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# Receptor modeling - scraps 

```{code-cell}
var('a12, a21, a13, a31, a23, a32, a34, a43')
d = {1: {2:a12, 3:a13}, 2: {1:a21, 3:a23}, 3: {2:a32, 1:a31, 4:a34}, 4: {3:a43}};
G = DiGraph(d,weighted=True)
G.plot(figsize=8,edge_labels=True,pos=vertex_positions,graph_border=True)
```

The weighted adjacency matrix for `G` is 

```{code-cell}
A = G.weighted_adjacency_matrix()
A
```

The Laplacian of `G` is:
```{code-cell}
L = G.laplacian_matrix()
L
```
This matrix is sometimes referred to as the `combinatorial Laplacian matrix` of the weighted directed graph `G`.


## Generator matrix and Laplacian 

The generator matrix `Q` for the Markov chain associated to `G` can be constructed from the weighted adjacency matrix `A` as folows.
```{code-cell}
Q = A - diagonal_matrix(sum(A.T))
```

The following code defines `e` to be column vector of ones.  This is used to show that each row of `Q` sums to zero.
```{code-cell}
e = matrix([1,1,1,1]).T
```

```{code-cell}
print(Q*e)
```

Multiplying on the left by the transpose of `e`, given by `e.T`, we see that each column of `Q` does not sum to zero.

```{code-cell}
print(e.T*Q)
```




## Three state model


As a simple example, consider a receptor model with three states arranged as follows.

```{code-cell}
G = DiGraph({0: {1:'a01'}, 1: {0:'a10', 2:'a12'}, 2: {1:'a21'}})
pos = {0: (0, 0), 1: (1, 0), 2: (2, 0)}
G.plot(figsize=4,edge_labels=True,pos=pos,graph_border=True)
```

```{code-cell}
G = DiGraph({'R': {'RL':'kap*L'}, 'RL': {'R':'kam', 'RLL':'kbp*L'}, 'RLL': {'RL':'kbm'}})
pos = {'R': (0, 0), 'RL': (1, 0), 'RLL': (2, 0)}
G.plot(figsize=4,edge_labels=True,pos=pos,graph_border=True,vertex_size=1000)
```


When both forward and reverse transitions are explicit, the state-transition diagram has the topology of a symmetric directed version of the [path graph](example_graphs:path_graph) with 3 vertices.

Here is the (undirected) path graph {math}`P_3`:

```{code-cell}
G = Graph({0: [1], 1: [2]})
G.plot(figsize=4)
```

The symmetric directed version is

```{code-cell}
:tags: ["hide-input"]
G = DiGraph({0: [1], 1: [0,2], 2: [1]})
G.plot(figsize=4)
```