# Optimal intercellular competition in senescence and cancer

Thomas C. T. Michaels  Department of Physics and Astronomy, Institute for the
Physics of Living Systems, University College London, Gower Street, London
WC1E 6BT, UK  Medical Research Council Laboratory for Molecular Cell Biology,
University College London, Gower Street, London, WC1E 6BT, UK  L. Mahadevan
Engineering and Applied Sciences, Physics, and Organismic and Evolutionary
Biology, Harvard University, Cambridge, MA 02138, USA

###### Abstract

Effective multicellularity requires both cooperation and competition between
constituent cells. Cooperation involves sacrificing individual fitness in
favor of that of the community, but excessive cooperation makes the community
susceptible to senescence and aging. Competition eliminates unfit senescent
cells via natural selection and thus slows down aging, but excessive
competition makes the community susceptible to cheaters, as exemplified by
cancer and cancer-like phenomena. These observations suggest that an optimal
level of intercellular competition in a multicellular organism maximizes
organismal vitality by delaying the inevitability of aging. We quantify this
idea using a statistical mechanical framework that leads to a generalized
replicator dynamical system for the population of cells that change their
vitality and cooperation due to somatic mutations that make them susceptible
to aging and/or cancer. By accounting for the cost of cooperation and strength
of competition in a minimal setting, we show that our model predicts an
optimal value of competition that maximizes vitality and delays the
inevitability of senescence or cancer.

The evolution of multicellularity is linked to the advantages of collective
physiology and behavior absent in unicellular life which include, but are not
limited to [1, 2, 3, 4]: division of labor, adaptation to varied environments,
efficient use of resources, creation and maintenance of extracellular
environmental niches, the collective inhibition of cell proliferation and
programmed cell death. All these benefits of multicellular life require
cooperation - the coordinated orchestration of functions that are essential
for the development and maintenance of a complex organism. Cooperation,
however, comes with a cost to individual cells that need to invest a part of
their resources into traits that contribute positively to organismal vitality
but reduce individual cell fitness [5, 6, 7].

Cooperation in multicellar organisms creates interdependence among cells,
which in turn leads to damage accumulation, gradual decay and aging. Indeed,
it has been shown that many complex systems with multiple connected components
(including both biologically evolved organisms and artificially engineered
systems) in general experience aging as a result of the interdependence
between the components [8, 9, 10, 11, 12, 13]. To ameliorate the consequences
of degradation induced by aging in both evolved and engineered systems
requires continuous maintenance and repair. In multicellular organisms, a
particular form of maintenance, controlled cell proliferation, increases the
risk of accumulating deleterious heritable somatic mutations that cause the
progressive decline of cellular function and, eventually, an irreversible
arrest of cell growth, i.e. cellular senescence [14, 15, 16]. The accumulation
of senescent cells results in a progressive loss of organismal vitality and a
number of aging-related pathologies [17, 16].

In multicellular organisms with renewable tissues, senescent cells can be
eliminated through natural selection as a result of the reduced fitness of
senescent cells compared to healthy cells [18, 19, 20, 5, 16, 17]. The
resulting intercellular competition serves to increase organismic vitality at
a cost associated with proliferation-driven renewal. The potential for the
breakdown of cellular cooperation driven by excessive proliferation can lead
to inappropriate cell survival, resource monopolization, abnormal cell
differentiation, or degradation of the extracellular environment, which are
considered hallmarks of cancer [21, 22, 2, 23, 24]. This is a form of cheating
that emerges in a competitive environment because uncooperative cancerous
cells enjoy a higher fitness relative to cooperative healthy cells and have an
advantage in selection [5]. Then individually uncooperative cells can thrive
(transiently) in a competitive environment, with deleterious consequences for
the (long-term) collective vitality of the organism [25].

Thus intercellular competition in multicellular organisms is a double-edged
sword: without competition, multicellularity is susceptible to senescence,
while too much competition can lead to cheating and cancer or cancer-like
phenomena [2, 23]. Recently, an elegant study [14] builds on this idea
suggesting that senescence and cancer are an inevitable consequence of the
dilemma posed by competition that cannot be too weak or too strong, and
spawned a series of commentaries on the generality of the conclusions [26, 27,
28]. Some questions that naturally arise in this context include the
possibility of a minimal analytic framework that might help to uncover the
essence of the arguments, while also posing the problem of whether there is an
optimal level of competition that maximizes organismal vitality by controlling
or delaying senescence without succumbing to cancer?

Here we attempt to answer this question in terms of an approach based on a
probabilistic master equation. We show that this leads to an analytically
tractable mathematical model for the dynamics of multicellular aging in terms
of a modified form of generalized replicator dynamics. Our solvable model
reveals the fundamental factors controlling the optimal level of intercellular
competition in terms of two parameters that characterize the base fitness and
vitality in the system. Our solution reveals the fundamental biophysical
factors controlling the level of intercellular competition and predict an
optimal value of this parameter that maximizes system vitality by delaying the
inevitability of senescence. By providing a minimal statistical mechanical
framework for the study of the collective vitality of a system, our results
may motivate the rational design of strategies for delaying aging in
biological, technical or social systems, where processes similar to those
considered here are at play.

Figure 1: Master equation for multicellular aging. a. Cell vigor $v$ and
cooperation $c$ define a two-dimensional coordinate system $(v,c)$ of cell
types. Progressive loss of vigor corresponds to cell senescence, while
progressive loss of cooperation corresponds to cancer. The dynamics of the
population $N(v,c,t)$ of cells of type $(v,c)$ at time $t$ are described in
terms of transition rates between different states
$(v,c)\to(v^{\prime},c^{\prime})$. The figure illustrates the different fluxes
away from and into state $(v,c)$: (i) each cell type $(v,c)$ proliferates with
state-dependent rate $f(v,c)$; (ii) mutations correspond to transitions that
lower vigor ($v\to v-1$, with rate $\mu_{v}$) or cooperation ($c\to c-1$, with
rate $\mu_{c}$). b. Fitness (competition) landscape for different cell types
$(v,c)$ for $k=0.3$. Senescent cells are non-competitive with $f(0,m)=0$.
Healthy cells are moderately competitive with $f(n,m)=f_{0}(1-k)$, while
cancer cells are the most competitive species with $f(n,0)=f_{0}$. In the
model it is important that $\partial f/\partial v>0$ but $\partial f/\partial
c<0$, i.e. the fitness landscape is tilted in favour of cancerous types. This
condition ensures that loss of cooperation in the form of cheating corresponds
to an advantage in selection.

## Dynamics of multicellular aging

To describe the dynamics of multicellular aging, we use a master equation
approach in the space of cell types, as an equivalent to the Price equation
formalism [29, 30, 31] (see SI Sec. S1 for a master equation formulation of
the Price equation). Following a previous study [14], we classify cell types
in terms of two traits: vigor $v$ and cooperation $c$. Vigor $v$ is used as a
general measure of cellular resources or function, e.g. metabolic activity.
Cooperation $c$ describes the fraction of resources that a cell devotes in
activities that favor the functioning of the multicellular organism, including
controlling homeostasis or maintaining the extracellular infrastructure. Since
it is measured in terms of the fraction of the vigor, we expect it to be an
intensive variable. Vigor and cooperation define a coordinate system of
discrete cell types $(v,c)$; in the simplest setting, we assume that $v$ and
$c$ take discrete values in the range $0\leq v\leq n$ and $0\leq c\leq m$
(Fig. 1(a)) (a formulation using continuous values of these variables does not
lead to results that are qualitatively different - see SI Sec. S2 and Fig.
S1). Cells with high vigor ($v=m$) and high cooperation ($c=m$) are ‘healthy’
(h), cell types that have lost vigor ($v=0$) are ‘senescent’ (s), while cells
with $c=0$ are ‘cancerous’ (c) [14]. Cells in state $(v=0,c=0)$ are both
senescent and cancerous (b). Cells with intermediate values of $v$ and $c$
represent types that are not fully degraded; this situation mimics the fact
that multiple mutations are necessary to induce senescence or cancer [32].

### Cell populations

To quantify the time evolution of the (average) population $N(v,c,t)$ of cells
in state $(v,c)$ at time $t$ we use a (mean field) master equation approach
[33]. This captures spontaneous transitions between states in the coordinate
system $(v,c)$. In our system transitions occur as a result of two effects
(Fig. 1(a)): (i) cell proliferation (i.e. intercellular competition) at a rate
$f(v,c)$ and (ii) somatic mutations, which are permanent changes of cell
genotype that lead to a gradual decline of vigor or cooperation phenotypes. In
the following we shall use the terms fitness and competition interchangeably
to denote the proliferation rate $f(v,c)$. We consider somatic mutations that
affect only one of the two traits $v$ or $c$ in single steps with (state-
dependent) rates $\mu_{v}v$, respectively, $\mu_{c}c$ [14], but note that our
framework can in principle be generalised to account for more complex
transitions. Then the master equation describing the time evolution of
$N(v,c,t)$ is given by:

$\displaystyle\frac{\partial N(v,c,t)}{\partial t}$
$\displaystyle=f(v,c)\,N(v,c,t)$ (1)
$\displaystyle+\mu_{v}(v+1)\,N(v+1,c,t)-\mu_{v}\,v\,N(v,c,t)$
$\displaystyle+\mu_{c}(c+1)\,N(v,c+1,t)-\mu_{c}\,c\,N(v,c,t)\,.$

The first term on the right-hand side of (1) describes the proliferation of
$N(v,c,t)$ with growth rate $f(v,c)$. The remaining terms in (1) describe the
effect of somatic mutations by means of reaction fluxes away from and into the
different cell types, i.e. $N(v,c,t)$ decreases due to transitions $v\to v-1$
and $c\to c-1$; conversely, $N(v,c,t)$ increases through transitions $v+1\to
v$ or $c+1\to c$.

### Cell fractions

Since it is more useful to consider the dynamics of population fractions
rather than the time evolution of various cell populations, we define the
fraction of cells in state $(v,c)$ at time $t$ as:

$\rho(v,c,t)=\frac{N(v,c,t)}{N(t)},\quad N(t)=\sum_{v,c}N(v,c,t)\,,$ (2)

where $N(t)$ is the total population of cells. We can then use (1) and derive
a dynamic equation for $\rho(v,c,t)$ (see SI Sec. S1) as:

$\displaystyle\frac{\partial\rho(v,c,t)}{\partial t}$
$\displaystyle=\big{(}f(v,c)-\overline{f}(t)\big{)}\,\rho(v,c,t)$
$\displaystyle+\mu_{v}(v+1)\,\rho(v+1,c,t)-\mu_{v}\,v\,\rho(v,c,t)$
$\displaystyle+\mu_{c}(c+1)\,\rho(v,c+1,t)-\mu_{c}\,c\,\rho(v,c,t)\,,$ (3a)
where $\overline{f}(t)=\sum_{v,c}f(v,c)\,\rho(v,c,t)$ (3b)

is the (time-dependent) average competition. Compared to (1), the key
difference of (3) lies in the first line, which describes the effect of
selection: cell types with $f>\overline{f}$ will be enriched by selection
(fitness is higher than average), while those with $f<\overline{f}$ will
decrease in frequency over time (fitness lower than average). (3) is thus an
extension of the replicator equation [34] that accounts for mutation fluxes
changing cell type. Note that while the master equation for cell populations
(1) is linear, the equation for cell fractions (3) is non linear. Also note
that the master equation is a deterministic equation describing the average
population or fraction of cells of a certain type, while $v$ and $c$ are
stochastic variables, capturing stochasticity in the appearance of mutations
or in cell proliferation. Alternative methods commonly used in population
genetics include PDE-based approaches such as the Fokker-Planck formalism
[35], which follow from the master equation through Taylor-series expansion in
$v$ and $c$; here we stay with the ME approach for generality.

We choose the mutation rates $\mu_{vv},\mu_{cc}$ to be linearly dependent on
$v$ and $c$ respectively for simplicity. This choice guarantees that in the
absence of selection the average vigor
$\overline{v}(t)=\sum_{v,c}v\,\rho(v,c,t)\propto e^{-\mu_{v}t}$ and
cooperation $\overline{c}(t)=\sum_{v,c}c\,\rho(v,c,t)\propto e^{-\mu_{c}t}$
decay exponentially with rates $\mu_{v}$, respectively, $\mu_{c}$, and
naturally ensures that mutation transitions do not reduce $v$ or $c$ below
zero (boundary condition at $v=0$, $c=0$).

Figure 2: Dynamics of multicellular aging. a. Time evolution of fractions of
healthy (h), senescent (s), cancerous (c) and both senescent and cancerous (b)
cells obtained as solution to the master equation (3) for $k=0.3$,
$\mu_{v}=10^{-3}$, $\mu_{c}=10^{-5}$, $f=0.004$ and $n=m=1$ (four-state
model). Solid lines indicate the numerical solution to (3), while dashed lines
indicate the exact analytical solution (given by (5)) for initial conditions
associated with $(v,c)=(1,1),(0,1),(1,0),(0,0)$) corresponding to the four-
state model (see texxt for details). b. Time evolution of the probabilities
$p_{v}(t)$ and $p_{c}(t)$, defined in (5), and graphical representation of the
timescales $\tau_{v}$ and $t_{1/2}$.  Figure 3: Organismal vitality and
optimal competition. a. Average organismal vitality $\overline{V}(T)$ at
observation time $T=3\mu_{v}^{-1}$ plotted against the cost of cooperation $k$
and the strength of cellular competition $f_{0}$. The solid line indicates the
position of the optimum line $f_{0}^{\star}(k)$. The plot is obtained by
numerical integration of (3) using the same parameters as in Fig. 2(a). b.
Optimal competition $f_{0}^{\star}(k)$ in terms of $k$ (this corresponds to a
top view of 3(a)). Points on the optimal line $f_{0}^{\star}(k)$ (scenario O)
balance the accumulation of senescent cells and development of cancer. Points
in the ($k$,$f_{0}$)-diagram that deviate from the optimal line (scenarios S
and C) correspond to systems that are either dominated by senescence or
cancer. c. Sensitivity of optimal conditions, defined as $|\partial
f_{0}^{\star}/\partial k|$. The plot is obtained by numerically evaluating the
derivative of the optimal line $f_{0}^{\star}(k)$ of 3(b) along $k$. Regions
of low sensitivity facilitate a robust approach to optimal conditions. d. Time
evolution of average organismal vitality $\overline{V}(t)$ for the scenarios
O, S and C defined in 3(b). e. Fractions of healthy, senescent and cancerous
cell types (h, s, c, and b) corresponding to scenarios O (left), S (middle)
and C (right). The optimal scenario O balances senescent cell accumulation and
cancer cell proliferation.

### Time evolution of cell fractions

To solve the master equation (3) we need to specify a closure relation for
$f(v,c)$. This procedure is analogous to path or contextual analysis of the
Price equation [36], where fitness, or other parameters entering the Price
equation, are partitioned into separate causes through regression equations
[29, 30, 31]. In the simplest setting we assume a linear relationship for
$f(v,c)$ (corresponding to a one-term Taylor expansion) by imposing two
minimal requirements: $f$ should increase with increasing $v$, and decrease
with increasing $c$, i.e.

$\frac{\partial f}{\partial v}>0,\frac{\partial f}{\partial c}<0\,.$ (4)

In Fig. 1(b), we depict the relative fitness of the different cell types:
senescent cells (s) have the lowest fitness with $f=0$, cancer cells (c) have
the highest fitness $f_{0}$, while healthy cells have an intermediate fitness
$f=f_{0}(1-k)$, where $k\in[0,1]$ is the cost of cooperation. A linear fitness
function that satisfies these requirements can then be written as
$f(v,c)=f_{0}k+f_{v}\,v-f_{c}\,c$, where $f_{v}=f_{0}(1-k)/n$ and
$f_{c}=-f_{0}k/m$, such that $f(n,m)=f_{0}(1-k)$, $f(n,0)=f_{0},f(0,m)=0$ and
$f(0,0)=f_{0}k$. We see that cells that have higher vigor or invest a smaller
fraction of their resources on sustaining the organism have an advantage in
selection compared to less vigorous or more cooperative types.

With the simple choice in (4), the master equation (3) can now be solved using
the method of generating functions [33] to obtain an exact analytical solution
for the evolution of the population fractions of cells (details in Materials
and Methods), yielding

$\displaystyle\rho(t,v,c)$
$\displaystyle=\textrm{Bin}\big{(}v,n,p_{v}(t)\big{)}\,\textrm{Bin}\big{(}c,m,p_{c}(t)\big{)}\,.$
(5a) Here $\textrm{Bin}(v,n,p)={{n}\choose{v}}\,p^{v}(1-p)^{n-v}$ denotes the
binomial distribution and the functions $p_{v}(t)$ and $p_{c}(t)$ are given by
$\displaystyle p_{v}(t)$
$\displaystyle=\frac{f_{v}-\mu_{v}}{f_{v}-\mu_{v}\,e^{-t/\tau_{v}}}\,,$ (5b)
$\displaystyle p_{c}(t)$
$\displaystyle=\frac{f_{c}+\mu_{c}}{f_{c}+\mu_{c}\,e^{t/\tau_{c}}}\,,$ (5c)

where $\tau_{v}=1/(f_{v}-\mu_{v})$ and $\tau_{c}=1/(f_{c}+\mu_{c})$ are the
two natural timescales that control the dynamics in our model. The analytical
solution reveals that $\rho(t,v,c)$ is the product of two independent binomial
distributions in vigor and cooperation spaces, with $p_{v}(t)$ and $p_{c}(t)$
representing the (time-dependent) probabilities of cells having one unit of
vigor or cooperation, respectively.

Fig. 2(a) illustrates the time evolution of the various cell fractions for
$n=m=1$. This corresponds to a four-state model where cells can be in one of
four states at time $t=0$: healthy (h, $(v,c)=(1,1)$), senescent (s,
$(v,c)=(0,1)$), cancerous (c, $(v,c)=(1,0)$), and both senescent and cancerous
(b, $(v,c)=(0,0)$). We also assume that $\mu_{v}\gg\mu_{c}$ capturing the
observation that only about one percent of human genes contributes to cancer
risk [37], so that $\mu_{c}/\mu_{v}\simeq 10^{-2}$

The separation of timescales between senescence-causing and cancer-causing
mutations implied by $\mu_{v}\gg\mu_{c}$ causes the resulting complex aging
dynamics to display two stages of kinetics, as reflected by the distinct
timescales controlling the time evolution of the probabilities $p_{v}$ and
$p_{c}$ (Fig. 2(b)). Initially there is a phase of characteristic timescale
$\tau_{v}$ where a buildup of senescent cells is observed as a result of
accumulation of senescence-causing mutations. In this rapidly varying initial
phase, a rapid pre-equilibrium is established between healthy and senescent
cells, with the fraction of healthy cells pre-equilibrating at
$p_{v}(\infty)^{n}=[1-\mu_{v}/(kf_{0})]^{n}$ and the fraction of senescent
cells approaching the maximal value $[\mu_{v}/(kf_{0})]^{n}$. This pre-
equilibrium reflects the interplay between selection $kf_{0}$ and mutation
forces $\mu_{v}$; stronger competition eliminates senescent cells more
effectively via natural selection hence increasing the pre-equilibrium
fraction of healthy cells. During this stage of dynamics, degradation events
causing cancer are negligible at leading order (i.e. $p_{c}\simeq 1$) and the
fraction of cancerous cells stays close to zero. After this rapid initial
phase, the solution develops into a second, slower phase (corresponding to the
slow manifold [38]), where senescent cells are slowly removed by the combined
action of selection and cancer-causing mutations and the fraction of cancerous
types is seen to increase with time. Cancerous cells display sigmoidal
kinetics, increasing slowly initially then more rapidly and eventually
reaching a plateau. At the end of this second phase of dynamics, the fractions
of healthy and senescent cells approach zero, while c and b cells form an
equilibrium with fractions $[1-\mu_{v}/(kf_{0})]^{n}$ and
$[\mu_{v}/(kf_{0})]^{n}$, respectively. A measure of the characteristic
timescale of cancer development is given by the time needed for $p_{c}(t)$ to
drop by a factor of two (the half-life), yielding
$t_{1/2}=\tau_{c}\ln(2+f_{c}/\mu_{c})$, i.e. it is directly proportional to
$\tau_{c}$ with a logarithmic correction that depends on the ratio
$f_{c}/\mu_{c}$. The half-life $t_{1/2}$ increases with decreasing $f_{c}$ or
$\mu_{c}$. In particular, for $\mu_{c}\to 0$ we find $t_{1/2}\to\infty$,
suggesting that in the absence of cancer-causing mutations, intercellular
competition is able to maintain the pre-equilibrium between healthy and
senescent cells indefinitely. In reality, for any value of $\mu_{c}>0$,
however small, the final ($t\to\infty$) state of the system consists of an
equilibrium between c and b cells and no healthy cells, in line with the idea
that above some threshold age, the mortality rate monotonically increases with
time [14].

### Organismal vitality and optimal competition

Having an understanding of the population dynamics using our master equation
formalism, we now turn to question the relative importance of the various
“microscopic” (cell-level) processes and their contribution to “macroscopic”
(organism-level) observables, and in particular the strength of intercellular
competition. To do so, a natural definition of a minimal model of average
organismal vitality is the population-weighted average

$\overline{V}(t)\equiv\sum_{v,c}V(v,c)\,\rho(v,c,t)\,,$ (6)

where $V(v,c)$ describes the contribution of cell type $(v,c)$ to the vitality
of the whole organism. Since vitality requires cells to be both vigorous and
cooperative, a simple choice for $V(v,c)$ is:

$V(v,c)=V_{0}\,v\,c,$ (7)

where $vc$ measures the amount of resources devoted to cooperative activities
and $V_{0}$ is a pre-factor that sets the units of vitality. This
multiplicative choice is guided by the thought that for a multicellular
organism to be vital, we need cells to be both vigorous and cooperative. Since
the mean of the binomial distribution $\textrm{Bin}(v,n,p)$ is
$\overline{v}=np$, for this choice of $V(v,c)$, the average vitality takes the
simple form
$\overline{V}(t)=V_{0}\,\overline{v}(t)\,\overline{c}(t)=V_{0}\,n\,m\,p_{v}(t)\,p_{c}(t)$.

Using the analytical solution for $\rho(v,c,t)$ given by [5] we optimize
$\overline{V}(T)$ with respect to the strength of intercellular competition
$f_{0}$, for times $T\gg 1/f_{0},1/\mu_{v}$, i.e. much larger than the
timescales of proliferation and mutation (the dependence of the results on the
choices of these parameters is considered later). Figure 3(a) shows a plot of
the average organismal vitality $\overline{V}(T)$ at the observation time
$T=3\mu_{v}^{-1}$ as a function of the cost of cooperation $k$ and the
strength of competition $f_{0}$. We see that $\overline{V}$ has a non-
monotonic behaviour with $f_{0}$. In Fig. 3(b), we show the distinct maximum
of vitality as a function of the base level of competition characterized by
$f_{0}$ and the cost of cooperation $k$, characterized by an optimal curve
$f_{0}^{\star}(k)$ (see SI Sec. S3) for asymptotic analytical expressions for
this optimal line $f_{0}^{\star}(k)$). Inspection of Fig. 3(b) shows that when
$k=0$ the optimal strength of competition diverges, i.e.
$f_{0}^{\star}\to\infty$. This is intuitive since for $k=0$ there is no cost
for cooperation. Cancer and healthy cells have therefore the same fitness,
which implies that cheating gives cancer cells no advantage in selection. As
we increase the cost of cooperation, $k$, the optimal competition
$f_{0}^{\star}(k)$ gradually decays indicating that the system cannot tolerate
high levels of intercellular competition when the cost of cooperation becomes
large. Eventually $f_{0}^{\star}(k)$ becomes zero at a critical value
$k=k_{\rm{crit}}$ and then stays identically zero for larger values
$k>k_{\rm{crit}}$. In this limit, the fitness of healthy cells is so low that
any non-zero amount of intercellular competition will result in a dominance of
cancer. The critical $k_{\rm{crit}}$ depends on a combination of the rates of
mutation (see SI Sec. S3 and Fig. S3 for the exact analytical expression for
$k_{\rm{crit}}$). When $\mu_{v}/\mu_{c}\to 0$ the critical $k_{\rm{crit}}\to
0$, while for $\mu_{v}/\mu_{c}\to\infty$ we have $k_{\rm{crit}}\to 1$. When
$\mu_{v}/\mu_{c}\to 1$ we find $k_{\rm{crit}}\to 1/2$. We note that the level
of optimal competition also depends on the rates of mutations causing
senescence $\mu_{v}$ and cancer $\mu_{c}$ and on the observation time $T$ (see
Fig. S2). Increasing $\mu_{v}$ requires a higher optimal level of competition
to counteract the stronger tendency of senescent cell accumulation, while
increasing $\mu_{c}$ causes $f_{0}^{\star}$ to decrease since the system is
more susceptible to cheating. Larger times $T$ require a lower level of
intercellular competition, reflecting a balance between shorter-term advantage
of cheating and longer-term detrimental effects of cheating on organismal
vitality.

The sensitivity of the value of the optimal competition is defined by the
derivative of the optimal line $f_{0}^{\star}(k)$ with respect to the cost of
cooperation $k$, i.e. $|\partial f_{0}^{\star}/\partial k|$. High sensitivity
means that optimal competition is easily affected by changes in $k$, i.e.
small fluctuations in $k$ will push the system away from the optimum. In
contrast, low sensitivity implies that optimal competition is relatively
insensitive to the choice of $k$, which facilitates a robust approach to the
optimum. In Fig. 3(c)), we show that to optimize vitality while maintaining
low sensitivity a plausible solution is to operate in the regime of low to
moderate $k$, as this choice yields a robust strategy with relatively high
vitality.

To understand the dynamics the populations as a function of the level of
competition, we note that in Fig. 3(b), the optimal curve $f_{0}^{\star}(k)$
divides $(k,f_{0})$-space into two separate regions, which we term the
senescent region and the cancerous region. As shown in Fig. 3(d), choosing the
parameters $(k,f_{0})$ along the optimal curve (scenario O) delays the loss of
organismal vitality, corresponds to a balance between senescence cell
accumulation and cancer proliferation. Compared to this situation, when
parameters are chosen in the senescent region (scenario S) the loss of
organismal vitality is dominated by accumulation of senescent cells (Fig.
3(e)), while in the cancerous region (scenario C) loss of vitality is driven
by the proliferation of cancer cells (Fig. 3(e)).

## Discussion

In this study, we have proposed a model for aging in terms of the dynamics of
a multicellular population characterized by its vigor and cooperation, and a
fitness parameter that is a function of these variables. Using a master-
equation based framework, we derived a replicator-like dynamical equation
accounting for fluxes due to mutations that lead to cancer and senescence.
Assuming a minimal closure relation for the fitness that is linear in the
vigor and cooperation, we are led to an analytic solution for the evolution of
the relative fraction of healthy, senescent and cancerous populations while
accounting for the interplay between competition and cooperation. Using our
model and a simple choice of organismal vitality that is multiplicative in the
vigor and cooperation, we then probed the optimal level of intercellular
competition that maximizes organismal vitality, balancing the cost of
cooperation and the strength of competition.

It might be interesting to interpret our result of an optimal level of
competition in terms of Parrondo’s paradox as highlighted in a commentary
[27], whereby combining losing strategies may lead to a winning strategy [39,
40, 41]. This idea has been invoked in a range of biological systems to
explain adaptation in areas ranging from genetics to ecology [42, 43, 44]. In
the context of multicellular aging, too little competition is a losing
strategy because it leads to senescent cell accumulation. On the other hand,
too much competition is also a losing strategy because even if senescent cells
are eliminated it leads to cheater-cell proliferation in the long term.
Following the logic of Parrondo’s paradox, we see that a fruitful approach to
delay multicellular aging combines both (losing) strategies to achieve a fine
balance that delays senescence and staves off cancer. In this optimal
scenario, competition is strategically reduced when cheater cells over-
proliferate and increased back when senescent cells over-accumulate. A similar
strategy may be realised with coexisting subpopulations of uncompetitive and
competitive cells if the ratio between the subpopulations is strategically
changed in response to the changing environment.

Our framework may be extended in multiple ways. Effective multicellularity not
only requires cooperation between cells, but also mechanisms for suppressing
conflict that result from it. This effect may be accounted for by introducing
a feedback between strength of competition and the current system state. To
investigate the impact of drugs that clear senescent or cancer cells [45], we
can envisage coupling the master replicator equation with dynamic equations
for an inhibitor that removes deleterious cell types. Generalizing our results
could thus provide a framework for interrogating the effect of strategies to
combat aging and cancer dynamics and how to optimize them. Furthermore, our
model considers detrimental mutations reducing vigor or cooperation as
irreversible. Even though proliferation arrest in senescence cells is
essentially irreversible, certain biological manipulations, including
inactivation of specific tumor suppressor genes, can reverse senescence [46].
The latter scenario could be studied by making transitions between cell types
in our model reversible, and more generally by including separate dynamic
equations modeling changes of $v$ and $c$ over evolutionary timescales to
model the effect of tumor suppressor genes. While we have limited ourselves
here to the balance between cooperation and competition during the later
stages of life associated with aging, similar questions arise during the
developmental stages of organisms when proliferation rates of cells are high
and thus susceptible to mutational errors [47]. Understanding the dynamic
regulation (and misregulation) of intercellular competition in early
development might also be amenable to our approach.

While we have limited ourselves to studying the role of discrete mutations, it
is not hard to generalize our framework to the case of continuously varying
vigor and cooperation. This leads to a Boltzmann-like equation for the
dynamics of cellular populations (see SI Sec. S2). Finally, it is worth noting
the natural appearance of extensive variables such as vigor (and vitality) and
intensive variables such as cooperation (and competition), suggesting natural
analogies to (non-equilibrium) thermodynamics. The relation between these
variables and various forms of closure relations will be explored in a
separate study.

We acknowledge support from the UCL Institute for the Physics of Living
Systems (TCTM), the Swiss National Science Foundation (TCTM), Peterhouse,
Cambridge (TCTM), the Simons Foundation (LM) and the Henri Seydoux Fund (LM).

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This immediately implies that anything in $\cJ^{\textcolor{red}{\cA}}$ acts as zero.

Looking at our string operators from <ref> above,
it is clear from (<ref>) that the dome ideal acts as zero whenever the anyon comes from $Z^\cA(\cX)$.
Indeed, if $X\in Z^\cA(\cX)$, then calculating in $Z(\cX)$, remembering that squiggly-drawn anyons represent $Z^\cA(\cX)^{\rev}$ as opposed to $Z^\cA(\cX)\subset Z(\cX)$ (and suppressing 4th roots),

\sum_{x\in \Irr(\cX)}
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\draw[thick] (0,0) -- (0,.7);
\filldraw[red] (0,.4) node[right, yshift=.15cm]{$\scriptstyle f$} circle (.05cm);
\filldraw[orange] (0,0) node[right,xshift=-.05cm]{$\scriptstyle \psi$} circle (.05cm);
\tikzmath{
\draw[red, thick] (-.4,0) arc (180:0:.4cm);
\draw[red, thick] (-.4,0) arc (-180:0:.4cm);
\draw[thick, knot, orange](0,-.7) --node[right]{$\scriptstyle X$} (0,-.4) -- (0,0);
\node[red] at (-.7,.3) {$\scriptstyle \Phi(a)$};
\draw[thick] (0,0) -- (0,.7);
\filldraw[red] (0,.4) node[right, yshift=.15cm]{$\scriptstyle f$} circle (.05cm);
\filldraw[orange] (0,0) node[right,xshift=-.05cm]{$\scriptstyle \psi$} circle (.05cm);
\tikzmath{
\draw[thick, orange](0,-.7) --node[right]{$\scriptstyle X$} (0,-.2);
\draw[red, thick] (-.2,.2) arc (180:0:.2cm);
\draw[red, thick] (-.2,.2) arc (-180:0:.2cm);
\draw[thick, knot] (0,-.2) -- (0,.7);
\filldraw[red] (0,.4) node[right, yshift=.15cm]{$\scriptstyle f$} circle (.05cm);
\filldraw[orange] (0,-.2) node[right,xshift=-.05cm]{$\scriptstyle \psi$} circle (.05cm);
\tikzmath{
\draw[thick, orange, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}](0,-.7) --node[right]{$\scriptstyle X$} (0,-.2);
\draw[red, thick] (-.2,.2) arc (180:0:.2cm);
\draw[red, thick] (-.2,.2) arc (-180:0:.2cm);
\draw[thick, knot] (0,-.2) -- (0,.7);
\node[red] at (-.5,.3) {$\scriptstyle \Phi(a)$};
\filldraw[red] (0,.4) node[right, yshift=.15cm]{$\scriptstyle f$} circle (.05cm);
\filldraw[orange] (0,-.2) node[right,xshift=-.05cm]{$\scriptstyle \psi$} circle (.05cm);
\,.

As in Remark <ref>, we can additionally see that isomorphism classes of $\Dome^{\textcolor{red}{\cA}}(\cX)$ are in bijection with boundary anyon types.

In the case of a twice-punctured sphere with internal $\cA$-edges,

\tikzmath[scale=1.5]{
\begin{scope}[xshift=-.4cm, yshift=.4cm]
\draw[thick, knot, thick] (1,1) -- (3,1);
\draw[thick, knot, thick] (1,2) -- (3,2);
\draw[thick, knot, thick] (1,3) -- (3,3);
\draw[thick, knot, thick] (1,1) -- (1,3);
\draw[thick, knot, thick] (3,1) -- (3,3);
\draw[thick, knot, thick] (2,1) -- (2,2.3);
\end{scope}
\begin{scope}[xshift=-.2cm, yshift=.2cm]
\draw[thick, knot, thick] (1,1) -- (3,1);
\draw[thick, knot, thick, red] (1,2) -- (3,2);
\draw[thick, knot, thick] (1,3) -- (3,3);
\draw[thick, knot, thick] (1,1) -- (1,3);
\draw[thick, knot, thick] (3,1) -- (3,3);
\draw[thick, knot, thick, red] (2,1) -- (2,3);
\end{scope}
\begin{scope}
\draw[thick, knot, thick] (1,1) -- (3,1);
\draw[thick, knot, thick, red] (1,2) -- (3,2);
\draw[thick, knot, thick] (1,3) -- (3,3);
\draw[thick, knot, thick] (1,1) -- (1,3);
\draw[thick, knot, thick] (3,1) -- (3,3);
\draw[thick, knot, thick, red] (2,1) -- (2,3);
\end{scope}
\begin{scope}[xshift=.2cm, yshift=-.2cm]
\draw[thick, knot, thick] (1,1) -- (3,1);
\draw[thick, knot, thick] (1,2) -- (3,2);
\draw[thick, knot, thick] (1,3) -- (3,3);
\draw[thick, knot, thick] (1,1) -- (1,3);
\draw[thick, knot, thick] (3,1) -- (3,3);
\draw[thick, knot, thick] (2,1) -- (2,2.3);
\end{scope}
\foreach \x in {1,2,3}{
\foreach \y in {1,2,3}{
\fill ($ (\x,\y) + (-.4,.4) $) circle (.035cm);
\fill ($ (\x,\y) + (.2,-.2) $) circle (.035cm);
\foreach \x in {1,2,3}{
\foreach \y in {1,3}{
\fill ($ (\x,\y) + (-.2,.2) $) circle (.035cm);
\fill (\x,\y) circle (.035cm);
\fill (1,2) circle (.035cm);
\fill ($ (1,2) + (-.2,.2) $) circle (.035cm);
\fill[red] (2,2) circle (.035cm);
\fill[red] ($ (2,2) + (-.2,.2) $) circle (.035cm);
\fill (3,2) circle (.035cm);
\fill ($ (3,2) + (-.2,.2) $) circle (.035cm);
\foreach \x in {1,2,3}{
\foreach \y in {1,3}{
\draw[thick] ($ (\x,\y) + (.2,-.2) $) -- ($ (\x,\y) + (-.4,.4) $);
\draw[thick] ($ (1,2) + (.2,-.2) $) -- ($ (1,2) + (-.4,.4) $);
\draw[thick] ($ (3,2) + (.2,-.2) $) -- ($ (3,2) + (-.4,.4) $);

the space of ground states is now the subspace of $\Tube(\cX)$ which is compatible with the $\textcolor{red}{\cA}$-bulk.
Since $\Dome^{\textcolor{red}{\cA}}(\cX)$ is just the quotient of $\Tube(\cX)$ obtained by imposing the compatibility relations $\cJ^{\textcolor{red}{\cA}}$, this is exactly the regular representation of $\Dome^{\textcolor{red}{\cA}}(\cX)$, i.e. $\Dome^{\textcolor{red}{\cA}}(\cX)$ as a $\Dome^{\textcolor{red}{\cA}}(\cX)-\Dome^{\textcolor{red}{\cA}}(\cX)$ bimodule.
An argument similar to Remark <ref> shows that anyon types are in bijective correspondence with $\Irr(\Rep(\Dome^{\textcolor{red}{\cA}}(\cX)))=\Irr(Z^{\cA}(\cX))$.

§.§ Sphere algebra, bulk excitations, and bulk enrichment of boundary

Suppose $\cA$ is an arbitrary UBFC and $\cX$ is an $\cA$-enriched UFC such that the unitary braided central functor $\Phi^Z:\cA\to Z(\cX)$ is fully faithful.
In this section, we analyze the (3+1)D bulk $\cA$-excitations and show they are $Z_2(\cA)$ by constructing representations of the sphere algebra $\Sphere(\cA)$, a further quotient of $\Tube(\cA)$.
This is consistent with <cit.> which states that $\Omega Z(\Sigma \cA)= Z_2(\cA)$.
By <cit.>, $Z^\cA(\cX)$ is canonically $Z_2(\cA)$-enriched as a UBFC, i.e., $Z_2(Z^\cA(\cX))=Z_2(\cA)$.
We realize this enrichment by using hopping operators to bring bulk excitations into the boundary.

The following definition was suggested to us by Corey Jones.

The sphere algebra $\Sphere(\cA)$ $\Sphere(\cA)$ the sphere algebra of $\cA$
is the quotient of $\Tube(\cA)$ by the $*$-ideal

\cJ:=
\left\langle\;\;
\tikzmath{
\draw[thick, orange] (-.3,-.3) --node[left]{$\scriptstyle c$} (-.3,-.7);
\draw[thick, orange] (.3,-.3) --node[right]{$\scriptstyle a$} (.3,-.7);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,.7);
\draw[thick, orange] (.3,.3) --node[right]{$\scriptstyle c$} (.3,.7);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\phi$}
\tikzmath{
\draw[thick, orange] (-.3,-.3) node[left, yshift=-.2cm]{$\scriptstyle c$} arc (-180:0:.6cm) -- (.9,.3) node[right]{$\scriptstyle \overline{c}$} arc (0:180:.3cm);
\draw[knot, thick, orange] (.3,-.3) -- (.3,-.9) --node[right]{$\scriptstyle a$} (.3,-1.3);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,1.1);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\phi$}
\,,\,
\tikzmath{
\draw[thick, orange] (-.3,-.3) --node[left]{$\scriptstyle c$} (-.3,-.7);
\draw[thick, orange] (.3,-.3) --node[right]{$\scriptstyle a$} (.3,-.7);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,.7);
\draw[thick, orange] (.3,.3) --node[right]{$\scriptstyle c$} (.3,.7);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\psi$}
\tikzmath{
\draw[thick, orange] (.3,-.3) -- (.3,-.9) --node[right]{$\scriptstyle a$} (.3,-1.3);
\draw[knot, thick, orange] (-.3,-.3) node[left, yshift=-.2cm]{$\scriptstyle c$} arc (-180:0:.6cm) -- (.9,.3) node[right]{$\scriptstyle \overline{c}$} arc (0:180:.3cm);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,1.1);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\psi$}
\;\;\right\rangle.

Similar to the dome algebra, $\Sphere(\cA)$ corresponds to thinking of $\cA$ as an $\cA\boxtimes \cA^{\rev}$-enriched UFC.

\tikzmath{
\pgfmathsetmacro{\xDome}{.4};
\pgfmathsetmacro{\yDome}{.15};
\coordinate (a) at (0,0);
\coordinate (b) at ($ (a) + 4*(\xDome,0) $);
\coordinate (c) at ($ (a) + (\xDome,0) $);
\draw[thick, orange] ($ (a) + 2*(\xDome,0) + (240:{\xDome}) $) .. controls ++(240:1cm) and ++(-60:1cm) .. ($ (a) + 2*(\xDome,0) + (-60:\xDome)$) ;
\node at ($ (a) + 2*(\xDome,0) - 8*(0,\yDome) $) {\textcolor{orange}{\scriptsize{$c$}}};
\draw[thick, orange] (a) --node[below=-.1cm] {\scriptsize{$a$}} ($ (a) + (\xDome,0) $);
\draw[thick, orange] ($ (a) + 3*(\xDome,0) $) --node[below=-.1cm] {\scriptsize{$b$}} ($ (a) + 4*(\xDome,0) $);
\draw[very thick, dotted] (a) arc (180:0:{5*\xDome} and {5*\yDome});
\draw[very thick, knot] (a) arc (-180:0:{5*\xDome} and {5*\yDome});
\draw[very thick] (a) arc (-180:180:{5*\xDome});
\draw[very thick, dotted] (b) arc (180:0:{\xDome} and {\yDome});
\draw[very thick] (b) arc (-180:0:{\xDome} and {\yDome});
\draw[very thick] (b) arc (-180:180:{\xDome});
\draw[very thick, dotted] (c) arc (180:0:{\xDome} and {\yDome});
\draw[very thick] (c) arc (-180:0:{\xDome} and {\yDome});
\draw[very thick] (c) arc (-180:180:{\xDome});
\node at ($ (c) + (\xDome,0) $) {$\scriptstyle \phi$};
\,\,\simeq\,\,
\tikzmath{
\pgfmathsetmacro{\xDome}{.4};
\pgfmathsetmacro{\yDome}{.15};
\coordinate (a) at (0,0);
\coordinate (b) at ($ (a) + 4*(\xDome,0) $);
\coordinate (c) at ($ (a) + (\xDome,0) $);
\node at ($ (a) + 8.5*(\xDome,0) $) {\textcolor{orange}{\scriptsize{$c$}}};
\draw[thick, orange] (a) --node[below=-.1cm] {\scriptsize{$a$}} ($ (a) + (\xDome,0) $);
\draw[thick, orange] ($ (a) + 3*(\xDome,0) $) --node[below=-.1cm] {\scriptsize{$b$}} ($ (a) + 4*(\xDome,0) $);
\draw[very thick, dotted] (a) arc (180:0:{5*\xDome} and {5*\yDome});
\draw[very thick] (a) arc (-180:0:{5*\xDome} and {5*\yDome});
\draw[very thick] (a) arc (-180:180:{5*\xDome});
\draw[very thick, dotted] (b) arc (180:0:{\xDome} and {\yDome});
\draw[very thick] (b) arc (-180:0:{\xDome} and {\yDome});
\draw[very thick] (b) arc (-180:180:{\xDome});
\draw[very thick, dotted] (c) arc (180:0:{\xDome} and {\yDome});
\draw[very thick] (c) arc (-180:0:{\xDome});
\node at ($ (c) + (\xDome,0) $) {$\scriptstyle \phi$};
\draw[thick, orange, knot] ($ (b) + (\xDome,0) + (-163:{3.2*\xDome} and {3.4*\yDome}) $) arc (-163:140:{3.2*\xDome} and {3.4*\yDome});
\draw[thick, orange, knot, densely dotted] ($ (b) + (\xDome,0) + (140:{3.2*\xDome} and {3.4*\yDome}) $) arc (140:163:{3.2*\xDome} and {3.4*\yDome});
\draw[very thick] (c) arc (-180:0:{\xDome} and {\yDome});
\draw[very thick] (c) arc (180:0:{\xDome});
\,\,\simeq\,\,
\tikzmath{
\pgfmathsetmacro{\xDome}{.4};
\pgfmathsetmacro{\yDome}{.15};
\coordinate (a) at (0,0);
\coordinate (b) at ($ (a) + 4*(\xDome,0) $);
\coordinate (c) at ($ (a) + (\xDome,0) $);
\draw[very thick, dotted] (a) arc (180:0:{5*\xDome} and {5*\yDome});
\draw[thick, orange, knot] ($ (a) + 2*(\xDome,0) + (60:{\xDome}) $) .. controls ++(60:1cm) and ++(120:1cm) .. ($ (a) + 2*(\xDome,0) + (120:\xDome)$) ;
\node at ($ (a) + 2*(\xDome,0) + 8*(0,\yDome) $) {\textcolor{orange}{\scriptsize{$c$}}};
\draw[thick, orange] (a) --node[below=-.1cm] {\scriptsize{$a$}} ($ (a) + (\xDome,0) $);
\draw[thick, orange] ($ (a) + 3*(\xDome,0) $) --node[below=-.1cm] {\scriptsize{$b$}} ($ (a) + 4*(\xDome,0) $);
\draw[very thick] (a) arc (-180:0:{5*\xDome} and {5*\yDome});
\draw[very thick] (a) arc (-180:180:{5*\xDome});
\draw[very thick, dotted] (b) arc (180:0:{\xDome} and {\yDome});
\draw[very thick] (b) arc (-180:0:{\xDome} and {\yDome});
\draw[very thick] (b) arc (-180:180:{\xDome});
\draw[very thick, dotted] (c) arc (180:0:{\xDome} and {\yDome});
\draw[very thick] (c) arc (-180:0:{\xDome} and {\yDome});
\draw[very thick] (c) arc (-180:180:{\xDome});
\node at ($ (c) + (\xDome,0) $) {$\scriptstyle \phi$};

The algebra $\Sphere(\cA)$ is actually abelian.
Indeed, consider the algebra maps:

\begin{equation}
\label{eqn:SphereAlgMaps}
\tikzmath{
\draw[thick, orange] (-.3,-.3) --node[left]{$\scriptstyle c$} (-.3,-.7);
\draw[thick, orange] (.3,-.3) --node[right]{$\scriptstyle a$} (.3,-.7);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,.7);
\draw[thick, orange] (.3,.3) --node[right]{$\scriptstyle c$} (.3,.7);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\phi$}
\overset{\lambda}{\longmapsto}
\delta_{a=b}
\cdot
\tikzmath{
\draw[thick, orange] (-.3,-.3) node[left, yshift=-.2cm]{$\scriptstyle c$} arc (-180:0:.6cm) -- (.9,.3) node[right]{$\scriptstyle \overline{c}$} arc (0:180:.3cm);
\draw[knot, thick, orange] (.3,-.3) -- (.3,-.9) --node[right]{$\scriptstyle a$} (.3,-1.3);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,1.1);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\phi$}
\qquad\text{and}\qquad
\tikzmath{
\draw[thick, orange] (-.3,-.3) --node[left]{$\scriptstyle c$} (-.3,-.7);
\draw[thick, orange] (.3,-.3) --node[right]{$\scriptstyle a$} (.3,-.7);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,.7);
\draw[thick, orange] (.3,.3) --node[right]{$\scriptstyle c$} (.3,.7);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\psi$}
\overset{\rho}{\longmapsto}
\delta_{a=b}
\cdot
\tikzmath{
\draw[thick, orange] (.3,-.3) -- (.3,-.9) --node[right]{$\scriptstyle a$} (.3,-1.3);
\draw[knot, thick, orange] (-.3,-.3) node[left, yshift=-.2cm]{$\scriptstyle c$} arc (-180:0:.6cm) -- (.9,.3) node[right]{$\scriptstyle \overline{c}$} arc (0:180:.3cm);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle b$} (-.3,1.1);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$\psi$}
\end{equation}

We see that if $\phi: ca\to ac$ and $\psi: da\to ad$, then $\phi\psi - \psi\phi \in \cJ$:

\begin{align*}
\phi\psi - \psi\phi
\phi\psi
- \lambda(\phi)\psi + \lambda(\phi)\psi
- \lambda(\phi)\lambda(\psi) + \underbrace{\lambda(\phi)\lambda(\psi)}_{=\lambda(\psi)\lambda(\phi)}
- \lambda(\psi)\phi + \lambda(\psi)\phi
\\
\big(\phi - \lambda(\phi)\big)\psi
\lambda(\phi)\big(\psi - \lambda(\psi)\big)
\lambda(\psi)\big(\lambda(\phi)-\phi\big)
\big(\lambda(\psi)-\psi\big)\phi
\in \cJ.
\end{align*}

The next lemma gives an abstract characterization of $\Sphere(\cA)$.

The following $*$-algebras are isomorphic:

* $\Sphere(\cA)$,

* $T_2/(T_2\cap \cJ)$ where $T_2\subset \Tube(\cX)$ is the subalgebra spanned by $\phi:ca\to bc$ for $a,b,c\in \Irr(\cA)$ such that $a=b\in Z_2(\cA)$, and

* $\bbC^{\Irr(Z_2(\cA))}$.

First, given
$a,b,c\in \Irr(\cA)$
$\phi : ca\to bc$ in $\Tube(\cA)$,
if $a\neq b$, then $\phi \in \cJ$.
If $a\neq b$, then $\lambda(\phi)=0$, so
$\phi = \phi-\lambda(\phi)\in \cJ$.

Second, suppose $a\notin \Irr(Z_2(\cA))$, and pick $c\in \Irr(\cA)$ whose $S$-matrix entry $S_{a,c}\neq d_ad_c$.
We claim that $\id_a\in \cJ$.

\frac{S_{a,c}
\cdot
\id_a
\tikzmath{
\draw[thick, orange] (0,0) -- (0,.5);
\draw[thick, red, knot] (0,0) circle (.3cm);
\draw[thick, orange, knot] (0,0) -- (0,-.5);
\node[red] at (-.45,0) {$\scriptstyle c$};
\node[orange] at (.15,0) {$\scriptstyle a$};
\tikzmath{
\draw[thick, orange] (0,-.5) -- (0,.5);
\draw[thick, red, knot] (0,0) circle (.3cm);
\node[red] at (-.45,0) {$\scriptstyle c$};
\node[orange] at (.15,0) {$\scriptstyle a$};
\lambda(\beta_{a,c})
\rho(\beta_{a,c})
\lambda(\beta_{a,c})
\beta_{a,c}
\beta_{a,c}
\rho(\beta_{a,c})
\in\cJ.

By the second isomorphism theorem, $T_2/(T_2\cap \cJ) \cong (T_2+\cJ)/\cJ$.
By the two facts proven above, every representative of an element of $\Sphere(\cA)$ in $\Tube(\cA)$ can be taken from $T_2$.
This means the $*$-algebra map $(T_2+\cJ)/\cJ \to \Sphere(\cA)$
given by
$x+\cJ\mapsto x+\cJ$
is onto and thus an isomorphism.

By the two facts proven above, the map $\delta_a\mapsto \id_a+T_2\cap \cJ$ is a surjective $*$-algebra map $\bbC^{\Irr(Z_2(\cA))}\to T_2/(T_2\cap \cJ)$.
The inverse is given by $\lambda|_{T_2}$, which descends to $T_2/(T_2\cap \cJ)$ since $T_2\cap \cJ \subset \ker(\lambda|_{T_2})$.

The proof of the following theorem is similar to Theorem <ref> and omitted.
The equivalences of (1)-(4) below are all well known; e.g., see <cit.>.

Suppose $\cA$ is a braided fusion category.
The following are equivalent.

* $a\in Z_2(\cA)$, the Müger center of $\cA$,

* The braidings $\beta_{a,b}$ and $\beta_{b,a}^{-1}$
agree for all $b\in \cA$,

* The two canonical lifts of $a\in\cA$ to the Drinfeld center $Z(\cA)$ agree,

* The braidings $\beta_{a,b}$ and $\beta_{b,a}^{-1}$ have the same matrix units, i.e.,
for every
$m\in \cA(a\to c)$,
$n\in \cA(a\to d)$,
$f\in \cA(b c\to e)$,
$g\in \cA(d b\to e)$
for $c,d,e\in \Irr(\cA)$,

\tikzmath{
\draw[thick, red] (-.3,-.45) node[left, yshift=.2cm]{$\scriptstyle a$} .. controls ++(90:.3cm) and ++(-90:.3cm) .. (.3,.45);
\draw[knot, thick, orange] (.2,-1.4) -- (.2,-.45) .. controls ++(90:.3cm) and ++(-90:.3cm) .. (-.2,.45) --node[left]{$\scriptstyle b$} (-.2,1.4);
\draw (.3,.95) --node[right]{$\scriptstyle c$} (.3,1.4);
\draw (-.3,-.95) --node[left]{$\scriptstyle d$} (-.3,-1.4);
\draw (0,2) --node[right]{$\scriptstyle e$} (0,2.4);
\draw (0,-2) --node[right]{$\scriptstyle e$} (0,-2.4);
\roundNbox{fill=white}{(0,1.7)}{.3}{.2}{.2}{$f$}
\roundNbox{fill=white}{(.3,.7)}{.25}{0}{0}{$m$}
\roundNbox{fill=white}{(-.3,-.7)}{.25}{0}{0}{$n^\dag$}
\roundNbox{fill=white}{(0,-1.7)}{.3}{.2}{.2}{$g^\dag$}
\tikzmath{
\draw[thick, orange] (.2,-1.4) -- (.2,-.45) .. controls ++(90:.3cm) and ++(-90:.3cm) .. (-.2,.45) --node[left]{$\scriptstyle b$} (-.2,1.4);
\draw[thick, red, knot] (-.3,-.45) node[left, yshift=.2cm]{$\scriptstyle a$} .. controls ++(90:.3cm) and ++(-90:.3cm) .. (.3,.45);
\draw (.3,.95) --node[right]{$\scriptstyle c$} (.3,1.4);
\draw (-.3,-.95) --node[left]{$\scriptstyle d$} (-.3,-1.4);
\draw (0,2) --node[right]{$\scriptstyle e$} (0,2.4);
\draw (0,-2) --node[right]{$\scriptstyle e$} (0,-2.4);
\roundNbox{fill=white}{(0,1.7)}{.3}{.2}{.2}{$f$}
\roundNbox{fill=white}{(.3,.7)}{.25}{0}{0}{$m$}
\roundNbox{fill=white}{(-.3,-.7)}{.25}{0}{0}{$n^\dag$}
\roundNbox{fill=white}{(0,-1.7)}{.3}{.2}{.2}{$g^\dag$}

* The representation of the tube algebra $\Tube(\cA)$ associated to $(a,\beta_a)$ descends to a representation of the sphere algebra $\Sphere(\cA)$.

While the sphere algebra is similar to the dome algebra, representations of the former correspond to $Z_2(\cA)$, while representations of the latter gives $Z^\cA(\cA)$, the centralizer $Z(\cA)$ of $A$ in $Z(\cA)$.
The canonical quotient map $\Dome^\cA(\cA)\twoheadrightarrow \Sphere(\cA)$ corresponds to the inclusion $Z_2(\cA)\hookrightarrow Z^\cA(\cA)$.

Similar to (<ref>) above, we have a $\Tube(\cA)$-action on any state with a localized bulk excitation.
Here, localized means that the bulk Hamiltonian is violated at (at most) 5 terms, namely one edge term and the 4 neighboring plaquette terms.
By a Morita equivalence argument similar to that in <ref>, it again suffices to consider the $\Tube(\cA)$-action at a single edge $\ell$ connected to one vertex of the form

\tikzmath{
\draw[thick, orange] (-.3,-.3) --node[left]{$\scriptstyle d$} (-.3,-.7);
\draw[thick, orange] (.3,-.3) --node[right]{$\scriptstyle b$} (.3,-.7);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle c$} (-.3,.7);
\draw[thick, orange] (.3,.3) --node[right]{$\scriptstyle d$} (.3,.7);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$f$}
\rhd
\tikzmath[scale=1.5]{
\draw[thick, red, xshift=-.4cm, yshift=.4cm] (.7,.7) grid (3.3,2.3);
\begin{scope}[xshift=-.2cm, yshift=.2cm]
\draw[thick, knot, thick, red] (.7,1) -- (3.3,1);
\draw[thick, knot, thick, red] (.7,2) -- (3.3,2);
\draw[thick, knot, thick, red] (1,.7) -- (1,2.3);
\draw[thick, knot, thick, red] (3,.7) -- (3,2.3);
\draw[thick, knot, thick, red] (2,.7) -- (2,1.3);
\draw[thick, knot, thick, red] (2,2.3) -- (2,2);
\end{scope}
\draw[thick, knot, red] (.7,.7) grid (3.3,2.3);
\foreach \x in {1,2,3}{
\foreach \y in {1,2}{
\draw[thick, red] ($ (\x,\y) + (.2,-.2) $) -- ($ (\x,\y) + (-.6,.6) $);
\foreach \x in {.8,1.8,2.8}{
\foreach \y in {1.2,2.2}{
\fill[red] (\x,\y) circle (.03cm);
\foreach \x in {.6,1.6,2.6}{
\foreach \y in {1.4,2.4}{
\fill[red] (\x,\y) circle (.03cm);
\foreach \x in {1,2,3}{
\foreach \y in {1,2}{
\fill[red] (\x,\y) circle (.03cm);
\node[red] at (1.9,1.5) {$\scriptstyle a$};
\delta_{a=b}
\left(\frac{d_c}{d_a}\right)^{1/4}
\cdot
\tikzmath[scale=1.5]{
\draw[thick, red, xshift=-.4cm, yshift=.4cm] (.7,.7) grid (3.3,2.3);
\begin{scope}[xshift=-.2cm, yshift=.2cm]
\draw[thick, knot, thick, red] (.7,1) -- (3.3,1);
\draw[thick, knot, thick, red] (.7,2) -- (3.3,2);
\draw[thick, knot, thick, red] (1,.7) -- (1,2.3);
\draw[thick, knot, thick, red] (3,.7) -- (3,2.3);
\draw[thick, knot, thick, red] (2,.7) -- (2,1.3);
\draw[thick, knot, thick, red] (2,2.3) -- (2,2);
\end{scope}
\foreach \x in {.6,1.6,2.6}{
\foreach \y in {1.4,2.4}{
\fill[red] (\x,\y) circle (.03cm);
\draw[thick, knot, red] (.7,.7) grid (3.3,2.3);
\foreach \x in {1,2,3}{
\foreach \y in {1}{
\draw[thick, red] ($ (\x,\y) + (.2,-.2) $) -- ($ (\x,\y) + (-.6,.6) $);
\filldraw[orange, thick, fill=gray!30, xshift=-.2cm, yshift=.2cm] (1.5,1.1) -- (1.8,1.1) to[out=0,in=180] (2,1.3) to[out=0,in=180] (2.2,1.1) -- (2.8,1.1) arc (-90:0:.1cm) -- (2.9,1.8) arc (0:90:.1cm) -- (1.2,1.9) arc (90:180:.1cm) -- (1.1,1.2) arc (180:270:.1cm) -- (1.5,1.1);
\draw[thick, knot, red] (.7,.7) grid (3.3,2.3);
\foreach \x in {1,2,3}{
\foreach \y in {2}{
\draw[thick, red] ($ (\x,\y) + (.2,-.2) $) -- ($ (\x,\y) + (-.6,.6) $);
\foreach \x in {.8,1.8,2.8}{
\foreach \y in {1.2,2.2}{
\fill[red] (\x,\y) circle (.03cm);
\foreach \x in {1,2,3}{
\foreach \y in {1,2}{
\fill[red] (\x,\y) circle (.03cm);
\node[orange] at (1.3,1.4) {$\scriptstyle d$};
\draw[thick, red] (1.8,1.5) -- (1.8,1.7);
\filldraw[orange] (1.8,1.5) node[right, yshift=.1cm]{$\scriptstyle f$} circle (.05cm);

One now applies the argument of (<ref>) to see that this equal to the action of

\tikzmath{
\draw[thick, orange] (-.3,-.3) node[left, yshift=-.2cm]{$\scriptstyle d$} arc (-180:0:.6cm) -- (.9,.3) node[right]{$\scriptstyle \overline{d}$} arc (0:180:.3cm);
\draw[knot, thick, orange] (.3,-.3) -- (.3,-.9) --node[right]{$\scriptstyle b$} (.3,-1.3);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle c$} (-.3,1.1);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$f$}
\qquad\text{ or }\qquad
\tikzmath{
\draw[thick, orange] (.3,-.3) -- (.3,-.9) --node[right]{$\scriptstyle b$} (.3,-1.3);
\draw[knot, thick, orange] (-.3,-.3) node[left, yshift=-.2cm]{$\scriptstyle d$} arc (-180:0:.6cm) -- (.9,.3) node[right]{$\scriptstyle \overline{d}$} arc (0:180:.3cm);
\draw[thick, orange] (-.3,.3) --node[left]{$\scriptstyle c$} (-.3,1.1);
\roundNbox{}{(0,0)}{.3}{.2}{.2}{$f$}
\,,

and thus this $\Tube(\cA)$-action descends to an action of $\Sphere(\cA)$.
This argument also proves that any other similar $\Tube(\cA)$-action one writes down at this edge
(for example, we could glue in the $d$ string around $f$ to a different loop around the boundary)
will be equal to this one.

Suppose $\cX$ is an $\cA$-enriched UFC, where the braided central functor $\Phi^Z:\cA\to Z(\cX)$ is full.
The Müger center of the enriched center $Z^\cA(\cX)$ is $Z_2(\cA)$.

The above lemma clearly fails when the braided functor $\cA\to Z(\cX)$ is not full.
For example, consider $\cA=\Rep(G)$ and $\cX=\Vect$ and $\cA\to Z(\cX)=\Vect$ is the forgetful functor.

Given a bulk excitation $a\in \Irr(Z_2(\cA))$,
for every
$\psi: \Phi(a)\to x$ where $x\in \Irr(\cX)$,
we can define the bulk-to-boundary hopping operator

B_oB_pB_qB_r T_\pi^{(a,\beta_a)}(\psi)\pi_k^1

as in the cartoon below.

\tikzmath[scale=2]{
\draw[thick, red, xshift=-.4cm, yshift=.4cm] (.7,.7) grid (3.3,3.3);
\filldraw[thick, orange, rounded corners=5pt, fill=gray!30] (1.65,1.7) -- (1.65,2.3) -- (1.75,2.1) -- (1.75,1.3) -- (1.65,1.5) -- (1.65,1.7);
\begin{scope}[xshift=-.2cm, yshift=.2cm]
\draw[thick, knot, thick, red] (.7,1) -- (3.3,1);
\draw[thick, knot, thick, red] (.7,2) -- (3.3,2);
\draw[thick, knot, thick, red] (.7,3) -- (3.3,3);
\draw[thick, knot, thick, red] (1,.7) -- (1,3.3);
\draw[thick, knot, thick, red] (3,.7) -- (3,3.3);
\draw[thick, knot, thick, red] (2,.7) -- (2,1);
\draw[thick, knot, thick, red] (2,3.3) -- (2,2);
\end{scope}
\foreach \x in {1,2,3}{
\foreach \y in {1,2,3}{
\fill[red] ($ (\x,\y) + (-.4,.4) $) circle (.025cm);
\filldraw[thick, orange, knot, rounded corners=5pt, fill=gray!30] (.9,1.3) rectangle (1.7,2.1);
\filldraw[thick, orange, knot, rounded corners=5pt, fill=gray!30] (1.9,1.3) rectangle (2.7,2.1);
\filldraw[thick, orange, rounded corners=5pt, xshift=.2cm, yshift=-.2cm, fill=gray!30] (1.65,1.7) -- (1.65,2.3) -- (1.75,2.1) -- (1.75,1.3) -- (1.65,1.5) -- (1.65,1.7);
\draw[thick, cyan, knot, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}] (1.8,1.2) --node[right, xshift=-.07cm]{$\scriptstyle a$} (1.8,2.1) -- (2,2.1) -- (2.2,1.9);
\draw[thick, knot, red] (.7,.7) grid (3.3,3.3);
\draw[thick, cyan, knot, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}] (2.2,1.9) -- (2.2,2.15) ;
\begin{scope}[xshift=.2cm, yshift=-.2cm]
\draw[thick, knot, thick] (.7,1) -- (3.3,1);
\draw[thick, knot, thick] (.7,2) -- (3.3,2);
\draw[thick, knot, thick] (.7,3) -- (3.3,3);
\draw[thick, knot, thick] (1,.7) -- (1,3.3);
\draw[thick, knot, thick] (3,.7) -- (3,3.3);
\draw[thick, knot, thick] (2,.7) -- (2,2);
\draw[thick, knot, thick] (2,3.3) -- (2,3);
\draw[thick, knot, thick] (2,2.35) -- (2,2.55);
\end{scope}
\foreach \x in {1,2,3}{
\foreach \y in {2,3}{
\draw[thick, red] ($ (\x,\y) + (.2,-.2) $) -- ($ (\x,\y) + (-.6,.6) $);
\draw[thick, red] ($ (1,1) + (.2,-.2) $) -- ($ (1,1) + (-.6,.6) $);
\draw[thick, red] ($ (2,1) + (.2,-.2) $) -- ($ (2,1) + (-.3,.3) $);
\draw[thick, red] ($ (3,1) + (.2,-.2) $) -- ($ (3,1) + (-.3,.3) $);
\foreach \x in {1,2,3}{
\foreach \y in {1,2,3}{
\fill[red] ($ (\x,\y) + (-.2,.2) $) circle (.025cm);
\fill[red] (\x,\y) circle (.025cm);
\fill ($ (\x,\y) + (.2,-.2) $) circle (.025cm);
\filldraw[cyan] (2.2,2.15) circle (.025cm) node[right]{$\scriptstyle \psi$};

That is, since $\Sphere(\cA)\cong \bbC^{\Irr(Z_2(\cA))}$, the $\Sphere(\cA)$-representation at our bulk edge $\ell$ and vertex $v$ must be labelled by the simple object $a$.
We apply $\pi^1_{k}$ at the boundary edge $k$, we extend $a$ along the path $\pi$
placing $\psi$ at the end to get a simple $x\in \Irr(\cX)$ on the boundary,
and finally we apply the 4 bulk plaquette operators $B_oB_pB_qB_r$ which meet the bulk edge $\ell$.

By (<ref>), the bulk-to-boundary hopping operator $H^{(a,\beta_a)}_\pi(\psi)$ takes vectors in the $\Sphere(\cA)$-representation corresponding to $a\in \Irr(Z_2(\cA))$
to vectors in the $\Dome^{\textcolor{red}{\cA}}(\cX)$-representation corresponding to $\Phi^Z(a)$,
thus showing $Z^\cA(\cX)$ is a $Z_2(\cA)$-enriched UBFC.

The above discussion is sub-optimal, as the hopping operator depends on a choice of $\psi$.
This problem occurs because we have pushed $a\in\cA$ into $\cX$ via $\Phi$, and $\Sphere(\cA)$ is not actually a quotient of $\Dome^{\textcolor{red}{\cA}}(\cX)$, but only Morita equivalent to a quotient.
If we instead use a model for the boundary where boundary excitations can be hosted at a single vertex as in Remark <ref> above,

\tikzmath{
\draw[thick] (-.5,0) node[left]{$\scriptstyle w$} -- (.5,0) node[right]{$\scriptstyle z$};
\draw[thick] (0,-.5) node[below]{$\scriptstyle x$} -- (0,.5) node[above]{$\scriptstyle y$};
\filldraw (0,0) circle (.05cm);
\draw[thick, red] (0,0) -- (-.4,.4) node[left]{$\scriptstyle a$};
\draw[thick, blue, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}] (0,0) -- (.4,-.4) node[right]{$\scriptstyle X$};
\node at (-.2,-.2) {$\scriptstyle v$};
\draw[purple!50, very thin] (-.25,.5) -- (.25,-.5);
{\draw[purple!50, very thin, -stealth] ($ (-.15,.3) - (.12,.06)$) to ($ (-.15,.3) + (.12,.06)$);}
{\draw[purple!50, very thin, -stealth] ($ (.15,-.3) - (.12,.06)$) to ($ (.15,-.3) + (.12,.06)$);}
\leftrightarrow
\cH_v:=
\bigoplus_{\substack{
x\in \Irr(\cX)
\\
X\in \Irr(Z^\cA(\cX))
\cX(\Phi(a)wx \to yzF(X))

we can avoid the choice of $\psi$ altogether.
Indeed, $\bbC^{\Irr(Z_2(\cA))}$ is an honest quotient of $\bbC^{\Irr(Z^{\cA}(\cX))}$, which is the algebra of local operators (Morita equivalent to $\Dome^{\textcolor{red}{\cA}}(\cX)$) acting at this type of ancilla.
In this setting, we can define the bulk-to-boundary hopping operator by

B_oB_pB_qB_r T_\pi^{(a,\beta_a)}

as in the cartoon below:

\tikzmath[scale=2]{
\draw[thick, red, xshift=-.4cm, yshift=.4cm] (.7,.7) grid (3.3,2.3);
\filldraw[thick, orange, rounded corners=5pt, fill=gray!30] (1.65,1.7) -- (1.65,2.3) -- (1.75,2.1) -- (1.75,1.3) -- (1.65,1.5) -- (1.65,1.7);
\begin{scope}[xshift=-.2cm, yshift=.2cm]
\draw[thick, knot, thick, red] (.7,1) -- (3.3,1);
\draw[thick, knot, thick, red] (.7,2) -- (3.3,2);
\draw[thick, knot, thick, red] (1,.7) -- (1,2.3);
\draw[thick, knot, thick, red] (3,.7) -- (3,2.3);
\draw[thick, knot, thick, red] (2,.7) -- (2,1);
\draw[thick, knot, thick, red] (2,2.3) -- (2,2);
\end{scope}
\foreach \x in {1,2,3}{
\foreach \y in {1,2}{
\fill[red] ($ (\x,\y) + (-.4,.4) $) circle (.025cm);
\filldraw[thick, orange, knot, rounded corners=5pt, fill=gray!30] (.9,1.3) rectangle (1.7,2.1);
\filldraw[thick, orange, knot, rounded corners=5pt, fill=gray!30] (1.9,1.3) rectangle (2.7,2.1);
\filldraw[thick, orange, rounded corners=5pt, xshift=.2cm, yshift=-.2cm, fill=gray!30] (1.65,1.7) -- (1.65,2.3) -- (1.75,2.1) -- (1.75,1.3) -- (1.65,1.5) -- (1.65,1.7);
\draw[thick, cyan, knot, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}] (1.8,1.2) --node[right, xshift=-.07cm]{$\scriptstyle a$} (1.8,2.1) -- (2,2.1) -- (2.5,1.6);
\draw[thick, knot, red] (.7,.7) grid (3.3,2.3);
\draw[thick, knot, xshift=.2cm, yshift=-.2cm] (.7,.7) grid (3.3,2.3);
\foreach \x in {1,2,3}{
\foreach \y in {2}{
\draw[thick, red] ($ (\x,\y) + (.2,-.2) $) -- ($ (\x,\y) + (-.6,.6) $);
\draw[thick, red] ($ (1,1) + (.2,-.2) $) -- ($ (1,1) + (-.6,.6) $);
\draw[thick, red] ($ (2,1) + (.2,-.2) $) -- ($ (2,1) + (-.3,.3) $);
\draw[thick, red] ($ (3,1) + (.2,-.2) $) -- ($ (3,1) + (-.3,.3) $);
\foreach \x in {1,2,3}{
\foreach \y in {1}{
\draw[thick, blue, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}] ($ (\x,\y) + (.2,-.2) $) -- ($ (\x,\y) + (.4,-.4) $);
\draw[thick, blue, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}] ($ (1,2) + (.2,-.2) $) -- ($ (1,2) + (.4,-.4) $);
\draw[thick, blue, decorate, decoration={snake, segment length=1mm, amplitude=.2mm}] ($ (3,2) + (.2,-.2) $) -- ($ (3,2) + (.4,-.4) $);
\foreach \x in {1,2,3}{
\foreach \y in {1,2}{
\fill[red] ($ (\x,\y) + (-.2,.2) $) circle (.025cm);
\fill[red] (\x,\y) circle (.025cm);
\fill ($ (\x,\y) + (.2,-.2) $) circle (.025cm);

Now using the isomorphism $\Sphere(\cA)\cong \bbC^{\Irr(Z_2(\cA))}$ from Remark <ref>, we see that this hopping operator intertwines the $\Sphere(\cA)$-action coming from the bulk with the $\bbC^{\Irr(Z_2(\cA))}\subset \bbC^{\Irr(Z^\cA(\cX))}$-action hosted on the blue edges.

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