)}-r^{2}\left(\dot{\vartheta}^{2}+\sin^{2}\vartheta\dot{\phi}^{2}\right).$
(56)

At this point, we will consider the motion on the equatorial plane by taking
$\vartheta=\pi/2$. Hence, we can find the equations of motion from the
Hamilton-Jacobi equation such that

$E=F(r)\dot{t},\qquad\quad L=r^{2}\dot{\phi},$ (57)

where $E$ (energy) and $L$ (angular momentum) are the conserved quantities. In
the null geodesics analysis we have, $g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$
and so we find

$\dot{r}^{2}+F(r)\dfrac{L^{2}}{r^{2}}=E^{2}.$ (58)

Now, by introducing a new variable $u=1/r$ to write the orbital equation in
terms of the radii $r_{\eta}$ and $r_{\theta}$, we obtain

$\left(\dfrac{du}{d\phi}\right)^{2}=\dfrac{1}{b^{2}}-\left(1-8\pi\eta^{2}\right)\left(1-ur_{\eta}\right)\left(1-ur_{\theta}\right)u^{2},$
(59)

here $b=L/E$ is the impact parameter defined as the perpendicular distance
(measured at infinity) between the geodesic and a parallel line that passes
through the origin. Hence, by differentiating Eq. (59), we find

$\dfrac{d^{2}u}{d\phi^{2}}=\dfrac{\left(1-8\pi\eta^{2}\right)u}{2}\left[ur_{\eta}\left(1-ur_{\theta}\right)+ur_{\theta}\left(1-ur_{\eta}\right)-2\left(1-ur_{\eta}\right)\left(1-ur_{\theta}\right)\right].$
(60)

The behavior of the geodesic lines for different values of the impact
parameter $b$ can be obtained by solving the equations (59) and (60)
numerically. In Fig. 9 we present the change of the geodesic lines for
different values of the parameters $b$, $\Theta$ and $\eta$. In the graphics,
we have a black disk that represents the limit of the event horizon, the
internal dotted circle is the radius for the photon sphere (critical radius),
and the external dashed circle is the critical impact parameter (shadow).
However, by fixing $\eta$ and increasing the values of the parameter $\Theta$
we see a reduction in the effect of the black hole on the light beam, while
for fixed $\Theta$ and varying $\eta$ we have an increase in this effect.

To determine the critical radius ($r_{c}$) and the critical impact parameter
($b_{c}$), we impose conditions $du/d\phi=0$ and $d^{2}u/d\phi^{2}=0$, to
obtain

$\displaystyle r_{c}$ $\displaystyle=$
$\displaystyle\dfrac{1}{4}\left[3\left(r_{\eta}+r_{\theta}\right)+\sqrt{9\left(r_{\eta}+r_{\theta}\right)^{2}-32r_{\eta}r_{\theta}}\right],$
(61) $\displaystyle b_{c}$ $\displaystyle=$
$\displaystyle\dfrac{r_{c}^{2}}{\sqrt{\left(1-8\pi\eta^{2}\right)\left(r_{c}-r_{\eta}\right)\left(r_{c}-r_{\theta}\right)}}.$
(62)

Therefore, for high frequencies the absorption is determined by

$\sigma_{abs}^{hf}=\pi
b^{2}_{c}=\dfrac{\pi\left(3M+\sqrt{9M^{2}+K_{1}}\right)^{4}}{8\left(1-8\pi\eta^{2}\right)^{3}\left(3M^{2}+K_{2}+M\sqrt{9M^{2}+K_{1}}\right)},$
(63)

where

$\displaystyle
K_{1}=-\frac{64M\left(1-8\pi\eta^{2}\right)\sqrt{\theta}}{\sqrt{\pi}}+\frac{128\left(1-8\pi\eta^{2}\right)^{2}{\theta}}{{\pi}},$
(64) $\displaystyle
K_{2}=-\dfrac{16M\left(1-8\pi\eta^{2}\right)\sqrt{\theta}}{\sqrt{\pi}}+\dfrac{32\left(1-8\pi\eta^{2}\right)^{2}\theta}{\pi}.$
(65)

Observe that making $\eta=\theta=0$, we have $\sigma_{abs}^{hf}=27\pi M^{2}$
which is the result for the Schwarzschild black hole.

For $\eta\neq 0$ and $\theta=0$, the absorption becomes

$\displaystyle\sigma_{abs}^{hf}=27\pi M^{2}(1+24\pi\eta^{2}+\cdots).$ (66)

Hence by varying the values of $\eta$, the absorption value at the high
frequency limit has its value increased as shown in Fig. 6. For $\eta=0$ and
$\theta\neq 0$, we obtain

$\displaystyle\sigma_{abs}^{hf}=27\pi M^{2}-72\pi
M\sqrt{\frac{\theta}{\pi}}+\cdots.$ (67)

Here the absorption at the high frequency limit has its value reduced when we
vary $\theta$ as shown in Fig. 6 .

Figure 9: Null geodesics in polar coordinates for different values of $\Theta$
and $\eta$, assuming values for the impact parameter between 4 and 7. The
radius of the horizon corresponds to the radius of the disk in black, the
inner black ring is the critical radius or the radius of the sphere of photos
while the outer ring corresponds to the critical impact parameter.

Now, we will calculate the size of the black hole shadow using the geodetic
study and expressing it via celestial coordinates.

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\lim\limits_{r_{o}\to\infty}\left[-r_{o}^{2}\sin\theta_{o}\dfrac{d\phi}{dr}\Bigr{\rvert}_{\theta=\theta_{o}}\right],$
(68) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle\lim\limits_{r_{o}\to\infty}\left[r_{o}^{2}\dfrac{d\theta}{dr}\Bigr{\rvert}_{\theta=\theta_{o}}\right],$
(69)

where $\left(r_{o},\theta_{o}\right)$ is the observer’s position at infinity.
In this case, for an observer in the equatorial plane, that is in
$\theta_{o}=\pi/2$, we have the following relation

$R_{s}\equiv\sqrt{\alpha^{2}+\beta^{2}}=b_{c}.$ (70)

Then

$\displaystyle
R_{s}=\left[\dfrac{\left(3M+\sqrt{9M^{2}+K_{1}}\right)^{4}}{8\left(1-8\pi\eta^{2}\right)^{3}\left(3M^{2}+K_{2}+M\sqrt{9M^{2}+K_{1}}\right)}\right]^{1/2}.$
(71)

Figure 10: Influence of the $\Theta$ and $\eta$ parameters on the shadow
radius.

For $\theta$ and $\eta$ small, we get

$\displaystyle
R_{s}=3\sqrt{3}M\left(1+12\pi\eta^{2}\right)-4\sqrt{3}(1+4\pi\eta^{2})\sqrt{\frac{\theta}{\pi}}+\cdots.$
(72)

Note that the global monopole causes an increase in the shadow radius when we
increase the $\eta$ parameter as we can see in Fig. 10. On the other hand,
noncommutativity has the effect of reducing the shadow radius. Besides, for
$\theta=0$ in (72), we have

$\displaystyle R_{s}\approx 3\sqrt{3}M\left(1+12\pi\eta^{2}\right).$ (73)

Here the shadow radius has its value increased when we vary the $\eta$
parameter.

For $\eta=0$ in (72), we obtain

$\displaystyle R_{s}=3\sqrt{3}M-4\sqrt{3}\sqrt{\frac{\theta}{\pi}}+\cdots.$
(74)

Hence the shadow radius has its value reduced when we vary the $\Theta$
parameter.

Next, taking the null mass limit in (71) we find that the shadow radius is
non-zero, given by

$\displaystyle R_{smin}\approx\frac{4}{3}(1+12\pi\eta^{2})M_{min}.$ (75)

Therefore, we can see that by varying the $\eta$ parameter with a fixed
$\Theta$ the shadow radius increases as we can see in Fig. 11 and Fig. 12.

Figure 11: Influence of the $\Theta$ and $\eta$ parameters on the shadow
radius.

Figure 12: Influence of the $\Theta$ and $\eta$ parameters on the shadow
radius.

###  III.2 Classic Analysis

Rewriting the equation of the orbit (59) in terms of $r$, we have

$\dfrac{d\phi}{dr}=\dfrac{1}{r^{2}}\left(\dfrac{1}{b^{2}}-\dfrac{(1-8\pi\eta^{2})}{r^{2}}+\dfrac{2M}{r^{3}}-\dfrac{8M}{r^{4}}\sqrt{\dfrac{\theta}{\pi}}\right)^{-1/2}.$
(76)

To calculate the total angle deflected by the beam we have to integrate the
above equation. One way to integrate analytically is to take the limit
$M/r<<1$. Before doing this expansion let us organize the equation by making
changes of variables

$\displaystyle y$ $\displaystyle=$
$\displaystyle\dfrac{1}{r}\left(\delta-\dfrac{M}{r}+\dfrac{(\tilde{\theta}-1)}{2}\dfrac{M^{2}}{r^{2}}\right),$
(77) $\displaystyle y^{2}$ $\displaystyle=$
$\displaystyle\dfrac{1}{r^{2}}\left[\left(\delta-\dfrac{M}{r}\right)^{2}+\left(\delta-\dfrac{M}{r}\right)\dfrac{(\tilde{\theta}-1)M^{2}}{r^{2}}+\dfrac{(\tilde{\theta}-1)^{2}}{4}\dfrac{M^{4}}{r^{4}}\right],$
(78) $\displaystyle\dfrac{dy}{dr}$ $\displaystyle=$
$\displaystyle-\dfrac{\delta}{r^{2}}+\dfrac{2M}{r^{3}}-\dfrac{3(\tilde{\theta}-1)M^{2}}{2r^{4}},$
(79)

where we define $\tilde{\theta}=\dfrac{8}{M}\sqrt{\dfrac{\theta}{\pi}}$ and
$\delta=1-8\pi\eta^{2}$, so the equation (76) is rewritten in this new
variable as

$\displaystyle\small\dfrac{d\phi}{dy}$ $\displaystyle=$
$\displaystyle\left(-\delta+\dfrac{2M}{r}-\dfrac{3(\tilde{\theta}-1)}{2}\dfrac{M^{2}}{r^{2}}\right)^{-1}\left\\{\dfrac{1}{b^{2}}-\dfrac{y^{2}}{\delta^{2}}+\dfrac{1}{\delta
r^{2}}\left[(\delta-1)\left(-\delta+\dfrac{2M}{r}-\dfrac{\tilde{\theta}M^{2}}{r^{2}}\right)\right.\right.$
(80) $\displaystyle-$ $\displaystyle\left.\left.\dfrac{M^{2}}{\delta
r^{2}}\left(\delta-1+(\tilde{\theta}-1)\dfrac{M}{\delta
r}-(\tilde{\theta}-1)^{2}\dfrac{M^{2}}{4\delta
r^{2}}\right)\right]\right\\}^{-1/2}.$

Now expanding the above equation at the limit $M/r<<1$ and organizing the
terms, we find

$\displaystyle\dfrac{d\phi}{dy}=\dfrac{b}{\delta^{2}\sqrt{1-\dfrac{y^{2}b^{2}}{\delta^{2}}}}\left\\{\delta+\dfrac{2M}{r}-\dfrac{2\left(-4+3\delta(\tilde{\theta}-1)\right)}{\delta^{2}}\dfrac{M^{3}}{r^{3}}+\dfrac{\left(8-3(\tilde{\theta}-1)\delta\right)}{2\delta}\dfrac{M^{2}}{r^{2}}+\dfrac{b^{2}\delta^{3}(1-\delta)}{2\left(b^{2}y^{2}-\delta^{2}\right)r^{2}}+\cdots\right\\}.$
(81)

We can identify the $y$ in terms of $M/r$ and a $y^{2}$ in $M^{2}/r^{2}$, so
we have

$\dfrac{d\phi}{dy}=\dfrac{b}{\sqrt{1-\dfrac{y^{2}b^{2}}{\delta^{2}}}}\left[\delta+\dfrac{2My}{\delta}+\dfrac{\left(12-3(\tilde{\theta}-1)\delta\right)M^{2}y^{2}}{2\delta^{3}}\right].$
(82)

The total angle deflected by the beam is given by

$\displaystyle\Delta\phi$ $\displaystyle=$ $\displaystyle
2\int_{0}^{y_{0}}\dfrac{b}{\sqrt{1-\dfrac{y^{2}b^{2}}{\delta^{2}}}}\left[\delta+\dfrac{2My}{\delta}+\dfrac{\left(12-3(\tilde{\theta}-1)\delta\right)M^{2}y^{2}}{2\delta^{3}}\right]dy,$
(83) $\displaystyle\Delta\phi$ $\displaystyle=$
$\displaystyle-\dfrac{M\sqrt{1-b^{2}y_{0}^{2}/\delta^{2}}\left(8\delta^{2}-3My_{0}(4+\delta-\tilde{\theta})\right)}{2b\delta^{3}}+\dfrac{4b^{2}\delta^{2}+3M^{2}(4+\delta-\delta\tilde{\theta})\arcsin(by_{0}/\delta)}{2b^{2}\delta^{2}}+\dfrac{4M}{b\delta}.$
(84)

Knowing that at the limit $M/r<<1$ we have that $y_{0}\rightarrow\delta/b$ and
the result becomes

$\Delta\phi=\dfrac{4M}{b\delta}+\pi+\dfrac{3\pi(4+\delta(1-\tilde{\theta}))M^{2}}{4b^{2}\delta^{2}}.$
(85)

Finally, the deflection angle $\chi=\Delta\phi-\pi$ in terms of
$\sqrt{\theta}$ and $\eta^{2}$ is given by

$\chi=\dfrac{4M}{b(1-8\pi\eta^{2})}+\dfrac{3\pi(5-8\pi\eta^{2})M^{2}}{4b^{2}(1-8\pi\eta^{2})^{2}}-\dfrac{6\pi
M}{b^{2}(1-8\pi\eta^{2})}\sqrt{\dfrac{\theta}{\pi}}.$ (86)

With the relationship of the deflection angle and the impact parameter, we can
combine this result with the classical differential scattering section
equation

$\dfrac{d\sigma_{clas}}{d\Omega}=\dfrac{b}{\sin\vartheta}\Big{|}\dfrac{db}{d\vartheta}\Big{|},$
(87)

to get the following result at the limit of small $\vartheta$

$\displaystyle\dfrac{d\sigma_{clas}}{d\Omega}\approx\dfrac{16M^{2}}{(-1+8\pi\eta^{2})^{2}\vartheta^{4}}+\dfrac{3\pi
M^{2}(5-8\pi\eta^{2})}{4(-1+8\pi\eta^{2})^{2}\vartheta^{3}}-\dfrac{6\pi
M}{(-1+8\pi\eta^{2})^{2}\vartheta^{3}}\sqrt{\dfrac{\theta}{\pi}}.$ (88)

We can see in the equation (88) that the influence of the non-commutative
parameter is very small Fig 13, since the monopole term will have a
contribution to the classical case even in the limit of small angles Fig 13 .

Figure 13: Classical scattering differential cross section.

In Fig. 14 we show the behavior of the curves for the classical and numerical
case. As expected the results for classical analysis fit well in the numerical
plot for small angles.

Figure 14: Classical and numerical scattering differential cross section. For
the numerics we assume $M\omega=1$ in (a) and $M\omega=2$ in (b).

##  IV Conclusions

In summary, in this paper, we investigate the process of massless scalar wave
scattering due to a noncommutative black hole with a global monopole through
the partial wave method. We have obtained, at the low frequency limit, the
phase shift analytically. The effects of the noncommutativity cause a
reduction in the differential scattering/absorption cross section and shadow
radius, while the presence of the global monopole has the effect of increasing
the value of such quantities. In addition, we show that these quantities are
non-zero when we take the limit of the zero mass parameter. Rather, they are
proportional to a minimum mass. Moreover, we have also checked the
analytically obtained results by numerically solving the radial equation for
arbitrary frequencies.

###### Acknowledgements.

We would like to thank CNPq, CAPES and CNPq/PRONEX/FAPESQ-PB (Grant no.
165/2018), for partial financial support. MAA, FAB and EP acknowledge support
from CNPq (Grant nos. 306398/2021-4, 312104/2018-9, 304852/2017-1).

## References

  * (1) S. W. Hawking, “Black holes in general relativity,” Commun. Math. Phys. 25, 152-166 (1972) doi:10.1007/BF01877517
  * (2) T. Padmanabhan,“Gravity and the thermodynamics of horizons,” Phys. Rept. 406, 49-125 (2005) [arXiv:gr-qc/0311036 [gr-qc]].
  * (3) R. J. Szabo, “Symmetry, gravity and noncommutativity,” Class. Quant. Grav. 23, R199-R242 (2006) doi:10.1088/0264-9381/23/22/R01 [arXiv:hep-th/0606233 [hep-th]].
  * (4) S. Ansoldi, P. Nicolini, A. Smailagic and E. Spallucci, “Noncommutative geometry inspired charged black holes,” Phys. Lett. B 645, 261-266 (2007) doi:10.1016/j.physletb.2006.12.020 [arXiv:gr-qc/0612035 [gr-qc]].
  * (5) P. Nicolini, “Noncommutative Black Holes, The Final Appeal To Quantum Gravity: A Review,” Int. J. Mod. Phys. A 24, 1229-1308 (2009) doi:10.1142/S0217751X09043353 [arXiv:0807.1939 [hep-th]].
  * (6) P. Nicolini, A. Smailagic and E. Spallucci, “Noncommutative geometry inspired Schwarzschild black hole,” Phys. Lett. B 632, 547-551 (2006) doi:10.1016/j.physletb.2005.11.004 [arXiv:gr-qc/0510112 [gr-qc]].
  * (7) K. Nozari and S. H. Mehdipour, “Hawking Radiation as Quantum Tunneling from Noncommutative Schwarzschild Black Hole,” Class. Quant. Grav. 25, 175015 (2008) doi:10.1088/0264-9381/25/17/175015 [arXiv:0801.4074 [gr-qc]].
  * (8) A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects, Cambridge University Press, 2000.
  * (9) T. W. B. Kibble, “Topology of Cosmic Domains and Strings,” J. Phys. A 9, 1387-1398 (1976) doi:10.1088/0305-4470/9/8/029
  * (10) E. R. Bezerra de Mello and C. Furtado, “The Nonrelativistic scattering problem by a global monopole,” Phys. Rev. D 56, 1345-1348 (1997) doi:10.1103/PhysRevD.56.1345
  * (11) J. P. M. Pitelli and P. S. Letelier, “Quantum Singularities Around a Global Monopole,” Phys. Rev. D 80, 104035 (2009) doi:10.1103/PhysRevD.80.104035 [arXiv:0911.2626 [gr-qc]].
  * (12) M. Sharif and S. Iftikhar, “Null Geodesics and Strong Field Gravitational Lensing of Black Hole with Global Monopole,” Adv. High Energy Phys. 2015, 854264 (2015) doi:10.1155/2015/854264
  * (13) R. Shaikh, “Black hole shadow in a general rotating spacetime obtained through Newman-Janis algorithm,” Phys. Rev. D 100, no.2, 024028 (2019) doi:10.1103/PhysRevD.100.024028 [arXiv:1904.08322 [gr-qc]].
  * (14) S. Haroon, K. Jusufi and M. Jamil, “Shadow Images of a Rotating Dyonic Black Hole with a Global Monopole Surrounded by Perfect Fluid,” Universe 6, no.2, 23 (2020) doi:10.3390/universe6020023 [arXiv:1904.00711 [gr-qc]].
  * (15) M. Barriola and A. Vilenkin, “Gravitational Field of a Global Monopole,” Phys. Rev. Lett. 63, 341 (1989) doi:10.1103/PhysRevLett.63.341
  * (16) M. A. Anacleto, F. A. Brito, S. J. S. Ferreira and E. Passos, “Absorption and scattering of a black hole with a global monopole in f(R) gravity,” Phys. Lett. B 788, 231 (2019) doi:10.1016/j.physletb.2018.11.020 [arXiv:1701.08147 [hep-th]].
  * (17) P. K. Townsend, Black holes: Lecture notes, [arXiv:gr-qc/9707012 [gr-qc]].
  * (18) J. A. H. Futterman, F. A. Handler and R. A. Matzner, Scattering from black holes (Cambridge University Press, England, 1988) doi:10.1017/CBO9780511735615
  * (19) Q. Li, C. Ma, Y. Zhang, Z. W. Lin and P. F. Duan, “Shadow, absorption and Hawking radiation of a Schwarzschild black hole surrounded by a cloud of strings in Rastall gravity,” Eur. Phys. J. C 82, no.7, 658 (2022) doi:10.1140/epjc/s10052-022-10623-3
  * (20) Y. Xing, Y. Yang, D. Liu, Z. W. Long and Z. Xu, “The ringing of quantum corrected Schwarzschild black hole with GUP,” Commun. Theor. Phys. 74, no.8, 085404 (2022) doi:10.1088/1572-9494/ac7cdc [arXiv:2204.11262 [gr-qc]].
  * (21) G. S. Bisnovatyi-Kogan and O. Y. Tsupko, “Analytical study of higher-order ring images of the accretion disk around a black hole,” Phys. Rev. D 105, no.6, 064040 (2022) doi:10.1103/PhysRevD.105.064040 [arXiv:2201.01716 [gr-qc]].
  * (22) X. X. Zeng, G. P. Li and K. J. He, “ The shadows and observational appearance of a noncommutative black hole surrounded by various profiles of accretions,” Nucl. Phys. B 974, 115639 (2022) doi:10.1016/j.nuclphysb.2021.115639 [arXiv:2106.14478 [hep-th]].
  * (23) M. F. Mourad and M. Abdelgaber, “Spherically symmetric AdS black holes with smeared mass distribution,” Mod. Phys. Lett. A 36, no.05, 2150029 (2021) doi:10.1142/S0217732321500292
  * (24) S. K. Jha and A. Rahaman, “Lorentz violation and noncommutative effect on superradiance scattering off Kerr-like black hole and on the shadow of it,” [arXiv:2111.02817 [gr-qc]].
  * (25) Q. Li, C. Ma, Y. Zhang, Z. W. Lin and P. F. Duan, “Gray-body factor and absorption of the Dirac field in ESTGB gravity,” Chin. J. Phys. 77, 1269-1277 (2022) doi:10.1016/j.cjph.2022.03.027
  * (26) D. J. Gogoi and U. D. Goswami, “Quasinormal modes and Hawking radiation sparsity of GUP corrected black holes in bumblebee gravity with topological defects,” JCAP 06, no.06, 029 (2022) doi:10.1088/1475-7516/2022/06/029 [arXiv:2203.07594 [gr-qc]].
  * (27) R. Karmakar, D. J. Gogoi and U. D. Goswami, “Quasinormal modes and thermodynamic properties of GUP-corrected Schwarzschild black hole surrounded by quintessence,” [arXiv:2206.09081 [gr-qc]].
  * (28) N. J. L. S. Lobos and R. C. Pantig, “Generalized Extended Uncertainty Principle Black Holes: Shadow and lensing in the macro- and microscopic realms,” [arXiv:2208.00618 [gr-qc]].
  * (29) O. Y. Tsupko, “Shape of higher-order photon rings: analytical description with polar curve,” [arXiv:2208.02084 [gr-qc]].
  * (30) X. X. Zeng, M. I. Aslam and R. Saleem, “The Optical Appearance of Charged Four-Dimensional Gauss-Bonnet Black Hole with Strings Cloud and Non-Commutative Geometry Surrounded by Various Accretions Profiles,” [arXiv:2208.06246 [gr-qc]].
  * (31) M. Heydari-Fard and M. Heydari-Fard, “Null geodesics and shadow of 4D Einstein–Gauss–Bonnet black holes surrounded by quintessence,” Int. J. Mod. Phys. D 31, no.08, 2250066 (2022) doi:10.1142/S0218271822500663 [arXiv:2109.02059 [gr-qc]].
  * (32) M. Heydari-Fard, M. Heydari-Fard and H. R. Sepangi, “Null geodesics and shadow of hairy black holes in Einstein-Maxwell-dilaton gravity,” Phys. Rev. D 105, no.12, 124009 (2022) doi:10.1103/PhysRevD.105.124009 [arXiv:2110.02713 [gr-qc]].
  * (33) M. Khodadi, G. Lambiase and D. F. Mota, “No-hair theorem in the wake of Event Horizon Telescope,” JCAP 09, 028 (2021) doi:10.1088/1475-7516/2021/09/028 [arXiv:2107.00834 [gr-qc]].
  * (34) M. Fathi, M. Olivares and J. R. Villanueva, “Gravitational Rutherford scattering of electrically charged particles from a charged Weyl black hole,” Eur. Phys. J. Plus 136, no.4, 420 (2021) doi:10.1140/epjp/s13360-021-01441-9 [arXiv:2009.03404 [gr-qc]].
  * (35) H. Chen, H. Hassanabadi, B. C. Lütfüoğlu and Z. W. Long, “Quantum corrections to the quasinormal modes of the Schwarzschild black hole,” [arXiv:2203.03464 [gr-qc]].
  * (36) R. A. Matzner and M. P. Ryan, “Low Frequency Limit of Gravitational Scattering,” Phys. Rev. D 16, 1636-1642 (1977) doi:10.1103/PhysRevD.16.1636
  * (37) P. J. Westervelt, “Scattering of electromagnetic and gravitational waves by a static gravitational field - comparison between the classical (general-relativistic) and quantum field-theoretic results,” Phys. Rev. D 3, 2319-2324 (1971) doi:10.1103/PhysRevD.3.2319
  * (38) P. C. Peters, “Differential Cross-Sections for Weak Field Gravitational Scattering,” Phys. Rev. D 13, 775-777 (1976) doi:10.1103/PhysRevD.13.775
  * (39) N. G. Sanchez, “Scattering of scalar waves from a Schwarzschild black hole,” J. Math. Phys. 17, no.5, 688 (1976) doi:10.1063/1.522949
  * (40) N. G. Sanchez, “The Wave Scattering Theory and the Absorption Problem for a Black Hole,” Phys. Rev. D 16, 937-945 (1977) doi:10.1103/PhysRevD.16.937
  * (41) N. G. Sanchez, “Absorption and Emission Spectra of a Schwarzschild Black Hole,” Phys. Rev. D 18, 1030 (1978) doi:10.1103/PhysRevD.18.1030
  * (42) N. G. Sanchez, “Elastic Scattering of Waves by a Black Hole,” Phys. Rev. D 18, 1798 (1978) doi:10.1103/PhysRevD.18.1798
  * (43) W. K. De Logi and S. J. Kovacs, “Gravitational Scattering of Zero Rest Mass Plane Waves,” Phys. Rev. D 16, 237-244 (1977) doi:10.1103/PhysRevD.16.237
  * (44) C. Doran and A. Lasenby, “Perturbation theory calculation of the black hole elastic scattering cross-section,” Phys. Rev. D 66, 024006 (2002) doi:10.1103/PhysRevD.66.024006 [arXiv:gr-qc/0106039 [gr-qc]].
  * (45) S. R. Dolan, “Scattering of long-wavelength gravitational waves,” Phys. Rev. D 77, 044004 (2008) [arXiv:0710.4252 [gr-qc]].
  * (46) L. C. B. Crispino, S. R. Dolan and E. S. Oliveira, “Scattering of massless scalar waves by Reissner-Nordstrom black holes,” Phys. Rev. D 79, 064022 (2009) doi:10.1103/PhysRevD.79.064022 [arXiv:0904.0999 [gr-qc]].
  * (47) A. A. Starobinskil and S. M. Churilov, “Amplification of electromagnetic and gravitational waves scattered by a rotating ”black hole”,” Sov. Phys. JETP 65, no.1, 1-5 (1974).
  * (48) G. W. Gibbons, “Vacuum Polarization and the Spontaneous Loss of Charge by Black Holes,” Commun. Math. Phys. 44, 245-264 (1975) doi:10.1007/BF01609829
  * (49) D. N. Page, “Particle Emission Rates from a Black Hole: Massless Particles from an Uncharged, Nonrotating Hole,” Phys. Rev. D 13, 198-206 (1976) doi:10.1103/PhysRevD.13.198
  * (50) A. A. Starobinskii and S. M. Churilov, Zh. Eksp. Teor. Fiz. 65, 3 (1973).
  * (51) F. Moura, “Scattering of spherically symmetric $d$-dimensional $\alpha^{\prime}-$corrected black holes in string theory,” JHEP 1309, 038 (2013) doi:10.1007/JHEP09(2013)038 [arXiv:1105.5074 [hep-th]].
  * (52) E. Jung and D. K. Park, “Effect of scalar mass in the absorption and emission spectra of Schwarzschild black hole,” Class. Quant. Grav. 21, 3717-3732 (2004) doi:10.1088/0264-9381/21/15/007 [arXiv:hep-th/0403251 [hep-th]].
  * (53) E. Jung, S. Kim and D. K. Park, “Proof of universality for the absorption of massive scalar by the higher-dimensional Reissner-Nordstrom black holes,” Phys. Lett. B 602, 105-111 (2004) doi:10.1016/j.physletb.2004.09.067 [arXiv:hep-th/0409145 [hep-th]].
  * (54) C. Doran, A. Lasenby, S. Dolan and I. Hinder, “Fermion absorption cross section of a Schwarzschild black hole,” Phys. Rev. D 71, 124020 (2005) doi:10.1103/PhysRevD.71.124020 [arXiv:gr-qc/0503019 [gr-qc]].
  * (55) S. Dolan, C. Doran and A. Lasenby, “Fermion scattering by a Schwarzschild black hole,” Phys. Rev. D 74, 064005 (2006) doi:10.1103/PhysRevD.74.064005 [arXiv:gr-qc/0605031 [gr-qc]].
  * (56) J. Castineiras, L. C. B. Crispino and D. P. M. Filho, “Source coupled to the massive scalar field orbiting a stellar object,” Phys. Rev. D 75, 024012 (2007) doi:10.1103/PhysRevD.75.024012
  * (57) C. L. Benone, E. S. de Oliveira, S. R. Dolan and L. C. B. Crispino, “Absorption of a massive scalar field by a charged black hole,” Phys. Rev. D 89, no. 10, 104053 (2014) doi:10.1103/PhysRevD.89.104053 [arXiv:1404.0687 [gr-qc]].
  * (58) C. I. S. Marinho and E. S. de Oliveira, “Scattering of massless scalar waves from Schwarzschild-Tangherlini black holes on the brane,” arXiv:1612.05604 [gr-qc].
  * (59) S. R. Das, G. W. Gibbons and S. D. Mathur, “Universality of low-energy absorption cross-sections for black holes,” Phys. Rev. Lett. 78, 417 (1997) doi:10.1103/PhysRevLett.78.417 [hep-th/9609052].
  * (60) C. F. B. Macedo, L. C. B. Crispino and E. S. de Oliveira, “Scalar waves in regular Bardeen black holes: Scattering, absorption and quasinormal modes,” Int. J. Mod. Phys. D 25, no. 09, 1641008 (2016) doi:10.1142/S021827181641008X [arXiv:1605.00123 [gr-qc]].
  * (61) E. S. de Oliveira, “Scalar scattering from black holes with tidal charge,” Eur. Phys. J. C 78, no. 11, 876 (2018) doi:10.1140/epjc/s10052-018-6316-9 [arXiv:1805.04987 [gr-qc]].
  * (62) H. Hai, W. Yong-Jiu and C. Ju-Hua, “Absorption cross section of black holes with global monopole,” Chin. Phys. B 22, no. 7, 070401 (2013).
  * (63) L. C. B. Crispino, E. S. Oliveira and G. E. A. Matsas, “Absorption cross section of canonical acoustic holes,” Phys. Rev. D 76, 107502 (2007).
  * (64) S. R. Dolan, E. S. Oliveira and L. C. B. Crispino, “Scattering of sound waves by a canonical acoustic hole,” Phys. Rev. D 79, 064014 (2009)
  * (65) E. S. Oliveira, S. R. Dolan and L. C. B. Crispino, “Absorption of planar waves in a draining bathtub,” Phys. Rev. D 81, 124013 (2010).
  * (66) S. R. Dolan, E. S. Oliveira and L. C. B. Crispino, “Aharonov-Bohm effect in a draining bathtub vortex,” Phys. Lett. B 701, 485 (2011). doi:10.1016/j.physletb.2011.06.013
  * (67) M. A. Anacleto, F. A. Brito and E. Passos, “Analogue Aharonov-Bohm effect in a Lorentz-violating background,” Phys. Rev. D 86, 125015 (2012) doi:10.1103/PhysRevD.86.125015 [arXiv:1208.2615 [hep-th]].
  * (68) M. A. Anacleto, F. A. Brito and E. Passos, “Noncommutative analogue Aharonov-Bohm effect and superresonance,” Phys. Rev. D 87, no.12, 125015 (2013) doi:10.1103/PhysRevD.87.125015 [arXiv:1210.7739 [hep-th]].
  * (69) S. R. Dolan and E. S. Oliveira, “Scattering by a draining bathtub vortex,” Phys. Rev. D 87, no. 12, 124038 (2013) doi:10.1103/PhysRevD.87.124038 [arXiv:1211.3751 [gr-qc]].
  * (70) M. A. Anacleto, I. G. Salako, F. A. Brito and E. Passos, “Analogue Aharonov-Bohm effect in neo-Newtonian theory,” Phys. Rev. D 92, no. 12, 125010 (2015) doi:10.1103/PhysRevD.92.125010 [arXiv:1506.03440 [hep-th]];
  * (71) M. A. Anacleto, F. A. Brito, A. Mohammadi and E. Passos, “Aharonov-Bohm effect for a fermion field in the acoustic black hole ”spacetime”,” Eur. Phys. J. C 77, no.4, 239 (2017) doi:10.1140/epjc/s10052-017-4801-1 [arXiv:1606.09231 [hep-th]].
  * (72) M. A. Anacleto, F. A. Brito and E. Passos, “Gravitational Aharonov–Bohm effect due to noncommutative BTZ black hole,” Phys. Lett. B 743, 184-188 (2015) doi:10.1016/j.physletb.2015.02.056 [arXiv:1408.4481 [hep-th]]
  * (73) M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, “Absorption and scattering of a noncommutative black hole,” Phys. Lett. B 803, 135334 (2020) doi:10.1016/j.physletb.2020.135334 [arXiv:1907.13107 [hep-th]].
  * (74) M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, “Higher-derivative analogue Aharonov–Bohm effect, absorption and superresonance,” Int. J. Mod. Phys. A 35, no.21, 2050112 (2020) doi:10.1142/S0217751X20501122 [arXiv:1810.13356 [hep-th]].
  * (75) M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, “Quantum-corrected scattering and absorption of a Schwarzschild black hole with GUP,” Phys. Lett. B 810, 135830 (2020) doi:10.1016/j.physletb.2020.135830 [arXiv:2003.13464 [gr-qc]].
  * (76) M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, “Absorption and scattering by a self-dual black hole,” Gen. Rel. Grav. 52, no.10, 100 (2020) doi:10.1007/s10714-020-02756-1 [arXiv:2002.12090 [hep-th]].
  * (77) M. A. Anacleto, F. A. Brito, B. R. Carvalho and E. Passos, “Noncommutative correction to the entropy of BTZ black hole with GUP,” Adv. High Energy Phys. 2021, 6633684 (2021) doi:10.1155/2021/6633684 [arXiv:2010.09703 [hep-th]].
  * (78) M. A. Anacleto, F. A. Brito, S. S. Cruz and E. Passos, “Noncommutative correction to the entropy of Schwarzschild black hole with GUP,” Int. J. Mod. Phys. A 36, no.03, 2150028 (2021) doi:10.1142/S0217751X21500287 [arXiv:2010.10366 [hep-th]].
  * (79) J. A. V. Campos, M. A. Anacleto, F. A. Brito and E. Passos, “Quasinormal modes and shadow of noncommutative black hole,” Sci. Rep. 12, no.1, 8516 (2022) doi:10.1038/s41598-022-12343-w [arXiv:2103.10659 [hep-th]].
  * (80) D. R. Yennie, D. G. Ravenhall and R. N. Wilson, “Phase-Shift Calculation of High-Energy Electron Scattering,” Phys. Rev. 95, 500-512 (1954) doi:10.1103/PhysRev.95.500
  * (81) I. I. Cotaescu, C. Crucean and C. A. Sporea, “Partial wave analysis of the Dirac fermions scattered from Schwarzschild black holes,” Eur. Phys. J. C 76, no. 3, 102 (2016) doi:10.1140/epjc/s10052-016-3936-9 [arXiv:1409.7201 [gr-qc]].


How the Avidity of Polymerase Binding to the -35/-10 Promoter Sites Affects
Gene Expression

Tal Einav1,∗, Rob Phillips1,2,3,∗

1Department of Physics, California Institute of Technology, Pasadena, CA,
91125, USA

2Department of Applied Physics, California Institute of Technology, Pasadena,
CA, 91125, USA

3Division of Biology and Biological Engineering, California Institute of
Technology, Pasadena, CA, 91125, USA

*Corresponding author<EMAIL_ADDRESS>(512) 468-1994; 1200 E. California Blvd, MC 103-33, Pasadena, CA 91125 *Corresponding author<EMAIL_ADDRESS>(626) 395-3374; 1200 E. California Blvd, MC 128-95, Pasadena, CA 91125

## Abstract

Although the key promoter elements necessary to drive transcription in
Escherichia coli have long been understood, we still cannot predict the
behavior of arbitrary novel promoters, hampering our ability to characterize
the myriad of sequenced regulatory architectures as well as to design new
synthetic circuits. This work builds on a beautiful recent experiment by
Urtecho et al. who measured the gene expression of over 10,000 promoters
spanning all possible combinations of a small set of regulatory elements.
Using this data, we demonstrate that a central claim in energy matrix models
of gene expression – that each promoter element contributes independently and
additively to gene expression – contradicts experimental measurements. We
propose that a key missing ingredient from such models is the avidity between
the -35 and -10 RNA polymerase binding sites and develop what we call a
refined energy matrix model that incorporates this effect. We show that this
the refined energy matrix model can characterize the full suite of gene
expression data and explore several applications of this framework, namely,
how multivalent binding at the -35 and -10 sites can buffer RNAP kinetics
against mutations and how promoters that bind overly tightly to RNA polymerase
can inhibit gene expression. The success of our approach suggests that avidity
represents a key physical principle governing the interaction of RNA
polymerase to its promoter.

## Significance Statement

Cellular behavior is ultimately governed by the genetic program encoded in its
DNA and through the arsenal of molecular machines that actively transcribe its
genes, yet we lack the ability to predict how an arbitrary DNA sequence will
perform. To that end, we analyze the performance of over 10,000 regulatory
sequences and develop a model that can predict the behavior of any sequence
based on its composition. By considering promoters that only vary by one or
two elements, we can characterize how different components interact, providing
fundamental insights into the mechanisms of transcription.

## Introduction

Promoters modulate the complex interplay of RNA polymerase (RNAP) and
transcription factor binding that ultimately regulates gene expression. While
our knowledge of the molecular players that mediate these processes constantly
improves, more than half of all promoters in Escherichia coli still have no
annotated transcription factors in RegulonDB [1] and our ability to design
novel promoters that elicit a target level of gene expression remains limited.

As a step towards taming the vastness and complexity of sequence space, the
recent development of massively parallel reporter assays has enabled entire
libraries of promoter mutants to be simultaneously measured [2, 3, 4]. Given
this surge in experimental prowess, the time is ripe to reexamine how well our
models of gene expression can extrapolate the response of a general promoter.

A common approach to quantifying gene expression, called the energy matrix
model, assumes that every promoter element contributes additively and
independently to the total RNAP (or transcription factor) binding energy [3].
This model treats all base pairs on an equal footing and does not incorporate
mechanistic details of RNAP-promoter interactions such as its strong binding
primarily at the -35 and -10 binding motifs (shown in Fig. 1A). A newer method
recently took the opposite viewpoint, designing an RNAP energy matrix that
only includes the -35 element, -10 element, and the length of the spacer
separating them [5], neglecting the sequence composition of the spacer or the
surrounding promoter region.

Although these methods have been successfully used to identify important
regulatory elements in unannotated promoters [6] and predict evolutionary
trajectories [5], it is clear that there is more to the story. Even in the
simple case of the highly-studied lac promoter, such energy matrices show
systematic deviations from measured levels of gene expressions, indicating
that some fundamental component of transcriptional regulation is still missing
[7].

We propose that one failure of current models lies in their tacit assumption
that every promoter element contributes independently to the RNAP binding
energy. By naturally relaxing this assumption to include the important effects
of avidity, we can push beyond the traditional energy matrix analysis in
several key ways including: (i) We can identify which promoter elements
contribute independently or cooperatively without recourse to fitting, thereby
building an unbiased mechanistic model for systems that bind at multiple
sites. (ii) Applying this approach to RNAP-promoter binding reveals that the
-35 and -10 motifs bind cooperatively, a feature that we attribute to avidity.
Moreover, we show that models that instead assume the -35 and -10 elements
contribute additively and independently sharply contradict the available data.
(iii) We show that the remaining promoter elements (the spacer, UP, and
background shown in Fig. 1A) do contribute independently and additively to the
RNAP binding energy and formulate the corresponding model for transcriptional
regulation that we call a refined energy matrix model. (iv) We use this model
to explore how the interactions between the -35 and -10 elements can buffer
RNAP kinetics against mutations. (v) We analyze a surprising feature of the
data where overly-tight RNAP-promoter binding can lead to decreased gene
expression. (vi) We validate our model by analyzing the gene expression of
over 10,000 promoters in E. coli recently published by Urtecho et al. [8] and
demonstrate that our framework markedly improves upon the traditional energy
matrix analysis.

While this work focuses on RNAP-promoter binding, its implications extend to
general regulatory architectures involving multiple tight-binding elements
including transcriptional activators that make contact with RNAP (CRP in the
lac promoter [9]), transcription factors that oligomerize (as recently
identified for the xylE promoter [6]), and transcription factors that bind to
multiple sites on the promoter (DNA looping mediated by the Lac repressor
[10]). More generally, this approach of categorizing which binding elements
behave independently (without resorting to fitting) can be applied to
multivalent interactions in other biological contexts including novel
materials, scaffolds, and synthetic switches [11, 12].

## Results

### The -35 and -10 Binding Sites give rise to Gene Expression that Defies
Characterization as Independent and Additive Components

Decades of research have shed light upon the exquisite biomolecular details
involved in bacterial transcriptional regulation via the family of RNAP
$\sigma$ factors [13]. In this work, we restrict our attention to the
$\sigma^{70}$ holoenzyme [8], the most active form under standard E. coli
growth condition, whose interaction with a promoter includes direct contact
with the -35 and -10 motifs (two hexamers centered roughly 10 and 35 bases
upstream of the transcription start site), a spacer region separating these
two motifs, an UP element just upstream of the -35 motif that anchors the
C-terminal domain ($\alpha$CTD) of RNAP, and the background promoter sequence
surrounding these elements.

Urtecho et al. constructed a library of promoters composed of every
combination of eight -35 motifs, eight -10 motifs, eight spacers, eight
backgrounds (BG), and three UP elements (Fig. 1A) [8]. Each sequence was
integrated at the same locus within the E. coli genome and transcription was
quantified via DNA barcoding and RNA sequencing. One of the three UP elements
considered was the absence of an UP binding motif, and this case will serve as
the starting point for our analysis.

The traditional energy matrix approach used by Urtecho et al. posits that
every base pair of the promoter will contribute additively and independently
to the RNAP binding energy [8], which by appropriately grouping base pairs is
equivalent to stating that the free energy of RNAP binding will be the sum of
its contributions from the background, spacer, -35, and -10 elements (see
Appendix A). Hence, the gene expression (GE) is given by the Boltzmann factor

$\text{GE}\propto e^{-\beta(E_{\text{BG}}+E_{\text{Spacer}}+E_{\mathchar
45\relax 35}+E_{\mathchar 45\relax 10})}.$ (1)

Note that all $E_{j}$s represent free energies (with an energetic and entropic
component); to see the explicit dependence on RNAP copy number, refer to
Appendix A. Fitting the 32 free energies (one for each background, spacer,
-35, and -10 element) and the constant of proportionality in Eq. 1 enables us
to predict the expression of $8\times 8\times 8\times 8=4{,}096$ promoters.

Fig. 2A demonstrates that Eq. 1 leads to a poor characterization of these
promoters ($R^{2}=0.57$, parameter values listed in Appendix B), suggesting
that critical features of gene expression are missing from this model. One
possible resolution is to assumes that the level of gene expression saturates
for very strong promoters at $r_{\text{max}}$ and for very weak promoters at
$r_{0}$ (caused by background noise or serendipitous near-consensus sequences
[5]), namely,

$\text{GE}=\frac{r_{0}+r_{\text{max}}e^{-\beta(E_{\text{BG}}+E_{\text{Spacer}}+E_{\mathchar
45\relax 35}+E_{\mathchar 45\relax
10})}}{1+e^{-\beta(E_{\text{BG}}+E_{\text{Spacer}}+E_{\mathchar 45\relax
35}+E_{\mathchar 45\relax 10})}}.$ (2)

Since Eq. 2 still assumes that each promoter element contributes additively
and independently to the total RNAP binding energy, it also makes sharp
predictions that markedly disagrees with the data (see Appendix C). Inspired
by these inconsistencies, we postulated that certain promoter elements, most
likely the -35 and -10 sites, may not contribute synergistically to RNAP
binding.

To that end, we consider a model for gene expression shown in Fig. 1B where
RNAP can separately bind to the -35 and -10 sites. RNAP is assumed to elicit a
large level of gene expression $r_{\text{max}}$ when fully bound but the
smaller level $r_{0}$ when unbound or partially bound. Importantly, the
Boltzmann weight of the fully bound state contains the free energy
$E_{\text{int}}$ representing the avidity of RNAP binding to the -35 and -10
sites. Physically, avidity arises because unbound RNAP binding to either the
-35 or -10 sites gains energy but loses entropy, while this singly bound RNAP
attaching at the other (-10 or -35) site again gains energy but loses much
less entropy, as it was tethered in place rather than floating in solution.
Hence we expect $e^{-\beta E_{\text{int}}}\gg 1$, and including this avidity
term implies that RNAP no longer binds independently to the -35 and -10 sites.

Our coarse-grained model of gene expression neglects the kinetic details of
transcription whereby RNAP transitions from the closed to open complex before
initiating transcription. Instead, we assume that there is a separation of
timescales between the fast process of RNAP binding/unbinding to the promoter
and the other processes that constitute transcription. In the quasi-
equilibrium framework shown in Fig. 1B, gene expression is given by the
average occupancy of RNAP in each of its states, namely,

$\text{GE}=\frac{r_{0}+e^{-\beta(E_{\text{BG}}+E_{\text{Spacer}})}\left(r_{0}e^{-\beta
E_{\mathchar 45\relax 35}}+r_{0}e^{-\beta E_{\mathchar 45\relax
10}}+r_{\text{max}}e^{-\beta\left(E_{\mathchar 45\relax 35}+E_{\mathchar
45\relax
10}+E_{\text{int}}\right)}\right)}{1+e^{-\beta(E_{\text{BG}}+E_{\text{Spacer}})}\left(e^{-\beta
E_{\mathchar 45\relax 35}}+e^{-\beta E_{\mathchar 45\relax
10}}+e^{-\beta(E_{\mathchar 45\relax 35}+E_{\mathchar 45\relax
10}+E_{\text{int}})}\right)}.$ (3)

We call this expression a refined energy matrix model since it reduces to the
energy matrix Eq. 1 (with constant of proportionality $r_{\text{max}}E^{-\beta
E_{\text{int}}}$) in the limit where gene expression is negligible when the
RNAP is not bound ($r_{0}\approx 0$) and the promoter is sufficiently weak or
the RNAP concentration is sufficiently small that polymerase is most often in
the unbound state (so that the denominator $\approx 1$). The background and
spacer are assumed to contribute to RNAP binding in both the partially and
fully bound states, an assumption that we rigorously justify in Appendix D.

Figure 1: The bivalent nature of RNAP-promoter binding. (A) Gene expression
was measured for RNAP promoters comprising any combination of -35, -10,
spacer, UP, and background (BG) elements. (B) When no UP element is present,
RNAP makes contact with the promoter at the -35 and -10 sites giving rise to
gene expression $r_{0}$ when unbound or partially bound and $r_{\text{max}}$
when fully bound. (C) Having two binding sites alters the dynamics of RNAP
binding. $k_{\text{on}}$ represents the on-rate from unbound to partially
bound RNAP and $\tilde{k}_{\text{on}}$ the analogous rate from partially to
fully bound RNAP, while $k_{\text{off},j}$ denotes the unbinding rate from
site $j$. Figure 2: Gene expression of promoters with no UP element. Model
predictions using (A) an energy matrix (Eq. 1) where the -35 and -10 elements
independently contribute to RNAP binding and (B) a refined energy matrix (Eq.
3) where the two sites contribute cooperatively. Inset: The epistasis-free
nature of the energy matrix model makes sharp predictions about the gene
expression of the consensus -35 and -10 sequences that markedly disagree with
the data. Parameter values given in Appendix B.

Fig. 2B demonstrates that the refined energy matrix Eq. 3 is better able to
capture the system’s behavior ($R^{2}=0.91$) while only requiring two more
parameters ($r_{0}$ and $E_{\text{int}}$) than the energy matrix model Eq. 1.
The sharp boundaries on the left and right represent the minimum and maximum
levels of gene expression, $r_{0}=0.18$ and $r_{\text{max}}=8.6$, respectively
(see Appendix E). The refined energy matrix predicts that the top 5% of
promoters will exhibit expression levels of 7.6 (compared to 8.5 measured
experimentally) while the weakest 5% of promoters should express at 0.2
(compared to the experimentally measured 0.1). In addition, this model quickly
gains predictive power, as its coefficient of determination only slightly
diminishes ($R^{2}=0.86$) if the model is trained on only 10% of the data and
used to predict the remaining 90%.

### Epistasis-Free Models of Gene Expression Lead to Sharp Predictions that
Disagree with the Data

To further validate that the lower coefficient of determination of the energy
matrix approach (Eq. 1) was not an artifact of the fitting, we can utilize the
epistasis-free nature of this model to predict the gene expression of double
mutants from that of single mutants. More precisely, denote the gene
expression $\text{GE}^{(0,0)}$ of a promoter with the consensus -35 and -10
sequences (and any background or spacer sequence). Let $\text{GE}^{(1,0)}$,
$\text{GE}^{(0,1)}$, and $\text{GE}^{(1,1)}$ represent promoters (with this
same background and spacer) whose -35/-10 sequences are mutated/consensus,
consensus/mutated, and mutated/mutated, respectively, where “mutated” stands
for any non-consensus sequence. As derived in Appendix D, the gene expression
of these three later sequences can predict the gene expression of the promoter
with the consensus -35 and -10 without recourse to fitting, namely,

$\text{GE}^{(0,0)}=\text{GE}^{(1,1)}\frac{\text{GE}^{(0,1)}}{\text{GE}^{(1,1)}}\frac{\text{GE}^{(1,0)}}{\text{GE}^{(1,1)}}.$
(4)

The inset in Fig. 2A compares the epistasis-free predictions ($x$-axis, right-
hand side of Eq. 4) with the measured gene expression ($y$-axis, left-hand
side of Eq. 4). These results demonstrate that the simple energy matrix
formulation fails to capture the interaction between the -35 and -10 binding
sites. While this calculation cannot readily generalize to the refined energy
matrix model since it exhibits epistasis, it is analytically tractable for
weak promoters where the refined energy matrix model displays a marked
improvement over the traditional energy matrix model (see Appendix C).

### RNAP Binding to the UP Element occurs Independently of the Other Promoter
Elements

Having seen that the refined energy matrix model (Eq. 3) can outperform the
traditional energy matrix analysis on promoters with no UP element, we next
extend the former model to promoters containing an UP element. Given the
importance of the RNAP interactions with the -35 and -10 sites seen above,
Fig. 3A shows three possible mechanisms for how the UP element could mediate
RNAP binding. For example, the C-terminal could bind strongly and
independently so that RNAP has three distinct binding sites. Another
possibility is that the RNAP $\alpha$CTD binds if and only if the -35 binding
site is bound. A third alternative is that the UP element contributes
additively and independently to RNAP binding (analogous to the spacer and
background).

To distinguish between these possibilities, we analyze the correlations in
gene expression between every pair of promoter elements (UP and -35, spacer
and background, etc.) to determine the strength of their interaction. Each
model in Fig. 3A will have a different signature: The top schematic predicts
strong interactions between the -35 and -10, between the UP and -35, and
between the UP and -10; the middle schematic would give rise to strong
dependence between the -35 and -10 as well as between the UP and -10, while
the UP and -35 elements would be perfectly correlated; the bottom schematic
suggests that the UP elements will contribute independently of the other
promoter elements.

Figure 3: The interaction between RNAP and the UP element. (A) Possible
mechanisms by which the RNAP C-terminal can bind to the UP element (orange
segments represent strong binding comparable to the -35 and -10 motifs; gray
segments represent weak binding comparable to the spacer and background). The
data supports the bottom schematic (see Appendix D). (B) The corresponding
characterization of 8,192 promoters identical to those shown in Fig. 2 but
with one of two UP binding motifs. Red points represent promoters with a
consensus -35 and -10. Data was fit using the same parameters as in Fig. 2B
and fitting the binding energies of the two UP elements (parameter values in
Appendix B).

This analysis, which we relegate to Appendix D, demonstrates that the UP
element is approximately independent of all other promoter elements
($R^{2}\gtrsim 0.6$) as are the background and spacer, indicating that the
bottom schematic in Fig. 3A characterizes the binding of the UP element. This
leads us to the general form of transcriptional regulation by RNAP, shown in
Eq. 5.

$\text{GE}=\frac{r_{0}+e^{-\beta(E_{\text{BG}}+E_{\text{Spacer}}+E_{\text{UP}})}\left(r_{0}e^{-\beta
E_{\mathchar 45\relax 35}}+r_{0}e^{-\beta E_{\mathchar 45\relax
10}}+r_{\text{max}}e^{-\beta\left(E_{\mathchar 45\relax 35}+E_{\mathchar
45\relax
10}+E_{\text{int}}\right)}\right)}{1+e^{-\beta(E_{\text{BG}}+E_{\text{Spacer}}+E_{\text{UP}})}\left(e^{-\beta
E_{\mathchar 45\relax 35}}+e^{-\beta E_{\mathchar 45\relax
10}}+e^{-\beta(E_{\mathchar 45\relax 35}+E_{\mathchar 45\relax
10}+E_{\text{int}})}\right)}$ (5)

Fig. 3B demonstrates how the expression of all promoters containing one of the
two UP elements combined with each of the eight background, spacer, -35, and
-10 sequences ($2\times 8^{4}=8{,}192$ promoters) closely matches the model
predictions ($R^{2}=0.88$). Remarkably, since we used the same free energies
and gene expression rates from Fig. 2B, characterizing these 8,192 promoters
only required two additional parameters (the free energies of the two UP
elements). This result emphasizes how understanding each modular component of
gene expression can enable us to harness the combinatorial complexity of
sequence space.

### Sufficiently Strong RNAP-Promoter Binding Energy can Decrease Gene
Expression

Although the 12,288 promoters considered above are well characterized by Eq. 5
on average, the data demonstrate that the full mechanistic picture is more
nuanced. For example, Urtecho et al. found that gene expression (averaged over
all backgrounds and spacers) generally increases for -35/-10 elements closer
to the consensus sequences [8]. In terms of the gene expression models studied
above (Eqs. 1-3), promoters with fewer -35/-10 mutations have more negative
free energies $E_{\mathchar 45\relax 35}$ and $E_{\mathchar 45\relax 10}$
leading to larger expression. Yet the strongest promoters with the consensus
-35/-10 violated this trend, exhibiting less expression than promoters one
mutation away. Thus, Urtecho et al. postulated that past a certain point,
promoters that bind RNAP too tightly may inhibit transcription initiation and
lead to decreased gene expression.

The promoters with a consensus -35/-10 are shown as red points in Fig. 3B, and
indeed these promoters are all predicted to bind tightly to RNAP and hence
express at the maximum level $r_{\text{max}}=8.6$, placing them on the right-
edge of the data. Yet depending on their UP, background, and spacer, many of
these promoters exhibit significantly less gene expression then expected.
Motivated by this trend, we posit that the state of transcription initiation
can be characterized by a free energy $\Delta E_{\text{trans}}$ relative to
unbound RNAP that competes with the free energy $\Delta E_{\text{RNAP}}$
between fully bound and unbound RNAP (see Appendix E).

Figure 4: Gene expression is reduced when RNAP binds a promoter too tightly.
Measured gene expression versus the inferred promoter strength $\Delta
E_{\text{RNAP}}$ relative to the transcription initiation state $\Delta
E_{\text{trans}}=-6.2\,k_{B}T$ (stronger promoters on the right). The dashed
line shows the prediction of the refined energy matrix model.

Assuming the rate of transcription initiation is proportional to the relative
Boltzmann weights of these two states, the level of gene expression
$r_{\text{max}}$ in Eq. 5 will be modified to

$\frac{r_{\text{max}}+r_{0}e^{-\beta(\Delta E_{\text{RNAP}}-\Delta
E_{\text{trans}})}}{1+e^{-\beta(\Delta E_{\text{RNAP}}-\Delta
E_{\text{trans}})}}.$ (6)

As expected, this expression reduces to $r_{\text{max}}$ for promoters that
weakly bind RNAP ($e^{-\beta(\Delta E_{\text{RNAP}}-\Delta
E_{\text{trans}})}\ll 1$) but decreases for strong promoters until it reaches
the background level $r_{0}$ when the promoter binds so tightly that RNAP is
glued in place and unable to initiate transcription. Upon reanalyzing the gene
expression data with the inferred value $\Delta E_{\text{trans}}=-6.2\,k_{B}T$
(see Appendix E), we can plot the measured level of gene expression against
the predicted RNAP-promoter free energy $\Delta E_{\text{RNAP}}$ as shown in
Fig. 4 (stronger promoters to the right). We find that this revised model
captures the downwards trend in gene expression observed for the strongest
promoters, most of which contain a consensus -35/-10.

### The Bivalent Binding of RNAP Buffers its Binding Behavior against Promoter
Mutations

In this final section, we investigate how the avidity between the -35 and -10
sites changes the dynamics of RNAP binding. More specifically, we consider the
effective dissociation constant governing RNAP binding when both the -35 and
-10 sites are intact and compare it to the case where only one site is capable
of binding. To simplify this discussion, we focus exclusively on the case of
RNAP binding to the -35 and -10 motifs as shown in the rates diagram Fig. 1C,
absorbing the effects of the background, spacer, and UP elements into these
rates.

At equilibrium, there is no flux between the four RNAP states. We define the
effective dissociation constant

$K_{D}^{\text{eff}}=\frac{K_{\mathchar 45\relax 35}K_{\mathchar 45\relax
10}}{c_{0}+K_{\mathchar 45\relax 35}+K_{\mathchar 45\relax 10}}$ (7)

which represents the concentration of RNAP at which there is a 50% likelihood
that the promoter is bound (see Appendix F).
$K_{j}=\frac{k_{\text{off},j}}{k_{\text{on}}}$ stands for the dissociation
constant of free RNAP binding to the site $j$ and
$c_{0}=\frac{\tilde{k}_{\text{on}}}{k_{\text{on}}}=[\text{RNAP}]e^{-\beta
E_{\text{int}}}$ represents the increased local concentration of singly bound
RNAP transitioning to the fully bound state (i.e., $E_{\text{int}}$ and
$c_{0}$ are the embodiments of avidity in the language of statistical
mechanics and thermodynamics, respectively). Note that $K_{D}^{\text{eff}}$ is
a sigmoidal function of $K_{\mathchar 45\relax 10}$ with height $K_{\mathchar
45\relax 35}$ and midpoint at $K_{\mathchar 45\relax 10}=c_{0}+K_{\mathchar
45\relax 35}$.

Fig. 5 demonstrates how the effective RNAP dissociation constant
$K_{D}^{\text{eff}}$ changes when mutations to the -10 binding motif alter its
dissociation constant $K_{\mathchar 45\relax 10}$. When the -35 sequence is
weak (dashed lines, $k_{\text{off},\mathchar 45\relax 35}\to\infty$),
$K_{D}^{\text{eff}}\approx K_{\mathchar 45\relax 10}$ signifying that RNAP
binding relies solely on the strength of the -10 site. In the opposite limit
where RNAP tightly binds to the -35 sequence (solid lines), the cooperativity
$c_{0}$ and the dissociation constant $K_{\mathchar 45\relax 35}$ shift the
curve horizontally and bound the effective dissociation constant to
$K_{D}^{\text{eff}}\leq K_{\mathchar 45\relax 35}$. This upper bound may
buffer promoters against mutations, since achieving a larger effective
dissociation constant would require not only wiping out the -35 site but in
addition mutating the -10 site. Finally, in the case where the cooperativity
$c_{0}$ is large, $K_{D}^{\text{eff}}\approx\frac{K_{\mathchar 45\relax
10}K_{\mathchar 45\relax 35}}{c_{0}}$ indicating that as soon as one site of
the RNAP binds, the other is very likely to also bind, thereby giving rise to
the multiplicative dependence on the two $K_{D}$s.

To get a sense for how these numbers translate into physiological RNAP dwell
times on the promoter, we note that the lifetime of bound RNAP is given by
$\tau=\frac{1}{K_{D}^{\text{eff}}k_{\text{on}}}$ (see Appendix F). Using
$K_{D}^{\text{eff}}\approx 10^{-9}\,\text{M}$ for the strong T7 promoter [14]
and assuming a diffusion-limited on-rate
$10^{8}\frac{1}{\text{M}\cdot\text{s}}$ leads to a dwell time of
$10\,\text{s}$, comparable to the measured dwell time of RNAP-promoter in the
closed complex [15]. It would be fascinating if recently developed methods
that visualize real-time single-RNAP binding events probed the dwell time of
the promoter constructed by Urtecho et al. to see how well the predictions of
the refined energy matrix model match experimental measurements [15].

Figure 5: The dissociation between RNAP and the promoter. RNAP binding to a
promoter with a strong (solid lines, $K_{\mathchar 45\relax 35}=1\mu\text{M}$)
or weak (dashed, $K_{\mathchar 45\relax 35}\to\infty$) -35 sequence. $c_{0}$
represents the local concentration of singly bound RNAP.

## Discussion

While high-throughput methods have enabled us to measure the gene expression
of tens of thousands of promoters, they nevertheless only scratch the surface
of the full sequence space. A typical promoter composed of 200 bp has
$4^{200}$ variants (more than the number of atoms in the universe).
Nevertheless, by understanding the principles governing transcriptional
regulation, we can begin to cut away at this daunting complexity to design
better promoters.

In this work, we analyzed a recent experiment by Urtecho et al. measuring gene
expression of over 10,000 promoters in E. coli using the $\sigma^{70}$ RNAP
holoenzyme [8]. These sequences comprised all combinations of a small set of
promoter elements, namely, eight -10s, eight -35s, eight spacers, eight
backgrounds, and three UPs depicted in Fig. 1A, providing an opportunity to
deepen our understanding of how these elements interact and to compare
different quantitative models of gene expression.

We first analyzed this data using classic energy matrix models which posit
that each promoter element contributes independently to the RNAP-promoter
binding energy. As emphasized by Urtecho et al. and other groups, such energy
matrices poorly characterize gene expression (Fig. 2A, $R^{2}=0.57$) and offer
testable predictions that do not match the data (Appendix C), mandating the
need for other approaches [8, 7].

To meet this challenge, we first determined which promoter elements contribute
independently to RNAP binding (Appendix D). This process, which was done
without recourse to fitting, demonstrated that the -35 and -10 elements bind
in a concerted manner that we postulated is caused by avidity. In this
context, avidity implies that when RNAP is singly bound to either the -35 or
-10 sites, it is much more likely (compared to unbound RNAP) to bind to the
other site, similar to the boost in binding seen in bivalent antibodies [16]
or multivalent systems [17, 18, 12]. Surprisingly, we found that outside the
-35/-10 pair, the other components of the promoter contributed independently
to RNAP binding.

Using these findings, we developed a refined energy matrix model of gene
expression (Eq. 5) that incorporates the avidity of between the -35/-10 sites
as well as the independence of the UP/spacer/background interactions. This
model was able to characterize the 4,096 promoters with no UP element (Fig.
2B, $R^{2}=0.91$) and the 8,192 promoters containing an UP element (Fig. 3B,
$R^{2}=0.88$). These results surpass those of the traditional energy matrix
model (Fig. 2A, $R^{2}=0.57$), only requiring two additional parameters that
could be experimentally determined (e.g., the interaction energy
$E_{\text{int}}$ arising from the -35/-10 avidity and the level of gene
expression $r_{0}$ of a promoter with a scrambled -10 motif, a scrambled -35
motif, or with both motifs scrambled).

These promising findings suggest that determining which components bind
independently is crucial to properly characterize multivalent systems. It
would be fascinating to extend this study to RNAP with other $\sigma$ factors
[13] as well as to RNAP mutants with no $\alpha$CTD or that do not bind at the
-35 site [19, 20]. Our model would predict that polymerases in this last
category with at most one strong binding site should conform to a traditional
energy matrix approach.

Quantitative frameworks such as the refined energy matrix model explored here
can deepen our understanding of the underlying mechanisms governing a system’s
behavior. For example, while searching for systematic discrepancies between
our model prediction and the gene expression measurements, we found that
promoters predicted to have the strongest RNAP affinity did not exhibit the
largest levels of gene expression (thus violating a core assumption of nearly
all models of gene expression that we know of). This led us to posit a
characteristic energy for transcription initiation that reduces the expression
of overly strong promoters (Fig. 4). In addition, we explored how having
separate binding sites at the -35 and -10 elements buffers RNAP kinetics
against mutations; for example, no single mutation can completely eliminate
gene expression of a strong promoter with the consensus -35 and -10 sequence,
since at least one mutation in both the -35 and -10 motifs would be needed
(Fig. 5).

Finally, we end by zooming out from the particular context of transcription
regulation and note that multivalent interactions are prevalent in all fields
of biology [21], and our work suggests that differentiating between
independent and dependent interactions may be key to not only characterizing
overall binding affinities but to also understand the dynamics of a system
[22]. Such formulations may be essential when dissecting the much more
complicated interactions in eukaryotic transcription where large complexes
bind at multiple DNA loci [23, 24] and more broadly in multivalent scaffolds
and materials [11, 12].

## Methods

We trained both the standard and refined energy matrix models on 75% of the
data and characterized the predictive power on the remaining 25%, repeating
the procedure 10 times. The coefficient of determination $R^{2}$ was
calculated for $y_{\text{data}}=\log_{10}(\text{gene expression})$ to prevent
the largest gene expression values from dominating the result. The
supplementary Mathematica notebook contains the data analyzed in this work and
can recreate all plots.

## Acknowledgements

We thank Suzy Beeler, Vahe Galstyan, Peng (Brian) He, and Zofii Kaczmarek for
helpful discussions. This work was supported by the Rosen Center at Caltech
and the National Institutes of Health through 1R35 GM118043-01 (MIRA).

## References

  * [1] Socorro Gama-Castro, Heladia Salgado, Alberto Santos-Zavaleta, Daniela Ledezma-Tejeida, Luis Muniz-Rascado, Jair Santiago Garcia-Sotelo, Kevin Alquicira-Hernandez, Irma Martinez-Flores, Lucia Pannier, Jaime Abraham Castro-Mondragon, Alejandra Medina-Rivera, Hilda Solano-Lira, Cesar Bonavides-Martinez, Ernesto Perez-Rueda, Shirley Alquicira-Hernandez, Liliana Porron-Sotelo, Alejandra Lopez-Fuentes, Anastasia Hernandez-Koutoucheva, Victor Del Moral-Chavez, Fabio Rinaldi and Julio Collado-Vides  “RegulonDB version 9.0: high-level integration of gene regulation, coexpression, motif clustering and beyond.”  In _Nucleic acids research_ 44.D1, 2016, pp. D133–D143  DOI: 10.1093/nar/gkv1156
  * [2] Rupali P Patwardhan, Choli Lee, Oren Litvin, David L Young, Dana Pe’er and Jay Shendure  “High-resolution analysis of DNA regulatory elements by synthetic saturation mutagenesis”  In _Nature Biotechnology_ 27.12 Nature Publishing Group, 2009, pp. 1173–1175  DOI: 10.1038/nbt.1589
  * [3] J. B. Kinney, A. Murugan, C. G. Callan and E. C. Cox  “Using deep sequencing to characterize the biophysical mechanism of a transcriptional regulatory sequence”  In _Proceedings of the National Academy of Sciences_ 107.20, 2010, pp. 9158–9163  DOI: 10.1073/pnas.1004290107
  * [4] Fumitaka Inoue and Nadav Ahituv  “Decoding enhancers using massively parallel reporter assays”  In _Genomics_ 106.3 Academic Press, 2015, pp. 159–164  DOI: 10.1016/J.YGENO.2015.06.005
  * [5] Avihu H. Yona, Eric J. Alm and Jeff Gore  “Random sequences rapidly evolve into de novo promoters”  In _Nature Communications_ 9.1 Springer US, 2018, pp. 1530  DOI: 10.1038/s41467-018-04026-w
  * [6] Nathan M. Belliveau, Stephanie L. Barnes, William T. Ireland, Daniel L. Jones, Michael J. Sweredoski, Annie Moradian, Sonja Hess, Justin B. Kinney and Rob Phillips  “Systematic approach for dissecting the molecular mechanisms of transcriptional regulation in bacteria”  In _Proceedings of the National Academy of Sciences_ , 2018  DOI: 10.1073/pnas.1722055115
  * [7] Talitha Forcier, Andalus Ayaz, Manraj Gill and Justin B Kinney  “Precision measurements of regulatory energetics in living cells”  In _bioRxiv_ , 2018
  * [8] Guillaume Urtecho, Arielle D Tripp, Kimberly Insigne, Hwangbeom Kim and Sriram Kosuri  “Systematic Dissection of Sequence Elements Controlling $\sigma$70 Promoters Using a Genomically-Encoded Multiplexed Reporter Assay in E. coli”  In _Biochemistry_ , 2018, pp. acs.biochem.7b01069  DOI: 10.1021/acs.biochem.7b01069
  * [9] Thomas Kuhlman, Zhongge Zhang, Milton H Saier and Terence Hwa  “Combinatorial transcriptional control of the lactose operon of Escherichia coli.”  In _Proceedings of the National Academy of Sciences of the United States of America_ 104.14 National Academy of Sciences, 2007, pp. 6043–6048  DOI: 10.1073/pnas.0606717104
  * [10] James Q Boedicker, Hernan G Garcia, Stephanie Johnson and Rob Phillips  “DNA sequence-dependent mechanics and protein-assisted bending in repressor-mediated loop formation”  In _Physical Biology_ 10.6 IOP Publishing, 2013, pp. 066005  DOI: 10.1088/1478-3975/10/6/066005
  * [11] Chad T. Varner, Tania Rosen, Jacob T. Martin and Ravi S. Kane  “Recent Advances in Engineering Polyvalent Biological Interactions”  In _Biomacromolecules_ 16.1 American Chemical Society, 2015, pp. 43–55  DOI: 10.1021/bm5014469
  * [12] Gui-Hua Yan, Kun Wang, Zhuxue Shao, Lei Luo, Zheng-Mei Song, Jingqi Chen, Rong Jin, Xiaoyong Deng, Haifang Wang, Zhonglian Cao, Yuanfang Liu and Aoneng Cao  “Artificial antibody created by conformational reconstruction of the complementary-determining region on gold nanoparticles”  In _Proceedings of the National Academy of Sciences_ 115.1 National Academy of Sciences, 2018, pp. E34–E43  DOI: 10.1073/pnas.1713526115
  * [13] Andrey Feklístov, Brian D. Sharon, Seth A. Darst and Carol A. Gross  “Bacterial Sigma Factors: A Historical, Structural, and Genomic Perspective”  In _Annual Review of Microbiology_ 68.1 Annual Reviews, 2014, pp. 357–376  DOI: 10.1146/annurev-micro-092412-155737
  * [14] C J Dayton, D E Prosen, K L Parker and C L Cech  “Kinetic measurements of Escherichia coli RNA polymerase association with bacteriophage T7 early promoters.”  In _The Journal of biological chemistry_ 259.3, 1984, pp. 1616–21  URL: http://www.ncbi.nlm.nih.gov/pubmed/6363413
  * [15] Feng Wang, Sy Redding, Ilya J. Finkelstein, Jason Gorman, David R. Reichman and Eric C. Greene  “The promoter-search mechanism of Escherichia coli RNA polymerase is dominated by three-dimensional diffusion”  In _Nature Structural and Molecular Biology_ , 2013  DOI: 10.1038/nsmb.2472
  * [16] Joshua S. Klein and Pamela J. Bjorkman  “Few and Far Between: How HIV May Be Evading Antibody Avidity”  In _PLoS Pathogens_ 6.5 Garland Science Publishing, 2010, pp. e1000908  DOI: 10.1371/journal.ppat.1000908
  * [17] Sudeep Banjade and Michael K Rosen  “Phase transitions of multivalent proteins can promote clustering of membrane receptors”  In _eLife_ 3, 2014  DOI: 10.7554/eLife.04123
  * [18] Jun Huang, Xun Zeng, Natalia Sigal, Peder J Lund, Laura F Su, Huang Huang, Yueh-hsiu Chien and Mark M Davis  “Detection, phenotyping, and quantification of antigen-specific T cells using a peptide-MHC dodecamer.”  In _Proceedings of the National Academy of Sciences of the United States of America_ 113.13 National Academy of Sciences, 2016, pp. E1890–7  DOI: 10.1073/pnas.1602488113
  * [19] Ashok Kumar, Richard A. Malloch, Nobuyuki Fujita, David A. Smillie, Akira Ishihama and Richard S. Hayward  “The Minus 35-Recognition Region of Escherichia coli Sigma 70 is Inessential for Initiation of Transcription at an Extended Minus 10 Promoter”  In _Journal of Molecular Biology_ 232.2 Academic Press, 1993, pp. 406–418  DOI: 10.1006/JMBI.1993.1400
  * [20] Leonid Minakhin and Konstantin Severinov  “On the role of the Escherichia coli RNA polymerase sigma 70 region 4.2 and alpha-subunit C-terminal domains in promoter complex formation on the extended -10 galP1 promoter.”  In _The Journal of biological chemistry_ 278.32 American Society for BiochemistryMolecular Biology, 2003, pp. 29710–8  DOI: 10.1074/jbc.M304906200
  * [21] Ang Gao, Krishna Shrinivas, Paul Lepeudry, Hiroshi I Suzuki, Phillip A Sharp and Arup K Chakraborty  “Evolution of weak cooperative interactions for biological specificity.”  In _Proceedings of the National Academy of Sciences of the United States of America_ National Academy of Sciences, 2018, pp. 201815912  DOI: 10.1073/pnas.1815912115
  * [22] Jennifer D Stone, Maxim N Artyomov, Adam S Chervin, Arup K Chakraborty, Herman N Eisen and David M Kranz  “Interaction of streptavidin-based peptide-MHC oligomers (tetramers) with cell-surface TCRs.”  In _Journal of immunology_ 187.12 American Association of Immunologists, 2011, pp. 6281–6290  DOI: 10.4049/jimmunol.1101734
  * [23] Nicolas Goardon, Julie A Lambert, Patrick Rodriguez, Philippe Nissaire, Sabine Herblot, Pierre Thibault, Dominique Dumenil, John Strouboulis, Paul-Henri Romeo and Trang Hoang  “ETO2 coordinates cellular proliferation and differentiation during erythropoiesis”  In _The EMBO journal_ 25.2 European Molecular Biology Organization, 2006, pp. 357–66  DOI: 10.1038/sj.emboj.7600934
  * [24] Michael Levine, Claudia Cattoglio and Robert Tjian  “Looping Back to Leap Forward: Transcription Enters a New Era”  In _Cell_ 157.1 Cell Press, 2014, pp. 13–25  DOI: 10.1016/J.CELL.2014.02.009