# Effect of switching time scale of receptor activity on chemotactic
performance of Escherichia coli

Shobhan Dev Mandal and Sakuntala Chatterjee  Department of Theoretical
Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector 3,
Salt Lake, Kolkata 700106, India.

###### Abstract

In the chemotactic motion of Escherichia coli the switching of transmembrane
chemoreceptors between active and inactive states is one of the most important
steps of the signaling pathway. We study the effect of this switching time-
scale on the chemotactic performance of the cell. We quantify performance by
the chemotactic drift velocity of the cell. Our extensive numerical
simulations on a detailed theoretical model show that as the activity
switching rate increases, the drift velocity increases and then saturates. Our
data also show the mean duration of a downhill run decreases strongly with the
switching rate, while that of an uphill run decreases relatively slowly. We
explain this effect from temporal variation of activity along uphill and
downhill trajectories. We show that for large and small switching rates the
nature of activity variation show qualitatively different behaviors along a
downhill run but similar behavior along an uphill run. This results in a
stronger dependence of downhill run duration on the switching rate, and
relatively milder dependence for uphill run duration.

##  I Introduction

Chemotaxis refers to directed motion of an organism in response to a chemical
gradient eisenbachbook . Many prokaryotic organisms like Escherichia coli,
Salmonella typhimurium, Bacillus subtilis, Rhodobacter sphaeroides or
Serratia marcescens salmonella ; bsubtilis ; rhotobacter ; ariel2015swarming
and eukaryotic organisms like Chlamydomonas rheinhartii or Tritrichomonas
foetus chlam ; foetus  use their run-and-tumble motility to perform
chemotaxis. Among all these organisms the motion of Escherichia coli or E.coli
is the most well-studied one berg2008coli . In presence of a spatial
concentration gradient of nutrient or some other attractant, an E.coli cell
shows a tendency to move up the gradient towards region of higher
concentration adler1973method ; adler1973chemotaxis ; berg1972chemotaxis .
Each cell has about ten flagella associated with it via flagellar motors. When
these motors rotate in the counter-clockwise (CCW) direction, the flagella
form a helical bundle and propelled by this bundle the cell runs in one
specific direction berg1972chemotaxis . When few of the motors turn clockwise,
the helical bundle is dispersed and the cell tumbles. The chemotaxis motion of
E.coli is mainly achieved by modulating the tumbling bias of the cell such
that the runs up the gradient are extended and those down the gradient are
shortened de2004chemotaxis ; chatterjee2011chemotaxis .

The intracellular signaling network of E.coli consists of two principal
modules: sensing and adaptation tu2013quantitative . While the sensing module
controls the response of the cell to the attractant signal, the adaptation
module utilizes negative feedback mechanism to maintain the protein levels
close to their average values barkai1997robustness . These two modules are
coupled via the activity of the transmembrane chemoreceptors. These receptors
can switch between active and inactive states, depending on the local
attractant concentration and the methylation levels of the receptors.
Probability to find a receptor in active state, is defined as its activity. In
an active state, the receptors promote auto-phosphorylation of the cytoplasmic
protein CheA to form CheA-P, which in turn donates its phosphate group to CheY
and CheB. In the phosphorylated state, CheY-P causes an increase in the
tumbling bias eisenbach1996control . On the other hand, the phosphorylated
CheB-P demethylates the receptors which increases the probability of the
receptors to switch to inactive state. Once in the inactive state, CheA
phosphorylation drops, cutting down supply of phosphate group to CheY and
CheB. In the absence of CheY-P tumbling bias decreases and the cell swims
smoothly. Low abundance of CheB-P hinders demethylation and under the action
of methylating enzyme CheR, the receptors’ methylation levels go up which in
turn increases the probability to switch the receptors to active state
bren2000signals . A high local concentration of attractant in the extra-
cellular environment increases the probability of binding attractant molecules
to the receptors and these binding events decrease the activity of the
receptors. When the local concentration of attractant is low, there are only
few nutrient molecules to bind to the receptors and hence the activity of the
receptors remain high. This signaling pathway therefore ensures that the cell
swims smoothly when the local attractant concentration is high and tumbles
frequently when the attractant is present in low concentration.

One important aspect of the intracellular reaction network for E.coli
chemotaxis is the cooperativity among the chemoreceptors. There are few
thousand receptors in an E.coli cell and they are arranged in a hexagonal
array briegel2012bacterial . The neighboring receptors experience cooperative
interaction and they tend to form ‘signaling teams’ parkinson2015signaling .
All receptors belonging to a particular team switch their activity states in
unison. This results in amplification of the input signal coming from ligand
binding sourjik2004functional ; tu2013quantitative . Larger the team size,
higher is the amplification. This effect is responsible for the high
sensitivity observed for E.coli chemotaxis, where the cell is known to respond
to even very weak concentration gradient of the attractant duke1999heightened
; sourjik2004functional . Recent experiments have shown that receptor
cooperativity not only enhances the sensitivity, but also creates large
activity fluctuations inside the cell colin2017multiple ;
keegstra2017phenotypic . In other words, heightened sensitivity comes with a
cost of intracellular noise. This results in an optimum size of the signaling
team at which the chemotactic performance of the cell is at its best shobhan .
In our earlier work we explained this optimality as a result of competition
between sensing and adaptation modules of the signaling network shobhan . We
have also shown that this sensing-adaptation interplay gives rise to highly
non-trivial, interesting dynamics of the methylation and activity of the
receptor clusters shobhanmethyl .

The transition of a signaling team between the active and inactive states is
captured well by MWC model monod1965nature  which has been quite successful so
far in explaining a wide class of experimental data sourjik2004functional ;
keymer2006chemosensing ; mello2005allosteric . According to this model, the
free energy difference between the active state and inactive state of a
receptor depends logarithmically on the local attractant concentration and on
the methylation level of the receptor. For a receptor cluster the free energy
becomes sum of free energies of the individual receptors which form the
cluster. This free energy is assumed to control the synchronous transition of
all receptors in the cluster from one activity state to the other as per the
principle of local detailed balance colin2017multiple . At the biochemical
level these transitions are associated with certain conformational changes of
the receptor molecules parkinson2015signaling . In colin2017multiple
different values of this transition time-scale was considered in simulations
and its effect on activity fluctuations was studied. It was found that
although the amplitude of fluctuation does not depend on the above time-scale,
the the temporal fluctuation of activity increases as this time-scale becomes
shorter colin2017multiple .

In the present work, we investigate how different choices of the activity
switching time-scale affect the chemotactic performance. We characterize the
performance by chemotactic drift velocity, i.e. the velocity with which the
cell climbs up the attractant gradient. Clearly, a large value of this drift
velocity indicates a better chemotactic performance. Our extensive numerical
simulations on a detailed theoretical model show that the chemotactic drift
velocity increases and then saturates as the switching of activity becomes
faster. We explain this observation by detailed analysis of temporal variation
of activity along the cell trajectory. We show that the downhill runs are
strongly curtailed for faster activity switching and the nature of variation
of activity along a downhill run becomes qualitatively different for fast and
slow switches. On the other hand, uphill runs are less strongly affected by
the switching timescale and this asymmetry results in an enhancement of
chemotactic drift.

In the next section of this paper, we present our model. In Sec. III we
present our results followed by conclusions.

Figure 1: Schematic diagram of E.coli signalling pathway.

##  II Model description

Our model consists of three main parts: switching of activity of the
chemoreceptors, (de)methylation of the receptors by enzymes, and run-tumble
motility of the cell. In an E.coli cell there are few thousand receptors which
form dimers liu2012molecular ; briegel2012bacterial . Due to cooperative
interaction, three dimers form trimer of dimers (TDs) and $n$ such TDs form a
cluster or a signaling team. In our model, for simplicity, we assume all
clusters have the same size. Activity state of a cluster is denoted as
$\alpha$ which can take two values $0$ and $1$, according as the cluster is
inactive or active. From Monod-Wyman-Changeux (MWC) model monod1965nature  the
free energy difference between these two states can be written as

$F=3n\left(1+\log\frac{1+c(x)/K_{min}}{1+c(x)/K_{max}}\right)-\sum_{j=1}^{3n}m_{j}$
(1)

where $m_{j}$ is the methylation level of the $j$-th dimer in the cluster
which can take any integer value between $0$ and $8$. At the cell position $x$
the concentration of the attractant is $c(x)$ . The two parameters $K_{min}$
and $K_{max}$ set the limit of sensitivity, such that for $c(x)\ll K_{min}$ or
$c(x)\gg K_{max}$ the free energy becomes independent of $c(x)$ and the cell
is not able to sense the concentration variation. The transition rate of the
cluster from $\alpha=0$ state to $\alpha=1$ state is taken to be
$\dfrac{w_{\alpha}}{1+\exp(F)}$ and the rate for the reverse transition is
$\dfrac{w_{\alpha}\exp(F)}{1+\exp(F)}$ colin2017multiple . Here, $w_{\alpha}$
has the dimension of rate which sets the overall timescale of activity
switching process. We demonstrate in this work how different choices of
$w_{\alpha}$ yield different behavior for the system.

The total number of molecules of methylating enzyme CheR and demethylating
enzyme CheB are far too less (of the order of hundred) compared to the number
of receptor dimers (few thousand) whose methylation levels they modify
li2004cellular . However, an E.coli cell is known to show near-perfect
adaptation berg1975transient ; goy1977sensory . To explain this few mechanisms
like ‘assistance neighborhood’ or ’brachiation’ have been suggested and
experimentally verified levin2002binding ; endres2006precise ;
hansen2008chemotaxis ; kim2002dynamic ; li2005adaptational . In an assistance
neighborhood model, a single enzyme does not only modify the methylation level
of the dimer it is attached to, but also of neighboring dimers. In a
brachiation mechanism, an enzyme molecule can perform random walk on the
receptor array and modify methylation levels of the dimers along its
trajectory.

An unbound CheR molecule from the cell cytoplasm can bind to a receptor dimer
if no other enzyme molecule is bound to it already. The binding can take place
at the tether site or the modification site of the receptor dimer
feng1999enhanced ; wu1996receptor ; pontius2013adaptation . While both these
bindings are slow, tether binding is relatively faster schulmeister2008protein
; pontius2013adaptation  and hence we consider only tether binding in our
model. The binding takes place with the rate $w_{r}$ and once bound, a CheR
molecule raises the methylation level of the dimer by one unit provided
methylation level of the dimer is less than $8$ and the dimer belongs to an
inactive cluster. A bound CheR molecule can unbind from the dimer with rate
$w_{u}$ and after that it can either rebind to another dimer within the same
cluster, or it can return to the cell cytoplasm. The rebinding time-scale is
much faster than $w_{r}$ and the possibility of rebinding within the same
cluster allows multiple methylation reactions by a single enzyme molecule in
that cluster before the enzyme molecule returns to the bulk. This is an
effective way to capture the essence of both assistance neighborhood and
brachiation mechanisms within our simple model. A CheB molecule in the cell
cytoplasm can undergo phosphorylation with the rate $w_{p}$ by an active
receptor, and dephosphorylation of CheB-P happens with the rate $w_{dp}$. An
unbound CheB-P molecule can bind to a dimer and decrease its methylation level
by one unit if the dimer is active and its methylation level is non-zero. The
unbinding and rebinding process remains same as those for a CheR molecule.

The tumbling bias of the cell is controlled by the concentration of
phosphorylated CheY molecules. When present in high concentration, CheY-P
molecules bind to the flagellar motors and cause them to rotate in the
clockwise direction which causes tumbles. Denoting $Y_{P}$ as the fraction of
CheY molecules which are phosphorylated, the rate equation governing this
fraction is given by jiang2010quantitative ; flores2012signaling

$\frac{dY_{P}}{dt}=K_{Y}a(1-Y_{P})-K_{Z}Y_{P}$ (2)

where $K_{Y}$ and $K_{Z}$ are rates of phosphorylation and dephosphorylation
respectively and $a$ is the total activity of the cell, i.e. average activity
of all the signaling teams. In our simulation we directly use the steady state
value $Y_{P}=\frac{a}{a+K_{Z}/K_{Y}}$. A cell in CCW rotational state or in
run mode can switch to CW rotational state or tumble mode with rate
$\omega\exp(-G)$ where $G=\Delta_{1}-\frac{\Delta_{2}}{1+Y_{0}/Y_{P}}$ and the
opposite switch from tumble to run happens with a rate $\omega\exp(G)$
sneddon2012stochastic ; dufour2014limits ; micali2017drift . $Y_{0}=0.34$ is
the adapted value of $Y_{P}$ in our system. In Table 1 we list all parameter
values in our model.

We present here results for chemotaxis of a single cell in one dimension. We
expect similar results in two dimensions as well shobhan ; shobhanmethyl . We
consider a one dimensional box of length $L=2000\mu m$ with reflecting
boundary conditions. We set up a linear nutrient profile of the form
$c(x)=c_{0}(1+x/x_{0})$, with $c_{0}=200$ $\mu M$ and $x_{0}=40000$ $\mu m$.
This can be considered a weak gradient for which the steady state position
distribution of the cell position is approximately linear. The smallest time-
step in our simulation is $dt=0.01s$. During a run, the cell moves either
leftward (downhill) or rightward (uphill) with speed $v=20\mu m/s$
berg2008coli  such that in $dt$ time-step it travels a distance $vdt$. In the
tumble mode, the cell position does not change with time. After each tumble,
when the cell switches to run mode, the sign of $v$ is chosen at random. All
our measurements are done in steady state, i.e. after a long enough time, when
the cell has been able to adapt to the attractant profile present in the
medium.

##  III Results

###  III.1 Homogeneous attractant environment

To explicitly show the effect of activity switching rate on the temporal
fluctuations of activity, we show in Fig. 2 the time-series of the activity
state of one particular cluster. As expected, the time interval between two
successive switching events drops with increasing $w_{\alpha}$. For large
$w_{\alpha}$ when the activity switching happens very frequently, the total
number of active clusters fluctuates rapidly with time. The total activity
that determines the tumbling bias of the cell, can go from high to low value
in a short span of time. This means the residence time of the cell in run mode
(or in tumble mode) decreases as $w_{\alpha}$ increases. In Fig. 3 we verify
this for mean duration of a run. Note that while $w_{\alpha}$ affects the
temporal fluctuations of activity, the range of fluctuations does not show
much dependence on $w_{\alpha}$. This range is mainly set by the value of $n$,
the signaling team size shobhan ; colin2017multiple . Since the number of
receptors in the cell is fixed, as the team size becomes large, there are
fewer teams present. The total activity of the cell is then averaged over a
fewer number of teams which increases the activity fluctuations. We have
explicitly verified (data not shown here) that the activity distribution does
not show any significant dependence on $w_{\alpha}$. These results are in line
with colin2017multiple  where $w_{\alpha}$ was varied in simulation and it was
found that the temporal fluctuation of activity increases with $w_{\alpha}$
but the amplitude of fluctuation remains unchanged.

Figure 2: Typical time series of activity of a single cluster recorded over a
time-window ($40$ $s$) along the run-tumble trajectory of swimming cell in
homogeneous attractant environment. As the switching rate
($w_{{\color[rgb]{0,0,1}\alpha}}$) increases, the switching between two
activity states $0$ and $1$ happens more frequently.  Figure 3:  Mean run
duration ($\tau$) of the chemotactic cell in homogeneous attractant
environment as a function of switching rate ($w_{{\color[rgb]{0,0,1}\alpha}}$)
for three different values of the cluster size $n$. The mean run duration
decreases with $w_{{\color[rgb]{0,0,1}\alpha}}$. Each data point has been
averaged over at least $10^{6}$ histories. All simulation parameters are given
in Sec. II.

###  III.2 Chemotactic performance gets better for fast activity switching

We quantify chemotactic performance by the drift velocity $V$ which measures
how fast the cell climbs up the attractant gradient shobhan ; dev2018optimal ;
chatterjee2011chemotaxis . We define the drift velocity as
$V=\dfrac{\Delta}{\tau}$, where $\Delta$ denotes the net displacement of the
cell during a run and $\tau$ denotes average run duration. While a particular
run can be directed both uphill and downhill, the form of the free energy gap
between the two activity states of the cluster in Eq. 1 is such that downhill
(uphill) runs are associated with an increase (decrease) in activity, and
therefore, an increase (decrease) in the tumbling bias. This means the uphill
runs tend to get extended and downhill runs tend to get shortened, which
results in a drift motion up the gradient. A large value of $V$ indicates
faster motion of the cell up the gradient, which reflects a better chemotactic
performance shobhan ; dev2018optimal ; chatterjee2011chemotaxis . To measure
how reliably the cell is able to climb up the attractant gradient, we also
calculate the ratio between the net directed displacement $\Delta$ and its
standard deviation $\sigma_{\Delta}$. A large value of this ratio would also
imply a strong chemotactic performance. We are interested to find out how the
chemotactic performance depends on the activity switching time-scale. In Fig.
4a and 4b we present our simulation results for $V$ and
$\dfrac{\Delta}{\sigma_{\Delta}}$ for different values of $w_{\alpha}$ and
three different choices for the signaling team size $n$. We find that both
these quantities increase with $w_{\alpha}$ and then saturates. To explain
this variation, consider the limit $w_{\alpha}=0$ which means activity of the
receptors do not flip. This will naturally impair the chemotactic ability of
the cell and $V$ will be zero. It is expected therefore for small
$w_{\alpha}$, we should observe $V$ increasing with $w_{\alpha}$. However,
when $w_{\alpha}$ is very large, the average run duration becomes small and
then it is not obvious how $V$ is affected. To probe it further, we separately
measure average duration of a rightward (uphill) run and a leftward (downhill)
run as a function of $w_{\alpha}$. Our data in Fig. 5 left panel show that
both runs become shorter for large $w_{\alpha}$ as expected, but mean duration
$\tau_{L}$ of leftward runs shows a faster drop with $w_{\alpha}$ than
$\tau_{R}$ for rightward runs. This means the difference $\tau_{R}-\tau_{L}$
increases with $w_{\alpha}$ (see Fig. 5 right panel). In other words, net
displacement during a run increases, and mean run duration decreases with
$w_{\alpha}$, as a consequence of which $V$ increases. In the next subsection,
we present detailed explanation of this counter-intuitive behavior.

Figure 4: Chemotactic efficiency of the cell as a function of switching rate
$w_{\alpha}$. (a) Drift velocity ($V$) as function of switching
$w_{{\color[rgb]{0,0,1}\alpha}}$. Data for three different clustersize has
been presented. $V$ increases with $w_{\alpha}$ and then saturates at large
values of $w_{\alpha}$. (b) Ratio of net average displacement in a
run($\Delta$) and fluctuation of this displacement($\sigma_{\Delta}$) as a
function of switching rate $w_{\alpha}$. The ratio increases with $w_{\alpha}$
and then saturates at large values of $w_{\alpha}$. All simulation parameters
are same as Fig. 3. These data points are averaged over at least $3\times
10^{7}$ histories. Figure 5: Mean duration of an uphill run ($\tau_{R}$) and a
downhill run ($\tau_{L}$) and their difference vs $w_{\alpha}$. Data for three
different values of cluster sizes have been shown. The left panel shows that
both $\tau_{R}$ (blue square) and $\tau_{L}$ (green circle) decrease with
$w_{\alpha}$ but $\tau_{L}$ falls at faster rate than $\tau_{R}$. The right
panel shows the difference between $\tau_{R}$ and $\tau_{L}$ goes up with
$w_{\alpha}$ and reaches a saturation. All simulation parameters are same as
Fig. 3. These data points are averaged over at least $3\times 10^{7}$
histories.

###  III.3 Explanation behind better performance at large $w_{a}$

Large value of activity increases the tumbling bias of the cell. Therefore,
when the cell enters the tumble mode, the number of active receptor clusters
tends to be high. In the tumble mode the cell position does not change, which
means ligand part of the free energy in Eq. 1 remains fixed. The active
receptors get demethylated and $F$ increases, which increases the probability
of transition to inactive state. When $w_{\alpha}$ is high, more clusters flip
from active to inactive state and the total activity drops quickly. Therefore,
when the cell comes out of the tumble mode and starts a new run, its activity
has a low value when $w_{\alpha}$ is large. Let $a_{t\to r}$ denotes the
activity when the tumble to run switch takes place. In Fig. 6 we plot the
average of this quantity $\langle a_{t\to r}\rangle$ as a function of
$w_{\alpha}$ and show that it decreases with $w_{\alpha}$.

Figure 6: Average of the activity $a_{t\to r}$ at the instant of a tumble to
run switch as a function of $w_{\alpha}$. As $w_{\alpha}$ increases value of
$\langle a_{t\to r}\rangle$ drops monotonically. Error bars of the data have
magnitude of the order of $10^{-5}$ and hence they are actually smaller than
the symbol size in the plot. All simulation parameters are same as Fig. 3.
These data points are averaged over at least $5\times 10^{5}$ histories.

The fact that new runs start with lower activity values as $w_{\alpha}$
increases, has an interesting implication for the activity variation during
uphill and downhill runs, as shown below. Let $N^{R}(t)$ and $N^{L}(t)$
denote, respectively, the number of rightward (uphill) and leftward (downhill)
runs whose duration is larger than $t$. Clearly, both these quantities
decrease with $t$. Let $a^{R}_{i}(t)$ and $a^{L}_{i}(t)$ be the total activity
of the cell at time $t$ during $i$-th uphill and downhill run, respectively.
Now we can define the following quantities:

$\langle{\Delta
a^{R}(t)}\rangle=\sum_{i=1}^{N^{R}(t)}\frac{a^{R}_{i}(t)-a^{R}_{i}(0)}{N^{R}(t)}$
(3)

and

$\langle\Delta
a^{L}(t)\rangle=\sum_{i=1}^{N^{L}(t)}\frac{a^{L}_{i}(t)-a^{L}_{i}(0)}{N^{L}(t)}.$
(4)

In Fig. 7 we plot these quantities as a function of time for two different
values of $w_{\alpha}$. Top panel (Fig. 7a) shows the variation of $\Delta
a^{L}(t)$. For smaller $w_{\alpha}$ values (purple squares) when the runs
start with relatively high value of activity, due to ongoing demethylation,
activity drops at small $t$. Moreover, higher activity values make these runs
prone to tumble and thus these trajectories drop out of $N^{L}(t)$ population
quickly, which brings down the average activity. As a result, $\Delta
a^{L}(t)$ takes negative values for small $t$. As $t$ increases, however, drop
in ligand concentration along the downhill run tends to increase the activity
again. Due to these two opposing effects, $\Delta a^{L}(t)$ after reaching a
minimum starts slowly increasing again. Our data show that till moderate or
large $t$, $\Delta a^{L}(t)$ remains negative with its magnitude decreasing
with $t$ i.e. activity still decreases on an average during the run, but the
magnitude of the change becomes less. At very large $t$ change in activity
becomes positive but still has a small magnitude. The situation is quite
different for large $w_{a}$ (green circles). Here, initial activity at $t=0$
is low which prompts methylation and decreases $F$ (Eq. 1). Also, decreasing
ligand concentration along the trajectory causes $F$ to decrease. Large
$w_{\alpha}$ ensures that activity responds quickly to this free energy change
and more and more clusters flip from inactive to active state. $\Delta
a^{L}(t)$ is positive in this case and grows rapidly with time. The behavior
of $\Delta a^{L}(t)$ is thus completely different for small and large
$w_{\alpha}$. From Fig. 7a it is clear that the values of $\Delta a^{L}(t)$
for small and large $w_{\alpha}$ are widely apart for the range of time we
have observed.

Figure 7: Temporal variation of activity for $w_{\alpha}=0.75$ $s^{-1}$
(purple squares) and $w_{\alpha}=10$ $s^{-1}$ (green circles). (a) shows data
for downhill runs For and (b) shows data for uphill runs. For downhill runs
qualitative nature of variation is very different for the two $w_{\alpha}$
values, while for uphill runs the variation is qualitatively similar. All
simulation parameters are same as Fig. 3. These data points are averaged over
at least $4\times 10^{6}$ histories.

On the other hand, Fig. 7b shows that the qualitative behavior of $\Delta
a^{R}(t))$ is quite similar for large and small $w_{\alpha}$. For small
$w_{\alpha}$ (purple squares) runs start with larger activity and subsequent
demethylation together with increasing ligand concentration along the
trajectory cause the activity to decrease making $\Delta a^{R}(t))$ negative
with its magnitude increasing steadily with time. For large $w_{\alpha}$
(green circles) initial activity being small, receptors tend to methylate
which causes activity to increase sharply for small $t$. But for moderate or
large times, $c(x)$ variation takes over and causes activity to decrease with
time. Thus for both small and large $w_{\alpha}$ we find $\Delta a^{R}(t))$
shows similar temporal variation, except for very small times. For the entire
time-range we have observed, the values of $\Delta a^{R}(t))$ for small and
large $w_{\alpha}$ are relatively closer compared to Fig. 7a. In other words,
average change in activity for small and large $w_{\alpha}$ are very different
along a downhill run, but the difference is significantly less along an uphill
run. This explains why $\tau_{L}$ decreases more strongly with $w_{\alpha}$
compared to $\tau_{R}$ (Fig. 5).

##  IV Conclusions

In this paper we have investigated the effect of activity switching timescale
on the chemotactic performance of a single E.coli cell. We find that the
performance improves as the activity switches faster. We quantify performance
by (a) the chemotactic drift velocity that measures how rapidly the cell
manages to climb up the chemical concentration gradient, and (b) by the
reliability of its net uphill drift. In both cases we find better performance
for faster activity switches. More specifically, the average duration of a
downhill run decreases rapidly with the activity switching rate $w_{\alpha}$,
but that of an uphill run decreases comparatively slowly. This asymmetry
enhances the chemotactic drift. To explain this observation we monitor
temporal variation of activity during uphill and downhill runs. Our detailed
measurements show that during downhill runs, nature of activity variation
shows widely different behavior for small and large values of $w_{\alpha}$,
while the behavior is not so different during uphill runs. This results in
stronger dependence on $w_{\alpha}$ of downhill run duration which explains
the above effect.

It should be possible to experimentally verify some of our main results.
Direct measurement of receptor activity has already been possible using FRET
based experiments sourjik2002receptor ; sourjik2004functional ;
colin2017multiple . Although most experiments focus on tethered cells, and our
studies consider a swimming cell, it is possible to verify our conclusions
even for a tethered cell by appropriately engineered attractant environment.
By identifying the counter clockwise rotation of the flagellar motors as the
run mode and the clockwise rotation as the tumble mode, one can ramp up (down)
the attractant concentration to mimic an uphill (downhill) run and hold the
attractant level steady for a tumble. Using this set-up the activity variation
of the cell can be measured with time and our results on $\Delta a^{R}(t)$ and
$\Delta a^{L}(t)$ can be verified. The switching of a receptor between active
and inactive states is a result of a conformational change of the receptor
molecule parkinson2015signaling . An experimental control over this
conformational change timescale may provide some insights into its effect on
the chemotactic performance of the cell. Moreover, our study can be used to
estimate the activity switching rate for wild-type E.coli, which has not been
measured so far to the best of our knowledge. The range of values of drift
velocity measured in our simulations agree with experimental data for $V$
recorded under similar attractant environment ahmed2008experimental ;
micali2017drift . A comparison between the experiments and our simulations
yields $w_{\alpha}$ ranging between $0.5s^{-1}$ and $1s^{-1}$. A direct
experimental measurement of $w_{\alpha}$ can help to verify this prediction.

##  V Acknowledgements

SDM acknowledges a research fellowship [Grant No. 09/575(0122)/2019-EMR -I]
from the Council of Scientific and Industrial Research (CSIR), India. SC
acknowledges financial support from the Science and Engineering Research
Board, India (Grant No: MTR/2019/000946).

##  Appendix A Model parameters

Table 1: Parameter values used in model and simulations Symbol |  Description | Value | References
---|---|---|---
$N_{dim}$ | Total number of receptor dimers | $7200$ | pontius2013adaptation ; li2004cellular
$N_{R}$ | Total number of CheR protein molecules | $140$ | pontius2013adaptation ; li2004cellular
$N_{B}$ | Total number of CheB protein molecules | $240$ | pontius2013adaptation ; li2004cellular
$K_{min}$ | Minimum concentration receptor can sense |  $18$ $\mu M$ |  jiang2010quantitative , flores2012signaling
$K_{max}$ | Maximum concentration receptor can sense |  $3000$ $\mu M$ | flores2012signaling ; jiang2010quantitative
$\omega$ | Switching frequency of motor |  $1.3$ $s^{-1}$ | sneddon2012stochastic
$\Delta_{1}$ | Non-dimensional constant regulating motor switching | $10$ | sneddon2012stochastic
$\Delta_{2}$ | Non-dimensional constant regulating motor switching | $20$ | sneddon2012stochastic
$Y_{0}$ | Adapted value of the fraction of CheY-P protein | $0.34$ | sneddon2012stochastic
$K_{Y}$ | Phosphorylation rate of CheY molecule |  $1.7$ $s^{-1}$ | flores2012signaling ; tu2008modeling
$K_{Z}$ | Dephosphorylation rate of CheY molecule |  $2$ $s^{-1}$ | flores2012signaling ; tu2008modeling
$w_{r}$ | Binding rate of bulk CheR to tether site of an unoccupied dimer |  $0.068$ $s^{-1}$ | pontius2013adaptation ; schulmeister2008protein
$w_{b}$ | Binding rate of bulk CheB-P to tether site of an unoccupied dimer |  $0.061$ $s^{-1}$ | pontius2013adaptation ; schulmeister2008protein
$w_{u}$ | Unbinding rate of bound CheR and CheB-P |  $5$ $s^{-1}$ | pontius2013adaptation ; schulmeister2008protein
$k_{r}$ | Methylation rate of bound CheR |  $2.7$ $s^{-1}$ | pontius2013adaptation ; schulmeister2008protein
$k_{b}$ | Demethylation rate of bound CheB-P |  $3$ $s^{-1}$ | pontius2013adaptation ; schulmeister2008protein
$w_{p}$ | CheB phosphorylation rate |  $3$ $s^{-1}$ | pontius2013adaptation ; stewart2000rapid
$w_{dp}$ | CheB-P dephosphorylation rate |  $0.37$ $s^{-1}$ | pontius2013adaptation

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# A note on the codegree of finite groups

Mark L. Lewis  Department of Mathematical Sciences, Kent State University,
Kent, OH 44242, USA  Quanfu Yan111Corresponding author. Department of
Mathematical Sciences, Kent State University, Kent, OH 44242, USA

Abstract. Let $\chi$ be an irreducible character of a group $G,$ and
$S_{c}(G)=\sum_{\chi\in{\rm Irr}(G)}{\rm cod}(\chi)$ be the sum of the
codegrees of the irreducible characters of $G.$ Write ${\rm
fcod}(G)=\frac{S_{c}(G)}{|G|}.$ We aim to explore the structure of finite
groups in terms of ${\rm fcod}(G).$ On the other hand, we determine the lower
bound of $S_{c}(G)$ for nonsolvable groups and prove that if $G$ is
nonsolvable, then $S_{c}(G)\geqslant S_{c}(A_{5})=68,$ with equality if and
only if $G\cong A_{5}.$ Additionally, we show that there is a solvable group
so that it has the codegree sum as $A_{5}.$

Keywords: Character codegrees, Codegree sum, Nonsolvable groups.
MSC: 20C15, 20D05

##  1 Introduction

All groups considered in this note are finite. Let $G$ be a group and let
${\rm Irr}(G)$ be the set of complex irreducible characters of $G$. For
$\chi\in{\rm Irr}(G),$ define the number

${\rm cod}(\chi)=\frac{|G:\ker\chi|}{\chi(1)}$

to be the codegree of $\chi.$ This definition of codegree was first introduced
by Qian, Wang and Wei in [9]. There is a very large number of recent articles
on character coddegrees, for example [6, 10, 11, 3].

Many authors often explore group theory invariants related to character
degrees and study how these invariants may influence the structure of $G$. The
average character degree (denoted by ${\rm acd}(G)$) is such an invariant. For
example, it was proved in [7] by Isaacs, Loukaki and Moreto that if ${\rm
acd}(G)\leqslant 3$, ${\rm acd}(G)<\frac{3}{2}$ or ${\rm acd}(G)<\frac{4}{3},$
then $G$ is respectively solvable, supersolvable or nilpotent. Another closely
related invariant is $f(G)=\frac{\sum_{\chi\in{\rm Irr}(G)}\chi(1)}{|G|}.$
This invariant has been investigated in many papers, see for examples [12,
Chapter 11] and [13, 15, 14]. Recently, Pan, Dong and Yang in [14] proved that
if $f(G)>2/3$ or $f(G)>2$, then $G$ is nilpotent or supersolvable
respectively. Motivated by these papers, it is natural to explore the
character codegree versions of known results for character degrees.

In this paper, we denote the sum of codegrees of irreducible characters of $G$
by

$S_{c}(G)=\sum_{\chi\in{\rm Irr}(G)}{\rm cod}(\chi),$

and let ${\rm acod}(G)$ denote the average codegree of the irreducible
characters of $G.$ Indeed, Wang, Qian, Lv and Chen in [16] investigated the
invariant ${\rm acod}(G).$ They provided the lower bounds for ${\rm acod}(G)$
in non-solvable and non-supersolvable groups.

We consider the analog of $f(G)$ with the focus on the invariant ${\rm
fcod}(G)=\frac{S_{c}(G)}{|G|}.$ In particular, we see what can be said
regarding this invariant. Unfortunately, the examples constructed in Section 2
led us to believe that no similar results can be obtained. We show that there
exist families of non-solvable groups $\\{G\\}$ and $\\{H\\}$ such that ${\rm
fcod}(G)\rightarrow 0$ as $|G|\rightarrow\infty,$ and ${\rm
fcod}(H)\rightarrow\infty$ as $|H|\rightarrow\infty.$ This tells us that the
invariant ${\rm fcod}(G)$ has so many possibilities. Thus, additionally we
discuss the special case when ${\rm fcod}(G)=1$ and provide some observations.

On the other hand, we consider the codegree sum $S_{c}(G)$ and study how it
affects the structure of $G.$ Motivated by the results in [1], we considered
the question in [17]: Let $G$ be a group of order $n$ and $C_{n}$ a cyclic
group of order $n,$ is it true that $S_{c}(G)\leqslant S_{c}(C_{n})?$ We
showed that this inequality holds for many classes of groups such as nilpotent
groups, and it remains valid for any finite group whose order is divisible by
up to 99 primes. However, it should be noted that the assertion does not hold
true in all cases. In this note, we will consider the lower bound of
$S_{c}(G)$ and prove the following:

######  Theorem 1.1.

Let $G$ be a non-solvable group. Then $S_{c}(G)\geqslant S_{c}(A_{5})=68,$
with equality if and only if $G\cong A_{5}.$

From the above theorem, if $G$ is a non-solvable group such that
$S_{c}(G)=S_{c}(A_{5}),$ then $G\cong A_{5}.$ Since the alternating group
$A_{5}$ is the smallest nonabelian simple group, one may ask if it is still
true when $G$ is solvable. However, the conclusion is not always true because
we noticed that the solvable group $G=A_{4}\rtimes C_{4}$ (SmallGroup(48,30))
has the same codegree sum as $A_{5}.$

The work in this paper was completed by the second author (P.h.D student)
under the supervision of the first author at Kent State University. The
contents of this paper may appear as part of the second author’s P.h.D
dissertation.

##  2 The invariant ${\rm fcod}(G)$

We begin by proving the following lemma, which plays an important role in
constructing examples.

######  Lemma 2.1.

Let $G=A\times B$ be a direct product of $A$ and $B$, then $S_{c}(G)\leqslant
S_{c}(A)\cdot S_{c}(B)$ and hence ${\rm fcod}(G)\leqslant{\rm
fcod}(A)\cdot{\rm fcod}(B).$ Furthermore, the equality holds if $(|A|,|B|)=1$.

[Proof]Let $\chi\in{\rm Irr}(A)$ and $\psi\in{\rm Irr}(B)$, then
$\ker\chi\times\ker\psi\leqslant\ker(\chi\times\psi)$ and so ${\rm
cod}(\chi\times\psi)\leqslant{\rm cod}(\chi)\cdot{\rm cod}(\psi).$ It follows
from [5, Theorem 4.21] that the characters $\chi\times\psi$ for $\chi\in{\rm
Irr}(A)$ and $\psi\in{\rm Irr}(B)$ are exactly the irreducible characters of
$G$. Hence, $S_{c}(G)\leqslant S_{c}(A)\cdot S_{c}(B)$ and so ${\rm
fcod}(G)\leqslant{\rm fcod}(A)\cdot{\rm fcod}(B).$

Now assume $(|A|,|B|)=1$. Let $g=hk\in\ker(\chi\times\psi)$ for some $h\in A$
and $k\in B.$ Then

$(\chi\times\psi)(hk)=\chi(h)\psi(k)=\chi(1)\psi(1).$

It follows that $h\in{Z}(\chi)$ and $k\in{Z}(\psi)$. Let
$\chi_{{Z}(\chi)}=\chi(1)\lambda$ and $\psi_{Z(\psi)}=\psi(1)\mu$, where
$\lambda$ and $\mu$ are linear characters of $Z(\chi)$ and $Z(\psi)$,
respectively. So $\lambda(h)=\mu(k)^{-1}$. Let $o(g)$ denote the order of
$g\in G.$ If either $\lambda(h)\neq 1$ or $\mu(k)\neq 1$, then $o(h)=o(k)$,
contrary to $(|H|,|K|)=1$. It follows that $\lambda(h)=\mu(k)=1$ and therefore
$h\in\ker{\chi}$ and $k\in\ker\psi$. Hence,
$\ker\chi\times\ker\psi=\ker(\chi\times\psi)$, which implies that

${\rm cod}(\chi\times\psi)=\frac{|A\times
B:\ker(\chi\times\psi)|}{\chi\times\psi(1)}={\rm cod}(\chi)\cdot{\rm
cod}(\psi).$

Thus $S_{c}(G)=S_{c}(A)\cdot S_{c}(B)$ and ${\rm fcod}(G)={\rm
fcod}(A)\cdot{\rm fcod}(B)$, as wanted.

The hypothesis $(|A|,|B|)=1$ cannot be dropped, as shown by the examples:
$S_{c}(C_{2}\times C_{2})=7$ but $S_{c}(C_{2})\cdot S_{c}(C_{2})=9.$ Next we
compute the values of ${\rm fcod}(G)$ for some specific groups.

######  Example 2.2.

(1) If $G$ is a dihedral groups $D_{2^{n}}$ of order $2^{n}$ with $n\geqslant
3,$ then $S_{c}(G)=1+2\cdot 3+\sum_{i=3}^{n}2^{i-1}\cdot
2^{i-3}=7+\frac{4}{3}(2^{2n-4}-1)$ and so ${\rm fcod}(G)\rightarrow\infty$ as
$n\rightarrow\infty.$

(2) Let $G\cong C_{8}\rtimes C_{2}^{2}$ (SmallGroup(32,43)), then
$S_{c}(G)=31.$ Therefore, ${\rm fcod}(G)=\frac{31}{32}<1.$

(3) ${\rm fcod}(A_{5})=\frac{68}{60}>1$ and ${\rm
fcod}(A_{6})=\frac{311}{360}<1,$ where $A_{n}$ is the alternating group of
degree $n.$

Using the groups described above, we are able to construct some families of
groups as follows.

######  Proposition 2.3.

(1) There are families of solvable groups $\\{G\\}$ and $\\{H\\}$ such that
${\rm fcod}(G)\rightarrow 0$ as $|G|\rightarrow\infty,$ and ${\rm
fcod}(H)\rightarrow\infty$ as $|H|\rightarrow\infty.$

(2) There exist families of non-solvable groups $\\{G\\}$ and $\\{H\\}$ such
that ${\rm fcod}(G)\rightarrow 0$ as $|G|\rightarrow\infty,$ and ${\rm
fcod}(H)\rightarrow\infty$ as $|H|\rightarrow\infty.$

[Proof]Let $S$ be a group described in Example 2.2(2) and $G=S\times
S\times\cdot\cdot\cdot\times S$ be the direct product of $n$ copies of $S.$ It
follows from Lemma 2.1 that ${\rm fcod}(G)\leqslant({\rm
fcod}(S))^{n}=(\frac{31}{32})^{n}\rightarrow 0$ as $n\rightarrow\infty.$
Together with Example 2.2(1), the statement (1) follows.

Now we prove (2). Let $G$ be the direct product of $n$ copies of $A_{6}.$
Using a similar proof as above, we have that ${\rm fcod}(G)\rightarrow 0$ as
$|G|\rightarrow\infty.$ Now let $H=A_{5}\times C_{p}$, where $p>5$ is a prime.
Clearly, $H$ is nonsolvable. Then by Lemma 2.1, ${\rm fcod}(H)={\rm
fcod}(A_{5})\cdot{\rm
fcod}(C_{p})=\frac{68}{60}\cdot\frac{1+p(p-1)}{p}\rightarrow\infty$ as
$p\rightarrow\infty.$

From the above examples and proposition, we may see that the invariant ${\rm
fcod}(G)$ has so many possibilities. Next we will focus on the special case
when ${\rm fcod}(G)=1.$ It is clear that the symmetric group $S_{3}$ is the
smallest nontrivial group such that ${\rm fcod}(G)=1.$ It is natural to ask:
Are there infinitely many groups satisfying ${\rm fcod}(G)=1?$ How can we
describe the stucture of such groups?

We believe that the first question will have a positive answer based on the
following proposition. However, we cannot answer this question completely
because of the difficulties in the aspect the number theory.

######  Proposition 2.4.

Let $G=C_{q}\rtimes C_{p}$ be a non-abelian group of order $pq,$ where $p,q$
are distinct primes. Then ${\rm fcod}(G)=1$ if and only if $q=p^{2}-p+1.$

[Proof]It is obvious that $S_{c}(G)=1+(p-1)p+\frac{(q-1)q}{p}.$ Then ${\rm
fcod}(G)=1$ if and only if $1+(p-1)p+\frac{(q-1)q}{p}=pq,$ which is equivalent
to $q^{2}+(p^{2}-1)q+p(p^{2}-p+1)=(q-p)(q-(p^{2}-p+1))=0.$ The result follows
from $p\not=q$.

We do find some pairs of $(p,q),$ such as $(2,3),(3,7),(13,157),$ so that
$q=p^{2}-p+1.$ However we do not know if there are infinitely many such prime
pairs $(p,q).$ On the other hand, we observe that if $G$ is a $p$-group, then
${\rm fcod}(G)\not=1$.

######  Proposition 2.5.

Let $G$ be $p$-group. Then $S_{c}(G)\equiv 1({\rm mod}\ {p})$ and so ${\rm
fcod}(G)\not=1$.

[Proof]If $\chi\not=1_{G},$ we have $|G:{\rm ker}\chi|\geqslant
1+\chi(1)^{2}>\chi(1)^{2}$ and so $1\leqslant\chi(1)<|G:{\rm
ker}\chi|/\chi(1).$ Since $G$ is a $p$-group, $\chi(1)^{2}$ divides $|G:{\rm
ker}\chi|$ for all $\chi\in{\rm Irr}(G).$ It follows that $p|{\rm cod}(\chi)$
if $\chi\not=1_{G}.$ Hence, $S_{c}(G)\equiv 1({\rm mod}\ p)$ and so ${\rm
fcod}(G)\not=1,$ as wanted.

##  3 Lower bound of $S_{c}(G)$ for nonsolvable groups

In this section, we start by presenting some basic properties of the
codegrees, which will be used in our proof. By [9, Lemma 2.1] and [9, Lemma
2.3], we have

######  Lemma 3.1.

Let $\chi$ be an irreducible character of $G.$

(1) For any normal subgroup $N$ of $G$ with $N\leqslant{\rm ker}\chi$, the
codegree ${\rm cod}(\chi)$ is independent of the choice of such $N.$ In
particular, $S_{c}(G/N)\leqslant S_{c}(G)$ with equality if and only if $N=1.$

(2) If $S$ is a subnormal subgroup of $G$ and $\phi$ is an irreducible
constituent of $\chi_{S},$ then ${\rm cod}\phi\mid{\rm cod}\chi.$

(3) If $G$ is a nonabelian simple group, then for any distinct primes
$p,q\in\pi(G)$ where $\pi(G)$ denotes the prime divisors of $|G|,$ there
exists $\chi\in{\rm Irr}(G)$ so that $pq\mid{\rm cod}\chi.$

When working with nonsolvable groups, it is always necessary to understand
some properties about finite nonabelian simple groups. Here our proof relies
on the fact (see [4, Theorem 1]): If $G$ is a finite nonabelian simple group
with $|\pi(G)|=3$, then $G$ is isomorphic to one of the following groups:
$A_{5},$ $A_{6},$ $L_{2}(7),$ $L_{2}(8),$ $L_{2}(17),$ $L_{3}(3),$ $U_{3}(3)$
or $U_{4}(2).$

[Proof of Theorem 1.1] It suffices to prove that $G\cong A_{5}$ if
$S_{c}(G)\leqslant S_{c}(A_{5})=68.$ We work by induction on $|G|.$ Let $M$ be
a maximal normal subgroup of $G.$ Then $G/M$ is a simple group. If $G/M$ is
nonabelian simple, then $S_{c}(G/M)\leqslant S_{c}(G)=68$ and by induction we
have that $M=1$ and $G\cong A_{5},$ as desired.

Now assume that $G/M$ is abelian. Then $G/M\cong C_{p}$ for some prime $p.$
Let $N$ be a minimal normal subgroup of $G$ contined in $M.$ Consider
$\overline{G}=G/N.$ If $N$ is abelian, then $\overline{G}$ is nonsolvable.
Since $S_{c}(\overline{G})<S_{c}(G)\leqslant 68,$ by induction
$\overline{G}\cong A_{5}$ and so $S_{c}(\overline{G})=68,$ a contradiction. It
follows that $N$ is nonabelian. Then $N\cong S_{1}\times
S_{2}\times\cdot\cdot\cdot\times S_{k}$, where $S_{i}\cong S_{1}=S$ are
isomorphic nonabelian simple groups. We claim that $N\cong S\cong A_{5}.$ Here
we consider the set $\pi(S).$ Assume that $|\pi(S)|>3$ and let
$2=p_{0}<p_{1}<p_{2}<p_{3}$ be the first four smallest prime divisors of
$|S|.$ Then $p_{3}\geqslant 7.$ Applying Lemma 3.1, there are irreducible
characters $\chi_{1},\chi_{2},\chi_{3}\in{\rm Irr}(G)$ so that
$p_{2}p_{3}\mid{\rm cod}\chi_{1},p_{1}p_{3}\mid{\rm cod}\chi_{2}$ and
$p_{1}p_{2}\mid{\rm cod}\chi_{3}$ respectively. If $\chi_{i}=\chi_{j}$ for
some $i\not=j,$ then $p_{1}p_{2}p_{3}\mid{\rm cod}\chi_{i}.$ Notice that
$p_{1}p_{2}p_{3}\geqslant 3\cdot 5\cdot 7=105>68$. Then $S_{c}(G)>{\rm
cod}\chi_{i}\geqslant 105,$ which is a contradiction. Hence
$\chi_{i}\not=\chi_{j}$ for any $i\not=j\in\\{1,2,3\\}.$ It follows that
$S_{c}(G)\geqslant\sum_{i=1}^{3}{\rm cod}\chi_{i}\geqslant 3\cdot 5+3\cdot
7\cdot 35=71>68.$ This contradiction indicates that $|\pi(S)|=3.$ As we
mentioned before, all finite nonabelian simple groups whose orders have three
prime divisors have been classified. If $S$ is isomotphic to $A_{6},$
$L_{2}(8),$ $L_{2}(17),$ $L_{3}(3),$ $U_{3}(3)$ or $U_{4}(2),$ then by [2]
there is an irreducible character of $S$ having codegree greater than 68. It
follows by Lemma 3.1(2), $S_{c}(G)>68.$ Now assume that $S\cong L_{2}(7).$
Then by [2] there are irreducible characters $\phi_{1},\phi_{2}$ of $S$ so
that ${\rm cod}\phi_{1}=2^{3}\cdot 7$ and ${\rm cod}\phi_{1}=2^{3}\cdot 3.$ If
$\phi_{1}=\phi_{2},$ then $2^{3}\cdot 3\cdot 7\mid{\rm cod}\phi_{1}$ and so
$S_{c}(G)>68.$ If $\phi_{1}\not=\phi_{2},$ then $S_{c}(G)>2^{3}\cdot
7+2^{3}\cdot 3=80>68.$ Both cases are impossible. Hence the only possibility
is $S\cong A_{5}.$ A similar proof works to show that $N\cong S,$ thereby
indicating the claim. Since $N\cong A_{5},$ it follows from Lemma 3.1(2) that
$G$ has irreducible characters $\chi_{1},\chi_{2}$ such that $20|{\rm
cod}\chi_{1}$ and $15|{\rm cod}\chi_{2}.$ In particular, by [8, Theorem 1.1],
let $\theta\in{\rm Irr}(N)$ with $\theta(1)=5,$ then there exists
$\chi_{3}\in{\rm Irr}(G)$ such that $(\chi_{3})_{N}=\theta$ and so $12|{\rm
cod}\chi_{3}.$ It is obvious that any two of these three characters cannot be
the same. Hence by the assumption $S_{c}(G)\leqslant 68,$ we have that ${\rm
cod}\chi_{3}=12$ or $24.$ Notice that

${\rm cod}\chi_{3}=\frac{|G:N|\cdot|N|}{5\cdot|{\rm
ker}\chi_{3}|}=12\cdot\frac{|G:N|}{|{\rm ker}\chi_{3}|}.$

If $\frac{|G:N|}{|{\rm ker}\chi_{3}|}=1,$ then $G=N\times{\rm ker}\chi_{3}$
since $N\cap{\rm ker}\chi_{3}={\rm ker}(\chi_{3})_{N}={\rm ker}\theta=1.$ It
follows that $G/{\rm ker}\chi_{3}\cong N\cong A_{5}.$ Since
$|G:N|\geqslant|G:M|>1$, we have that ${\rm ker}\chi_{3}>1$ and hence
$S_{c}(G)>S_{c}(G/{\rm ker}\chi_{3})=68,$ a contradiction. Hence we may have
that $|G:{\rm ker}\chi_{3}|=2|N|=120.$ It follows that $G/{\rm
ker}\chi_{3}\cong S_{5}$ or $A_{5}\times C_{2},$ both of which have codegree
sum greater than 68. This completes the proof.

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