Tools and Examples - Part 3#
Enumerating allosteric parameters in receptor dimers#
This notebook illustrates some of the function definitions contained in receptor_tools.ipynb.
UNDER CONSTRUCTION#
The file receptor_tools.ipynb includes the function enumerate_allosteric_parameters(G) which takes a graph G and returns the \(\kappa\) and \(\eta\) values for a dimer model with the topology of the reduced graph power \(G^{(2)}\). The default method uses \(e_1, e_2, \ldots , e_v\) while method='alpha' gives single digit edge labels (\(b,c,\ldots\)).
%%capture
%run receptor_tools.ipynb
Below we will consider a receptor model with the topology of the house graph. Note that G=graph.HouseGraph() constructs a graph G with integer vertices.
G=graphs.HouseGraph()
G.show(figsize=4,graph_border=True)
The method = alpha has replaces the vertices with monomials. The spanning tree is constructed with BFS labeling. KappaEta is a list of lists giving the probability of each state as a symbolic expression which is displayed in LaTeX format. The polynomial ring A has the \(\kappa_i\) and \(\eta_{ij}\) as symbolic variables for \(0 \leq i,j \leq v-1\).
(G, T, KappaEta, A) = enumerate_allosteric_parameters(G,method='alpha',show=True)
| \(1\) | \(2 \kappa_{\mathit{b}}\) | \(2 \kappa_{\mathit{c}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{e}}\) |
| \(0\) | \(\kappa_{\mathit{b}}^{2} \eta_{\mathit{bb}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{c}} \eta_{\mathit{bc}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{c}} \kappa_{\mathit{d}} \eta_{\mathit{bc}} \eta_{\mathit{bd}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{e}} \eta_{\mathit{bc}} \eta_{\mathit{bd}} \eta_{\mathit{be}}\) |
| \(0\) | \(0\) | \(\kappa_{\mathit{c}}^{2} \eta_{\mathit{cc}}\) | \(2 \kappa_{\mathit{c}}^{2} \kappa_{\mathit{d}} \eta_{\mathit{cc}} \eta_{\mathit{cd}}\) | \(2 \kappa_{\mathit{c}}^{2} \kappa_{\mathit{d}} \kappa_{\mathit{e}} \eta_{\mathit{cc}} \eta_{\mathit{cd}} \eta_{\mathit{ce}}\) |
| \(0\) | \(0\) | \(0\) | \(\kappa_{\mathit{c}}^{2} \kappa_{\mathit{d}}^{2} \eta_{\mathit{cc}} \eta_{\mathit{cd}}^{2} \eta_{\mathit{dd}}\) | \(2 \kappa_{\mathit{c}}^{2} \kappa_{\mathit{d}}^{2} \kappa_{\mathit{e}} \eta_{\mathit{cc}} \eta_{\mathit{cd}}^{2} \eta_{\mathit{dd}} \eta_{\mathit{ce}} \eta_{\mathit{de}}\) |
| \(0\) | \(0\) | \(0\) | \(0\) | \(\kappa_{\mathit{c}}^{2} \kappa_{\mathit{d}}^{2} \kappa_{\mathit{e}}^{2} \eta_{\mathit{cc}} \eta_{\mathit{cd}}^{2} \eta_{\mathit{dd}} \eta_{\mathit{ce}}^{2} \eta_{\mathit{de}}^{2} \eta_{\mathit{ee}}\) |
The next example is reminicent of the cubic ternary complex model. Note that graphs.GridGraph([2,2,2]) has tuples as vertices. We use cannonical_label() to make the vertices integers. This is necessary because enumerate_allosteric_parameters() requires a graph G with integer vertices.
G=graphs.GridGraph([2,2,2])
G.show(figsize=6,graph_border=True,vertex_size=2000)
G=G.canonical_label()
G.show(figsize=6,graph_border=True)
(G, T, KappaEta, A) = enumerate_allosteric_parameters(G,method='alpha',show=True)
| \(1\) | \(2 \kappa_{\mathit{b}}\) | \(2 \kappa_{\mathit{c}}\) | \(2 \kappa_{\mathit{d}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{e}}\) | \(2 \kappa_{\mathit{d}} \kappa_{\mathit{f}}\) | \(2 \kappa_{\mathit{d}} \kappa_{\mathit{g}}\) | \(2 \kappa_{\mathit{d}} \kappa_{\mathit{g}} \kappa_{\mathit{h}}\) |
| \(0\) | \(\kappa_{\mathit{b}}^{2} \eta_{\mathit{bb}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{c}} \eta_{\mathit{bc}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{d}} \eta_{\mathit{bd}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{c}} \kappa_{\mathit{e}} \eta_{\mathit{bc}} \eta_{\mathit{be}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{d}} \kappa_{\mathit{f}} \eta_{\mathit{bd}} \eta_{\mathit{bf}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{d}} \kappa_{\mathit{g}} \eta_{\mathit{bd}} \eta_{\mathit{bg}}\) | \(2 \kappa_{\mathit{b}} \kappa_{\mathit{d}} \kappa_{\mathit{g}} \kappa_{\mathit{h}} \eta_{\mathit{bd}} \eta_{\mathit{bg}} \eta_{\mathit{bh}}\) |
| \(0\) | \(0\) | \(\kappa_{\mathit{c}}^{2} \eta_{\mathit{cc}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \eta_{\mathit{cd}}\) | \(2 \kappa_{\mathit{c}}^{2} \kappa_{\mathit{e}} \eta_{\mathit{cc}} \eta_{\mathit{ce}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{f}} \eta_{\mathit{cd}} \eta_{\mathit{cf}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{g}} \eta_{\mathit{cd}} \eta_{\mathit{cg}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{g}} \kappa_{\mathit{h}} \eta_{\mathit{cd}} \eta_{\mathit{cg}} \eta_{\mathit{ch}}\) |
| \(0\) | \(0\) | \(0\) | \(\kappa_{\mathit{d}}^{2} \eta_{\mathit{dd}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{e}} \eta_{\mathit{cd}} \eta_{\mathit{de}}\) | \(2 \kappa_{\mathit{d}}^{2} \kappa_{\mathit{f}} \eta_{\mathit{dd}} \eta_{\mathit{df}}\) | \(2 \kappa_{\mathit{d}}^{2} \kappa_{\mathit{g}} \eta_{\mathit{dd}} \eta_{\mathit{dg}}\) | \(2 \kappa_{\mathit{d}}^{2} \kappa_{\mathit{g}} \kappa_{\mathit{h}} \eta_{\mathit{dd}} \eta_{\mathit{dg}} \eta_{\mathit{dh}}\) |
| \(0\) | \(0\) | \(0\) | \(0\) | \(\kappa_{\mathit{c}}^{2} \kappa_{\mathit{e}}^{2} \eta_{\mathit{cc}} \eta_{\mathit{ce}}^{2} \eta_{\mathit{ee}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{e}} \kappa_{\mathit{f}} \eta_{\mathit{cd}} \eta_{\mathit{de}} \eta_{\mathit{cf}} \eta_{\mathit{ef}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{e}} \kappa_{\mathit{g}} \eta_{\mathit{cd}} \eta_{\mathit{de}} \eta_{\mathit{cg}} \eta_{\mathit{eg}}\) | \(2 \kappa_{\mathit{c}} \kappa_{\mathit{d}} \kappa_{\mathit{e}} \kappa_{\mathit{g}} \kappa_{\mathit{h}} \eta_{\mathit{cd}} \eta_{\mathit{de}} \eta_{\mathit{cg}} \eta_{\mathit{eg}} \eta_{\mathit{ch}} \eta_{\mathit{eh}}\) |
| \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(\kappa_{\mathit{d}}^{2} \kappa_{\mathit{f}}^{2} \eta_{\mathit{dd}} \eta_{\mathit{df}}^{2} \eta_{\mathit{ff}}\) | \(2 \kappa_{\mathit{d}}^{2} \kappa_{\mathit{f}} \kappa_{\mathit{g}} \eta_{\mathit{dd}} \eta_{\mathit{df}} \eta_{\mathit{dg}} \eta_{\mathit{fg}}\) | \(2 \kappa_{\mathit{d}}^{2} \kappa_{\mathit{f}} \kappa_{\mathit{g}} \kappa_{\mathit{h}} \eta_{\mathit{dd}} \eta_{\mathit{df}} \eta_{\mathit{dg}} \eta_{\mathit{fg}} \eta_{\mathit{dh}} \eta_{\mathit{fh}}\) |
| \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(\kappa_{\mathit{d}}^{2} \kappa_{\mathit{g}}^{2} \eta_{\mathit{dd}} \eta_{\mathit{dg}}^{2} \eta_{\mathit{gg}}\) | \(2 \kappa_{\mathit{d}}^{2} \kappa_{\mathit{g}}^{2} \kappa_{\mathit{h}} \eta_{\mathit{dd}} \eta_{\mathit{dg}}^{2} \eta_{\mathit{gg}} \eta_{\mathit{dh}} \eta_{\mathit{gh}}\) |
| \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(0\) | \(\kappa_{\mathit{d}}^{2} \kappa_{\mathit{g}}^{2} \kappa_{\mathit{h}}^{2} \eta_{\mathit{dd}} \eta_{\mathit{dg}}^{2} \eta_{\mathit{gg}} \eta_{\mathit{dh}}^{2} \eta_{\mathit{gh}}^{2} \eta_{\mathit{hh}}\) |
An example of a receptor model with more states. Using show=False suppresses the output. One can still work with the ring A, show the BFS spanning tree, and list the probability of each state.
(G, T, KappaEta, A) = enumerate_allosteric_parameters(graphs.PetersenGraph(),method='integer',show=False)
A.inject_variables()
Defining kappa_1, kappa_2, kappa_3, kappa_4, kappa_5, kappa_6, kappa_7, kappa_8, kappa_9, eta_11, eta_12, eta_22, eta_13, eta_23, eta_33, eta_14, eta_24, eta_34, eta_44, eta_15, eta_25, eta_35, eta_45, eta_55, eta_16, eta_26, eta_36, eta_46, eta_56, eta_66, eta_17, eta_27, eta_37, eta_47, eta_57, eta_67, eta_77, eta_18, eta_28, eta_38, eta_48, eta_58, eta_68, eta_78, eta_88, eta_19, eta_29, eta_39, eta_49, eta_59, eta_69, eta_79, eta_89, eta_99
T.show(figsize=6,graph_border=True,edge_labels=True)
from IPython.display import display, Math
for i,k in enumerate(flatten(KappaEta)):
if k != 0:
display(Math(latex(i)+':'+latex(k)))
\[\displaystyle 0 : 1\]
\[\displaystyle 1 : 2 \kappa_{\mathit{1}}\]
\[\displaystyle 2 : 2 \kappa_{\mathit{4}}\]
\[\displaystyle 3 : 2 \kappa_{\mathit{5}}\]
\[\displaystyle 4 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}}\]
\[\displaystyle 5 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{6}}\]
\[\displaystyle 6 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{3}}\]
\[\displaystyle 7 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{6}} \kappa_{\mathit{9}}\]
\[\displaystyle 8 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{6}} \kappa_{\mathit{7}} \kappa_{\mathit{9}}\]
\[\displaystyle 9 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{8}}\]
\[\displaystyle 11 : \kappa_{\mathit{1}}^{2} \eta_{\mathit{11}}\]
\[\displaystyle 12 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{4}} \eta_{\mathit{14}}\]
\[\displaystyle 13 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{5}} \eta_{\mathit{15}}\]
\[\displaystyle 14 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \eta_{\mathit{11}} \eta_{\mathit{12}}\]
\[\displaystyle 15 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}} \eta_{\mathit{11}} \eta_{\mathit{16}}\]
\[\displaystyle 16 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}}\]
\[\displaystyle 17 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{16}} \eta_{\mathit{19}}\]
\[\displaystyle 18 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}} \kappa_{\mathit{7}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{16}} \eta_{\mathit{17}} \eta_{\mathit{19}}\]
\[\displaystyle 19 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{8}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}} \eta_{\mathit{18}}\]
\[\displaystyle 22 : \kappa_{\mathit{4}}^{2} \eta_{\mathit{44}}\]
\[\displaystyle 23 : 2 \kappa_{\mathit{4}} \kappa_{\mathit{5}} \eta_{\mathit{45}}\]
\[\displaystyle 24 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{4}} \eta_{\mathit{14}} \eta_{\mathit{24}}\]
\[\displaystyle 25 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{4}} \kappa_{\mathit{6}} \eta_{\mathit{14}} \eta_{\mathit{46}}\]
\[\displaystyle 26 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{4}} \eta_{\mathit{14}} \eta_{\mathit{24}} \eta_{\mathit{34}}\]
\[\displaystyle 27 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{4}} \kappa_{\mathit{6}} \kappa_{\mathit{9}} \eta_{\mathit{14}} \eta_{\mathit{46}} \eta_{\mathit{49}}\]
\[\displaystyle 28 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{4}} \kappa_{\mathit{6}} \kappa_{\mathit{7}} \kappa_{\mathit{9}} \eta_{\mathit{14}} \eta_{\mathit{46}} \eta_{\mathit{47}} \eta_{\mathit{49}}\]
\[\displaystyle 29 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{4}} \kappa_{\mathit{8}} \eta_{\mathit{14}} \eta_{\mathit{24}} \eta_{\mathit{34}} \eta_{\mathit{48}}\]
\[\displaystyle 33 : \kappa_{\mathit{5}}^{2} \eta_{\mathit{55}}\]
\[\displaystyle 34 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{5}} \eta_{\mathit{15}} \eta_{\mathit{25}}\]
\[\displaystyle 35 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{5}} \kappa_{\mathit{6}} \eta_{\mathit{15}} \eta_{\mathit{56}}\]
\[\displaystyle 36 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{5}} \eta_{\mathit{15}} \eta_{\mathit{25}} \eta_{\mathit{35}}\]
\[\displaystyle 37 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{5}} \kappa_{\mathit{6}} \kappa_{\mathit{9}} \eta_{\mathit{15}} \eta_{\mathit{56}} \eta_{\mathit{59}}\]
\[\displaystyle 38 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{5}} \kappa_{\mathit{6}} \kappa_{\mathit{7}} \kappa_{\mathit{9}} \eta_{\mathit{15}} \eta_{\mathit{56}} \eta_{\mathit{57}} \eta_{\mathit{59}}\]
\[\displaystyle 39 : 2 \kappa_{\mathit{1}} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{5}} \kappa_{\mathit{8}} \eta_{\mathit{15}} \eta_{\mathit{25}} \eta_{\mathit{35}} \eta_{\mathit{58}}\]
\[\displaystyle 44 : \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}}^{2} \eta_{\mathit{11}} \eta_{\mathit{12}}^{2} \eta_{\mathit{22}}\]
\[\displaystyle 45 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{6}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{16}} \eta_{\mathit{26}}\]
\[\displaystyle 46 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}}^{2} \kappa_{\mathit{3}} \eta_{\mathit{11}} \eta_{\mathit{12}}^{2} \eta_{\mathit{22}} \eta_{\mathit{13}} \eta_{\mathit{23}}\]
\[\displaystyle 47 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{6}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{19}} \eta_{\mathit{29}}\]
\[\displaystyle 48 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{6}} \kappa_{\mathit{7}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{17}} \eta_{\mathit{27}} \eta_{\mathit{19}} \eta_{\mathit{29}}\]
\[\displaystyle 49 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}}^{2} \kappa_{\mathit{3}} \kappa_{\mathit{8}} \eta_{\mathit{11}} \eta_{\mathit{12}}^{2} \eta_{\mathit{22}} \eta_{\mathit{13}} \eta_{\mathit{23}} \eta_{\mathit{18}} \eta_{\mathit{28}}\]
\[\displaystyle 55 : \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}}^{2} \eta_{\mathit{11}} \eta_{\mathit{16}}^{2} \eta_{\mathit{66}}\]
\[\displaystyle 56 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{6}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{36}}\]
\[\displaystyle 57 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}}^{2} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{16}}^{2} \eta_{\mathit{66}} \eta_{\mathit{19}} \eta_{\mathit{69}}\]
\[\displaystyle 58 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}}^{2} \kappa_{\mathit{7}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{16}}^{2} \eta_{\mathit{66}} \eta_{\mathit{17}} \eta_{\mathit{67}} \eta_{\mathit{19}} \eta_{\mathit{69}}\]
\[\displaystyle 59 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{6}} \kappa_{\mathit{8}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{36}} \eta_{\mathit{18}} \eta_{\mathit{68}}\]
\[\displaystyle 66 : \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}}^{2} \kappa_{\mathit{3}}^{2} \eta_{\mathit{11}} \eta_{\mathit{12}}^{2} \eta_{\mathit{22}} \eta_{\mathit{13}}^{2} \eta_{\mathit{23}}^{2} \eta_{\mathit{33}}\]
\[\displaystyle 67 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{6}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{36}} \eta_{\mathit{19}} \eta_{\mathit{29}} \eta_{\mathit{39}}\]
\[\displaystyle 68 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{6}} \kappa_{\mathit{7}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{36}} \eta_{\mathit{17}} \eta_{\mathit{27}} \eta_{\mathit{37}} \eta_{\mathit{19}} \eta_{\mathit{29}} \eta_{\mathit{39}}\]
\[\displaystyle 69 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}}^{2} \kappa_{\mathit{3}}^{2} \kappa_{\mathit{8}} \eta_{\mathit{11}} \eta_{\mathit{12}}^{2} \eta_{\mathit{22}} \eta_{\mathit{13}}^{2} \eta_{\mathit{23}}^{2} \eta_{\mathit{33}} \eta_{\mathit{18}} \eta_{\mathit{28}} \eta_{\mathit{38}}\]
\[\displaystyle 77 : \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}}^{2} \kappa_{\mathit{9}}^{2} \eta_{\mathit{11}} \eta_{\mathit{16}}^{2} \eta_{\mathit{66}} \eta_{\mathit{19}}^{2} \eta_{\mathit{69}}^{2} \eta_{\mathit{99}}\]
\[\displaystyle 78 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}}^{2} \kappa_{\mathit{7}} \kappa_{\mathit{9}}^{2} \eta_{\mathit{11}} \eta_{\mathit{16}}^{2} \eta_{\mathit{66}} \eta_{\mathit{17}} \eta_{\mathit{67}} \eta_{\mathit{19}}^{2} \eta_{\mathit{69}}^{2} \eta_{\mathit{79}} \eta_{\mathit{99}}\]
\[\displaystyle 79 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{6}} \kappa_{\mathit{8}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{36}} \eta_{\mathit{18}} \eta_{\mathit{68}} \eta_{\mathit{19}} \eta_{\mathit{29}} \eta_{\mathit{39}} \eta_{\mathit{89}}\]
\[\displaystyle 88 : \kappa_{\mathit{1}}^{2} \kappa_{\mathit{6}}^{2} \kappa_{\mathit{7}}^{2} \kappa_{\mathit{9}}^{2} \eta_{\mathit{11}} \eta_{\mathit{16}}^{2} \eta_{\mathit{66}} \eta_{\mathit{17}}^{2} \eta_{\mathit{67}}^{2} \eta_{\mathit{77}} \eta_{\mathit{19}}^{2} \eta_{\mathit{69}}^{2} \eta_{\mathit{79}}^{2} \eta_{\mathit{99}}\]
\[\displaystyle 89 : 2 \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}} \kappa_{\mathit{3}} \kappa_{\mathit{6}} \kappa_{\mathit{7}} \kappa_{\mathit{8}} \kappa_{\mathit{9}} \eta_{\mathit{11}} \eta_{\mathit{12}} \eta_{\mathit{13}} \eta_{\mathit{16}} \eta_{\mathit{26}} \eta_{\mathit{36}} \eta_{\mathit{17}} \eta_{\mathit{27}} \eta_{\mathit{37}} \eta_{\mathit{18}} \eta_{\mathit{68}} \eta_{\mathit{78}} \eta_{\mathit{19}} \eta_{\mathit{29}} \eta_{\mathit{39}} \eta_{\mathit{89}}\]
\[\displaystyle 99 : \kappa_{\mathit{1}}^{2} \kappa_{\mathit{2}}^{2} \kappa_{\mathit{3}}^{2} \kappa_{\mathit{8}}^{2} \eta_{\mathit{11}} \eta_{\mathit{12}}^{2} \eta_{\mathit{22}} \eta_{\mathit{13}}^{2} \eta_{\mathit{23}}^{2} \eta_{\mathit{33}} \eta_{\mathit{18}}^{2} \eta_{\mathit{28}}^{2} \eta_{\mathit{38}}^{2} \eta_{\mathit{88}}\]