E XPLORING EPITHELIAL - CELL CALCIUM SIGNALING WITH GEOMETRIC AND TOPOLOGICAL DATA ANALYSIS
A BSTRACT
Spatio-temporal calcium imaging is widely used to study neuronal activity.
However the function of calcium signaling in epithelial cells is not well understood
We thus explore the underlying ﬂuorescence patterns using a combination of data geometry and topology.
We model the epithelial tissue as a graph and use a variant of geometric scattering transform (a multi-level wavelet transform on a graph) to describe the signaling at each time point.
We then embed these discriptors using the manifold learning method PHATE in order to capture the entire time-trajectory of calcium signaling.
Finally, we use persistent homology computed on the PHATE embedding to quantitatively characterize the trajectory.
We demonstrate that scattering coefﬁcients can effectively learn time point embeddings while PHATE can represent relationships between time points.
Persistent homology provides a way to quantify differences between signaling dynamics as evidenced by the differences in dynamics between wild type cells and cells that are stalled in a particular cell cycle phase
I NTRODUCTION
Calcium ions play a signiﬁcant role in transferring intercelluar signals and thus coordinating cellular behavior (Celli et al., 2011; Elias et al., 2002; Murata et al., 2018).
Traditionally, calcium signaling has been studied via imaging techniques, particularly in neurons, where calcium signals act as charge carriers and intracellular messengers.
However, calcium signaling is less well-studied in other tissues.
Here we study patterns of time-varying calcium signaling in the basal stem cell layer of the skin epidermis where the function of the calcium signaling is not well known (Murata et al., 2018).
Therefore, we pursue an unsupervised signaling pattern analysis here to try to derive hypotheses regarding signaling dynamics.
In particular, we study the connection between cell cycle and calcium signaling by comparing the dynamics of wildtype cells with cells that are cycle-arrested.
We model the epithelial tissue as a nearest neighbors cellular graph G = { V, E } with each vertex v i ∈
V being a cell, and ( v j , v k ) ∈
E if the cells v i and v j are spatially adjacent.
Since epithelial cells are arranged in planar spatial patterns, a graph where adjacent cells are those that touch at the cellular surfaces captures the tissue architecture.
Then the calcium activity level of the cell is regarded as a signal x on the graph, i.e., each vertex v i in the graph as activation value x ( v i ) representing the level of calcium activity.
With this representation in place, we characterize the dynamics of calcium signaling ﬁrst by deriving a descriptor of the spatial via geometric scattering transform (an extension of the original scattering transform on graphs) applied to the signal
A (Gao et al., 2019).
After capturing the spatial pattern at each time point, we then represent the temporal variation of the spatial pattern using a non-linear dimensionality reduction method called PHATE (Moon et al., 2019).
We choose PHATE because unlike many other dimensionality reduction methods, PHATE can preserve global multidimensional trajectory patterns in low dimensions without shattering their structure as tSNE/UMAP tend to do.
The new PHATE coordinates can be used as a metric space in which to compute the more typical persistent homology based on Vietoris–Rips complexes (Rieck et al., 2020).
This type of simplicial complex construction is based on increasing an (cid:15) threshold for connecting points based on distance, which in our case is the distance in the PHATE coordinate space.
We show that this method of analysis is useful for capturing the calcium signaling dynamics across a tissue, and allows us to visualize and quantitatively detect differences between complex signaling dynamics in the cellular state space.
M ETHOD 2. G EOMETRIC SCATTERING FOR TIME POINT EMBEDDING
A graph G = ( V, E ) consisting of n vertices V
= { v 1 , . .
. , v n } , and edges E ⊆ { ( v j , v k ) : 1 ≤ j, k ≤ n } can be constructed for epidermal cells based on their spatial locations.
We deﬁne the adjacency matrix of graph G as A and the corresponding degree matrix as D .
In this study we only considered unweighted graphs, which means A ( v j , v k )
=
if ( v j , v k ) ∈
E and 0 otherwise.
D is given by D ( v j , v j ) =
(cid:80)
[Equation] [Equation]
) , D
[Equation] [Equation]
) = 0 ,
[Equation]
(cid:54) =
j .
The raw signals x on the vertices are the ﬂuorescence intensity for each cell at each time point.
However, the relative signal intensity with respect to background signal (deﬁned as the minimum ﬂuorescence intensity for each cell during the 30 minutes experiment) better reﬂects the change of cell activities.
Therefore, the raw signals were transformed as: x = ( x − x min ) / x min and normalized.
Our graph wavelets are designed based on lazy random walk P = 12 (cid:0)
I + AD − 1 (cid:1) over graph:
[Equation]
The random walk P can be regarded as a low pass ﬁlter by retaining low frequency signals and suppressing high frequencies.
We thus recover high frequencies information through performing multi-scale wavelet transform.
The graph wavelet transform of signal x is constructed as Wx = { P 2 J x , Ψ j x : 1 ≤ j ≤ J } .
The spatial locations of cells remained unchanged during the experiments, but the signals kept changing along the time.
We thus embedded each time point by calculating scattering coefﬁcients for all the nodes at that time point.
The zero order scattering coefﬁcient were calculate as
[Equation]
As in Gao et al.
(2019), the zero order scattering features can be augmented by ﬁrst order scattering features via applying graph wavelets and extracting ﬁner description of high frequency response of signal x. Speciﬁcally, the ﬁrst order scattering coefﬁcients for each time point were calculated as
[Equation]
and the second order scattering coefﬁcients can be further obtained by
[Equation]
Finally, the zero order, ﬁrst order and second order scattering coefﬁcients were combined together as the embeddings for each time point.
The value of J were selected based on the diameter of graphs.
2. P ERSISTENT HOMOLOGY ANALYSIS OF PHATE TRAJECTORIES
The time point embeddings from geometric scattering form a matrix of t × m , where t is the number of time points in the data and m is the number of scattering coefﬁcients for each time point.
We can visualize these embeddings by applying PHATE.
We ﬁrst calculated a distance matrix D based on Euclidean distance and applied an α -decaying kernel K to generate an afﬁnity matrix as well as the diffusion operator P .
The optimal value t for diffusion is automatically chosen to be the knee point of the von Neumann entropy of P .
PHATE enables us to capture the variance from time point embedding while still preserving the trajectory and global relations.
The PHATE trajectory can thus reﬂect the time-varying patterns from calcium ﬂuorescence data.
To have a quantitatively description of the generated PHATE trajectories, we calculated their persistent homology.
The time point in the new PHATE coordinates form a point cloud X
and we calculated the persistent homology based on the Vietoris-Rips ﬁltration.
Deﬁne the Vietoris-Rips complex of X as the ﬁltered complex that contains a subset of X as a simplex if all pairwise distances in the subset are less than or equal to s , explicitly
[Equation]
[Equation]
The Vietoris-Rips persistence of X is the persistent simplicial homology of V R s ( X ) .
Finally we recorded the birth and death of topological features in the persistent diagram.
In the diagram, each point indicates a topological feature in the trajectory which appears at a certain birth time and disappears at a death time.
A point’s distance from the diagonal therefore represents the persistence of the associated topological feature.
A PPLICATIONS AND R ESULTS 3. S YNTHETIC DATASET
To validate the usefulness of our method, we ﬁrst tested it on a synthetic dataset we created.
We took a graph created from one of our calcium signalling samples with 1867 vertices (cells), and (1) we used a normalized graph Laplacian to diffuse a Dirac signal x centered on node i ( x i = 1 ) over the graph for 300 steps (illustrated in Appendix Fig. 7), which resulted in 300 signals, each more diffused than the previous.
Then we added normalized random noise (cid:15) ∼ N ( µ, σ 2 ) with µ =and σ = 0 .
001 to signals generated in (1) .
These perturbed signals are thus similar but not exactly the same to original ones.
For the Dirac signal x centered on node i , we further set x j = 1 so that during diffusion initially it was similar to signals in (1) but would eventually diffuse to different patterns.
We also diffused this signal for 300 steps.
Finally we combined signals from (2) and signals from (1) to form a new 600-step signal, and signals from (3) were also combined with signals from (1) to form another series of signals of 600 steps.
We calculated scattering coefﬁcients for each time step and used PHATE to visualize them.
As shown in Fig.
2a, time points with perturbed signals overlapped with time points with original signals, showing scattering transform and PHATE are invariant to small degrees of noise; while in
Fig. 2b, time points from two signal sources formed two branches with their starting points near each other in PHATE coordinates.
Thus from one end to the next this is akin to a signal condensing and then diffusing again.
As expected, this creates a loop-like structure in the PHATE graph.
3. C ALCIUM SIGNALING EXPERIMENT
We applied our method to data from optimized two-photon time-lapse imaging of live mice with a Ca 2+ -sensor in all epidermal cells expressing Keratin 14 (K14-Cre; Rosa-CAG-LSL-GCaMP6s)
(Chen et al., 2013; Mesa et al., 2018).
To follow the cytosolic calcium dynamics in all basal epithelial cells, we took images every two seconds for a duration of 30 minutes across multiple adult mice.
The spatial distribution with time-varying signals of cells form spatial-temporal patterns that may represent unique cell activities and interactions.
Spatial analysis of the signaling clusters during homeostasis revealed non radial patterns when initiating cells propagated signal to their neighbors (as shown in Appendix Fig.4).
These results seem to suggest that neighboring cells might have different competencies to participate in Ca 2+ signaling.
Given basal cells asynchronously transition through the cell cycle phases, we hypothesized that clusters of cells may have different abilities to propagate calcium depending on what phase of the cell cycle individual cells may be at.
To test this hypothesis, we genetically stall cells at G 1 stage to see if calcium signaling patterns are changed.
Under this system we wish to mathematically characterize the spatial-temporal patterns before and after interventions.
We tested our proposed method on this calcium signalling dataset which contains both wide type cells and cells that are stalled at G 1 stage.
These two types of cells generate different spatio-temporal ﬂuorescence patterns we tried to retrieve.
We ﬁrst showed that the geometric scattering transform for time point embedding captures the ﬂuorescence variability via the visualization through PHATE, then we compared the differences of
PHATE trajectories utilizing persistent homology in Fig.
3. C OMPARISON OF PHATE TRAJECTORIES ON TWO TYPES OF CELLS
Wild type cells were used as control group to cells that were stalled at G 1 stage which we referred as positive group below.
There were a total of four duplicates from control group and three duplicates from positive group, from which we built four and three graphs respectively.
We generated scattering coefﬁcients for each time point from each graph.
Then we applied PHATE and visualized the embeddings.
Each time point was colored based on the corresponding time point id.
In Fig.
3, we compared PHATE trajectories for time points from control group and positive group.
The time point embeddings from control group (Fig.
3e, 3f, 3g, and 3h) formed smooth trajectories along time, suggesting the change of signals over graph were generally steady.
This also matched with our observation that there were fewer single cell ﬂashing or small waves of ﬂashes in control group.
In contrast, PHATE for positive group in Fig.
3i, 3j, and 3k showed different and more complex patterns with loops or holes in the trajectories.
This may be caused by rapid change of signals over the graph: more single cell ﬂashes and shorter lingering time may contribute to this.
3. T OPOLOGICAL FEATURE ANALYSIS
We performed topological data analysis using persistent homology to quantatively (rather than qualitatively) compare PHATE trajectories.
The persistence diagram from control group (Fig. 3a, 3b, 3c, and 3d) and positive group (Fig.
3l, 3m, and 3n) were compared.
H 0 features represent connected components in the trajectory, H 1 features represent loops and H 2 represent voids.
There were fewer H 1 features from the persistent diagrams for control group, and many of them were created and died at a later stage.
The persistence of H 1 features were also in general shorter than those in positive group.
On the other hand, in positive groups most H 1 features usually appeared and disappeared at earlier stage.
As a comparison, we also included results from activation ﬁltration in
Appendix. A.6, which showed a similar pattern for positive and control groups but with much more noisy illustration.
These differences from persistence diagram suggest different topological features for PHATE trajectories and hence different calcium ﬂuorescence patterns in these two types of cells.
C ONCLUSION
We present an unsupervised approach for uncovering calcium signalling patterns from epithelial cells.
Our approach combines data geometry, graph signal processing as well as topology to represent signals and their time-varying dynamics.
These representations allowed us to explore the signaling dynamics in the novel context of epithelial cells, where regular dynamics are smoother and undergo larger loops than cell-cycle halted dynamics.
This suggests that cell cycle is related to signaling competency of cells.
Future work will involve associating topological features with signaling features.
A A PPENDIX A. D IFFUSION G EOMETRY
To better capture local afﬁnities for all pairs of points in a dataset, one of the common approaches is to construct a distance matrix using a predeﬁned distance function and apply kernel functions to transform these global distances to local similarities.
As showed in Coifman & Maggioni (2006), a graph can be built from discrete data with afﬁnities
[Equation]
However, the direct embedding of local afﬁnities can result in a loss of global structures.
Therefore a
Markov diffusion operator T deﬁned on the dataset can be used to travel along the intrinsic geometry structure of the data manifold and preserve the overall structure of data.
This diffusion operator T can be deﬁned as
[Equation]
=
[Equation]
Both D and Q are diagonal matrices calculated as D ii := (cid:80)
j ( M (cid:15) ) ij and Q ii :=
(cid:80) j ( K (cid:15) ) ij .
By calculating the t th power of the diffusion operator, we can capture multiscale local neighborhoods of data points up to scale t, where locality is considered via random walks that propagate over the intrinsic manifold geometry of the data.
A. G EOMETRIC SCATTERING
Geometric scattering transform is an unsupervised method for generating embeddings for graphstructured data (Gao et al., 2019).
It is constructed by applying a cascade of graph wavelet transform followed by nonlinear modulus operation such as absolute value nonlinearity (Gao et al., 2019; Gama et al., 2019).
Graph wavelets are designed based on lazy random walks P = 12 (cid:0) I + AD − 1 (cid:1) over graph as in Eq.
The multi-scale nature of graph wavelets allows geometric scattering transform to traverse the entire graph in one layer, which provides both local and global graph features.
Summation of the signal responses is used to to obtain invariant graph-level features.
Finally, the zero-order scattering coefﬁcients are calculated by taking statistical moments of the summation of signals, and the ﬁrst order features are obtained by applying graph wavelet which aggregates multiscale information of the graph.
Second order geometric scattering features can further augment ﬁrst order features by iterating the graph wavelet and absolute value transforms.
The collection of graph scattering features provides a rich set of multiscale invariants of the graph G and can be used under both supervised and unsupervised settings for graph embedding.
A. PHATE
PHATE is a dimensionality-reduction method that captures both local and global nonlinear structure through constructing a diffusion geometry (Moon et al., 2019).
It computes local similarities between data points, and diffuses through the data using a Markovian random-walk diffusion process to infer global relations and represent shape of the data.
Given a n × m data matrix N , where n is the number of data points and m the number of features,
PHATE ﬁrst computes the pairwise distance matrix M using a predeﬁned distance function φ with
M ij = φ ( N i, : , N j, : ) .
As we described in Sec.
A.1, an afﬁnity matrix W is computed to reﬂect local similarities.
Speciﬁcally, PHATE utilizes an α -decaying kernel with a locally-adaptive bandwidth (cid:15) k,i corresponding to the k -NN distance of the i -th data point.
The elements of W are given by:
[Equation]
The decaying factor α regulates the decay rate of the kernel (smaller α ⇒ kernel with lighter tails), α = 2 corresponding to the Gaussian.
The diffusion operator T can then be obtained by calculating the row-sum of the afﬁnity matrix with element T i,j giving the probability of moving from the i -th to the j -th data point in one time step.
The global structure of the data can be further learned through calculating the t th power of the diffusion operator T , which propagates afﬁnity of the data through diffusion up to a scale of t.
Finally, PHATE conducts dimensionality reduction while retaining diffusion geometry information obtained from T by deﬁning a pairwise potential distance as ID i,j = (cid:107) log T i, : − log T j, : (cid:107) 2 between corresponding t -step diffusion probability distributions of the two points.
This preserves the global context for each data point.
The resulting information distance matrix ID is ﬁnally embedded into a low dimensional (2D or 3D) space by metric multidimensional scaling (MDS), and makes it possible to visualize intrinsic geometric information from data.
A. P ERSISTENT HOMOLOGY
Simplicial homology calculates a sequence of homology groups for describing the topological features of a simplicial complex K and measuring its connectivity.
Persistent homology is an extension of simplicial homology that requires a simplicial complex K and an additional function f : K → R .
If f only attains a ﬁnite set of function values f 0 ≤ f 1 ≤ · · · ≤ . . .
f m − 1 ≤ f m , then K can be sorted according to these values, leading to a ﬁltration of K , i.e., a nested sequence of simplicial complexes ∅ =
K 0 ⊆
K 1 ⊆ · · · ⊆ K m − 1 ⊆
K m = K , with K i := { σ ∈ K | f
( σ ) ≤ f i } .
Filtrations model the evolution of K along f and permit tracking whether topological features are created or destroyed .
Persistent homology represents each topological feature as a tuple ( f i , f j ) ∈ R 2 , with i ≤ j and f i , f j ∈ im( f ) .
These tuples are collected in a persistence diagram D d , which contains the birth and death times of the features as a point, thus summarizing all d -dimensional topological features
. A. S NAPSHOTS OF FLASHING WAVES
Snapshots of a large ﬂashing wave was shown in Fig.
The signaling cluster revealed non-radial patterns when initiating cells propagated signal to their neighbors.
A. A CTIVATION FILTRATION
As a comparison to PHATE visualization of scattering coefﬁcients, we also calculated persistent diagram using activation ﬁltration for both control and positive groups.
The D 0 and D 1 features were shown in Fig.
5 and Fig.
6 respectively.
They were arranged in the same order as in Fig.
The birth time of D 0 features from positive groups were in general smaller than those in the control group, especially for samples demonstrated in Fig.
5e and Fig.
5g.
In addition, the death time of Dfeatures from Fig.
6 for positive groups were smaller than control group.
A. D IFFUSION OF A D IRAC SIGNAL ON GRAPH
Here we show an example of how a Dirac signal deﬁned on one of the cells is diffused by a normalized graph Laplacian in Fig.
The signal starts from one cell, then diffuses to its surrounding cells in a few steps.