Eight sites · rules to exact ODE

1 · Rules & state space

Eight slots, not eight chemical atoms: the anchored core has 8 C + 8 P + 8 L = 24 vertices. C/P/L and cp/cl/next/bound are abstract types. Exactly two next-neighbours per C and eight C in total imply 8, 4+4, or 3+5; connectedness is not imposed.

basal_bind

A free slot becomes occupied. NoEdge(P,L) tests that it is free. Per-map rate: k_on. Topology and all 24 vertices are preserved.

Full-size rule View in MolFoundry

cooperative_bind

A free slot next to an occupied slot becomes occupied. Two occupied neighbours give two matches. Per-map rate: k_coop. Topology and all 24 vertices are preserved.

Full-size rule View in MolFoundry

unbind

An occupied slot becomes free by deleting its bound edge. Per-map rate: k_off. Topology and all 24 vertices are preserved.

Full-size rule View in MolFoundry

Direct state-space expansion

Choose an initial graph and the number of rule applications. MolFoundry generates the successors; isomorphic states are merged. Depth counts events, not elapsed time.

Start Through depth 2

8 · initially all free · complete depth-3 expansion

00000000

0/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 0

00000010

1/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 1

00010010

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

00001010

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

00000110

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

00100010

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

00010110

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

00100110

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

00101001

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

00101010

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

00001110

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

Source state	Rule	Target state	Multiplicity/rate
g-16fbe06e041511	basal_bind	g-9d0019b1dc5b75	8
g-9d0019b1dc5b75	basal_bind	g-a611b88c54b79a	2
g-9d0019b1dc5b75	basal_bind	g-8426161b158ca6	2
g-9d0019b1dc5b75	basal_bind	g-4025d743f65523	2
g-9d0019b1dc5b75	basal_bind	g-a826c1634ae893	1
g-9d0019b1dc5b75	cooperative_bind	g-4025d743f65523	2
g-9d0019b1dc5b75	unbind	g-16fbe06e041511	1
g-a611b88c54b79a	basal_bind	g-64ba3311a86d37	2
g-a611b88c54b79a	basal_bind	g-ffa1f022b2f210	2
g-a611b88c54b79a	basal_bind	g-28ab1363db0948	2
g-a611b88c54b79a	cooperative_bind	g-64ba3311a86d37	2
g-a611b88c54b79a	cooperative_bind	g-ffa1f022b2f210	2
g-a611b88c54b79a	unbind	g-9d0019b1dc5b75	2
g-8426161b158ca6	basal_bind	g-699be0bb08ff4b	2
g-8426161b158ca6	basal_bind	g-64ba3311a86d37	2
g-8426161b158ca6	basal_bind	g-61b19ab04802bc	1
g-8426161b158ca6	basal_bind	g-28ab1363db0948	1
g-8426161b158ca6	cooperative_bind	g-64ba3311a86d37	2
g-8426161b158ca6	cooperative_bind	g-61b19ab04802bc	2
g-8426161b158ca6	unbind	g-9d0019b1dc5b75	2
g-4025d743f65523	basal_bind	g-64ba3311a86d37	2
g-4025d743f65523	basal_bind	g-61b19ab04802bc	2
g-4025d743f65523	basal_bind	g-ffa1f022b2f210	2
g-4025d743f65523	cooperative_bind	g-61b19ab04802bc	2
g-4025d743f65523	unbind	g-9d0019b1dc5b75	2
g-a826c1634ae893	basal_bind	g-699be0bb08ff4b	2
g-a826c1634ae893	basal_bind	g-ffa1f022b2f210	4
g-a826c1634ae893	cooperative_bind	g-ffa1f022b2f210	4
g-a826c1634ae893	unbind	g-9d0019b1dc5b75	2

4+4 · initially all free · complete depth-3 expansion

0000 + 0000

0/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 0

1000 + 0000

1/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 1

1100 + 0000

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

1010 + 0000

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

1000 + 1000

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

1101 + 0000

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

1100 + 1000

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

1010 + 1000

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

Source state	Rule	Target state	Multiplicity/rate
g-b2bd4a509f253a	basal_bind	g-6272b48698d6a8	8
g-6272b48698d6a8	basal_bind	g-c48cebe0d56d62	2
g-6272b48698d6a8	basal_bind	g-25b6c2c6c6a287	1
g-6272b48698d6a8	basal_bind	g-536c6c8d8c1d19	4
g-6272b48698d6a8	cooperative_bind	g-c48cebe0d56d62	2
g-6272b48698d6a8	unbind	g-b2bd4a509f253a	1
g-c48cebe0d56d62	basal_bind	g-eaf49b56b74c90	2
g-c48cebe0d56d62	basal_bind	g-00a3546be89459	4
g-c48cebe0d56d62	cooperative_bind	g-eaf49b56b74c90	2
g-c48cebe0d56d62	unbind	g-6272b48698d6a8	2
g-25b6c2c6c6a287	basal_bind	g-0f57c93fe6e834	4
g-25b6c2c6c6a287	basal_bind	g-eaf49b56b74c90	2
g-25b6c2c6c6a287	cooperative_bind	g-eaf49b56b74c90	4
g-25b6c2c6c6a287	unbind	g-6272b48698d6a8	2
g-536c6c8d8c1d19	basal_bind	g-0f57c93fe6e834	2
g-536c6c8d8c1d19	basal_bind	g-00a3546be89459	4
g-536c6c8d8c1d19	cooperative_bind	g-00a3546be89459	4
g-536c6c8d8c1d19	unbind	g-6272b48698d6a8	2

3+5 · initially all free · complete depth-3 expansion

000 + 00000

0/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 0

000 + 00001

1/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 1

100 + 00000

1/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 1

10010 + 000

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

00001 + 100

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

11000 + 000

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

110 + 00000

2/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 2

11100 + 000

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

10110 + 000

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

10010 + 100

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

00001 + 110

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

11000 + 100

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

111 + 00000

3/8 occupied slots. All 24 vertices View graph in MolFoundry

Minimum depth 3

Source state	Rule	Target state	Multiplicity/rate
g-2bf8bd68f6f1b6	basal_bind	g-0ad70f242c2fe3	5
g-2bf8bd68f6f1b6	basal_bind	g-e999a766a51577	3
g-0ad70f242c2fe3	basal_bind	g-42b102184158b1	2
g-0ad70f242c2fe3	basal_bind	g-d4ea8eb2456e19	3
g-0ad70f242c2fe3	basal_bind	g-e92c105e35b886	2
g-0ad70f242c2fe3	cooperative_bind	g-e92c105e35b886	2
g-0ad70f242c2fe3	unbind	g-2bf8bd68f6f1b6	1
g-e999a766a51577	basal_bind	g-d4ea8eb2456e19	5
g-e999a766a51577	basal_bind	g-a4aac820a36a97	2
g-e999a766a51577	cooperative_bind	g-a4aac820a36a97	2
g-e999a766a51577	unbind	g-2bf8bd68f6f1b6	1
g-42b102184158b1	basal_bind	g-0b1cd9a70036f5	1
g-42b102184158b1	basal_bind	g-8d51aa68597f1d	2
g-42b102184158b1	basal_bind	g-b6c557eb2fa069	3
g-42b102184158b1	cooperative_bind	g-0b1cd9a70036f5	2
g-42b102184158b1	cooperative_bind	g-8d51aa68597f1d	2
g-42b102184158b1	unbind	g-0ad70f242c2fe3	2
g-d4ea8eb2456e19	basal_bind	g-6fec61a0b08182	2
g-d4ea8eb2456e19	basal_bind	g-b6c557eb2fa069	2
g-d4ea8eb2456e19	basal_bind	g-d76fb6da4bc659	2
g-d4ea8eb2456e19	cooperative_bind	g-6fec61a0b08182	2
g-d4ea8eb2456e19	cooperative_bind	g-d76fb6da4bc659	2
g-d4ea8eb2456e19	unbind	g-0ad70f242c2fe3	1
g-d4ea8eb2456e19	unbind	g-e999a766a51577	1
g-e92c105e35b886	basal_bind	g-0b1cd9a70036f5	2
g-e92c105e35b886	basal_bind	g-8d51aa68597f1d	1
g-e92c105e35b886	basal_bind	g-d76fb6da4bc659	3
g-e92c105e35b886	cooperative_bind	g-0b1cd9a70036f5	2
g-e92c105e35b886	unbind	g-0ad70f242c2fe3	2
g-a4aac820a36a97	basal_bind	g-6fec61a0b08182	5
g-a4aac820a36a97	basal_bind	g-9a8aa5436a4890	1
g-a4aac820a36a97	cooperative_bind	g-9a8aa5436a4890	2
g-a4aac820a36a97	unbind	g-e999a766a51577	2

2 · Inputs & semantics

The input query P0 counts occupied slots.

Supplied · round 0

P0

Explicit supplied invariant	Consequence
Each C owns exactly one P and one L; every P/L has at most one C owner.	No sharing of one owned P/L between two C.
Each P/L has at most one bound partner; bound endpoints have the same C owner.	A slot has a Boolean occupied/free state.
Exactly eight C; each C has exactly two C neighbours; simple undirected graph.	Cycles of length ≥3; possible component partitions are 8, 4+4, 3+5.
Connectedness is NOT required.	Two rings are allowed in a single state.
The signature and invariant theory are input, not learned.	Inert orphan P/L vertices allowed by “at most” do not affect the anchored observables.

First composition step

For each of the three rules r, compare P0 ∘ r (observable after the event) with r ∘ P0 (observable before it), with the rightmost operation first. MolFoundry excludes overlaps violating the incidence bounds of Ω. The remaining composition list sizes are not ODE coefficients.

Rule	Before: r ∘ P0	After: P0 ∘ r	Signed jump after reduction
basal_bind	1	2	+P1
cooperative_bind	2	3	+P2
unbind	2	1	−P0

Thus dE[P0]/dt = k_on E[P1] + k_coop E[P2] − k_off E[P0]. The new counts P1 and P2 describe free slots and mixed neighbours.

Computed · round 1

P1

Computed · round 1

P2

3 · Observable discovery

1 → 3 → 4 → 6 → 10 → 16 → 26 → 38 → 45 → 47 → 48 → 48

Synthesis closes with 48 coordinates: the query and 47 additional linear combinations of graph counts. A later check finds one linear dependency.

Round 0 · 1 coordinates

Supplied · round 0

P0

Round 1 · 3 coordinates

Computed · round 1

P1

Computed · round 1

P2

Round 2 · 4 coordinates

Computed · round 2

P3

Round 3 · 6 coordinates

Computed · round 3

P4

First 3 of 4 terms shown.

Computed · round 3

P5

Round 4 · 10 coordinates

Computed · round 4

P6

Computed · round 4

P7

Computed · round 4

P8

First 3 of 8 terms shown.

Computed · round 4

P9

Round 5 · 16 coordinates

Computed · round 5

P10

Computed · round 5

P11

First 3 of 5 terms shown.

Computed · round 5

P12

First 3 of 11 terms shown.

Computed · round 5

P13

First 3 of 11 terms shown.

Computed · round 5

P14

First 3 of 4 terms shown.

Computed · round 5

P15

Round 6 · 26 coordinates

Computed · round 6

P16

Computed · round 6

P17

Computed · round 6

P18

Computed · round 6

P19

First 3 of 12 terms shown.

Computed · round 6

P20

First 3 of 12 terms shown.

Computed · round 6

P21

First 3 of 23 terms shown.

Computed · round 6

P22

Computed · round 6

P23

First 3 of 19 terms shown.

Computed · round 6

P24

Computed · round 6

P25

Round 7 · 38 coordinates

Computed · round 7

P26

Computed · round 7

P27

Computed · round 7

P28

First 3 of 24 terms shown.

Computed · round 7

P29

Computed · round 7

P30

First 3 of 33 terms shown.

Computed · round 7

P31

Computed · round 7

P32

First 3 of 30 terms shown.

Computed · round 7

P33

Computed · round 7

P34

First 3 of 32 terms shown.

Computed · round 7

P35

Computed · round 7

P36

First 3 of 25 terms shown.

Computed · round 7

P37

Round 8 · 45 coordinates

Computed · round 8

P38

First 3 of 22 terms shown.

Computed · round 8

P39

Computed · round 8

P40

Computed · round 8

P41

First 3 of 38 terms shown.

Computed · round 8

P42

First 3 of 38 terms shown.

Computed · round 8

P43

Computed · round 8

P44

Round 9 · 47 coordinates

Computed · round 9

P45

Computed · round 9

P46

Round 10 · 48 coordinates

Computed · round 10

P47

Round 11 · 48 coordinates

Closure reached.

4 · ODE & pJC

Let yᵢ(t) = E[Pᵢ(Xₜ)]. The exported equation is y′ = A(k)y, with 48 coordinates and one fixed rate-separated matrix for all three topologies. Each coordinate is an exact linear combination of raw injective counts. Signed coordinates need not be nonnegative; they are not probabilities.

\frac{d P_{0}}{d t} = - P_{0} k_{off} + P_{1} k_{on} + P_{2} k_{coop}

\frac{d P_{1}}{d t} = P_{0} k_{off} - P_{1} k_{on} - P_{2} k_{coop}

\frac{d P_{2}}{d t} = 2 P_{0} k_{off} + 2 P_{1} k_{on} - 2 P_{2} k_{off} - 2 P_{2} k_{on} - 2 P_{3} k_{coop}

All 48 exact equations

\frac{d P_{0}}{d t} = - P_{0} k_{off} + P_{1} k_{on} + P_{2} k_{coop}

\frac{d P_{1}}{d t} = P_{0} k_{off} - P_{1} k_{on} - P_{2} k_{coop}

\frac{d P_{2}}{d t} = 2 P_{0} k_{off} + 2 P_{1} k_{on} - 2 P_{2} k_{off} - 2 P_{2} k_{on} - 2 P_{3} k_{coop}

\frac{d P_{3}}{d t} = 2 P_{2} k_{coop} + 2 P_{2} k_{on} - 4 P_{3} k_{coop} - 2 P_{3} k_{off} - 3 P_{3} k_{on} - 2 P_{4} k_{coop} + P_{5} k_{off}

\frac{d P_{4}}{d t} = 4 P_{0} k_{off} + 2 P_{1} k_{on} - 2 P_{2} k_{off} - 2 P_{2} k_{on} - 2 P_{3} k_{off} + P_{3} k_{on} - 2 P_{4} k_{coop} - P_{4} k_{off} - 4 P_{4} k_{on} - 2 P_{5} k_{off} - P_{6} k_{on} + 2 P_{7} k_{coop} + P_{8} k_{off}

\frac{d P_{5}}{d t} = 2 P_{0} k_{coop} + 4 P_{0} k_{on} - 2 P_{2} k_{on} + 4 P_{3} k_{coop} + P_{3} k_{on} - P_{4} k_{coop} - P_{5} k_{coop} - 3 P_{5} k_{off} - 2 P_{5} k_{on} - P_{8} k_{coop} - P_{9} k_{coop}

\frac{d P_{6}}{d t} = - 3 P_{6} k_{on} + 3 P_{9} k_{off}

\frac{d P_{7}}{d t} = 6 P_{0} k_{off} + 2 P_{1} k_{on} - P_{10} k_{on} + 2 P_{11} k_{coop} + P_{12} k_{off} - 6 P_{2} k_{off} - 2 P_{2} k_{on} + P_{3} k_{on} + 3 P_{4} k_{off} - P_{4} k_{on} - 3 P_{5} k_{off} - P_{6} k_{on} - 2 P_{7} k_{coop} - P_{7} k_{off} - 5 P_{7} k_{on} + P_{8} k_{off} + 3 P_{9} k_{off}

\frac{d P_{8}}{d t} = - 6 P_{0} k_{coop} - 6 P_{0} k_{on} + P_{12} k_{coop} + P_{13} k_{coop} + 2 P_{14} k_{off} + 6 P_{2} k_{coop} + 6 P_{2} k_{on} - 3 P_{4} k_{coop} + 3 P_{4} k_{on} + 3 P_{5} k_{coop} + 3 P_{5} k_{on} - 2 P_{7} k_{coop} - 5 P_{8} k_{coop} - 2 P_{8} k_{off} - 3 P_{8} k_{on} - 3 P_{9} k_{coop} - 3 P_{9} k_{on}

\frac{d P_{9}}{d t} = 2 P_{15} k_{off} + P_{6} k_{on} - 2 P_{9} k_{coop} - P_{9} k_{off} - 2 P_{9} k_{on}

\frac{d P_{10}}{d t} = - 4 P_{10} k_{on} + 4 P_{16} k_{off}

\frac{d P_{11}}{d t} = 2 P_{1} k_{on} - P_{10} k_{on} - P_{11} k_{off} - 6 P_{11} k_{on} + P_{13} k_{off} - P_{17} k_{on} - 2 P_{18} k_{coop} + P_{19} k_{off} - 2 P_{2} k_{on} + P_{3} k_{on} - P_{4} k_{on} - P_{6} k_{on} - P_{7} k_{on}

\frac{d P_{12}}{d t} = - 12 P_{0} k_{coop} - 12 P_{0} k_{off} - 12 P_{0} k_{on} - 2 P_{11} k_{coop} - 5 P_{12} k_{coop} - 4 P_{12} k_{off} - 4 P_{12} k_{on} + P_{13} k_{coop} + 6 P_{13} k_{off} + 4 P_{14} k_{off} + 12 P_{15} k_{off} - 4 P_{16} k_{coop} - 8 P_{16} k_{off} - 4 P_{16} k_{on} - 4 P_{18} k_{off} + \frac{4 P_{19} k_{coop}}{3} + \frac{4 P_{19} k_{off}}{3} + 12 P_{2} k_{coop} + 12 P_{2} k_{off} + 12 P_{2} k_{on} - \frac{2 P_{20} k_{coop}}{3} + \frac{5 P_{20} k_{off}}{6} + \frac{P_{21} k_{off}}{2} - 12 P_{3} k_{off} - 2 P_{4} k_{coop} + 2 P_{4} k_{off} - 2 P_{4} k_{on} + 6 P_{5} k_{coop} + 6 P_{5} k_{off} + 6 P_{5} k_{on} - 4 P_{7} k_{coop} - 8 P_{7} k_{off} + 4 P_{7} k_{on} - 2 P_{8} k_{coop} + 2 P_{8} k_{off} - 2 P_{8} k_{on} - 6 P_{9} k_{coop} - 6 P_{9} k_{off} - 6 P_{9} k_{on}

\frac{d P_{13}}{d t} = 12 P_{0} k_{off} + P_{11} k_{coop} + \frac{P_{11} k_{off}}{2} + P_{12} k_{off} - 2 P_{13} k_{coop} - 5 P_{13} k_{off} - 4 P_{13} k_{on} + 4 P_{16} k_{off} + 2 P_{18} k_{off} + \frac{P_{19} k_{coop}}{3} - \frac{P_{19} k_{off}}{2} - 12 P_{2} k_{off} - \frac{2 P_{20} k_{coop}}{3} + P_{20} k_{off} - P_{22} k_{coop} + \frac{P_{22} k_{off}}{2} + P_{23} k_{off} + 2 P_{4} k_{off} - 6 P_{5} k_{off} + P_{7} k_{coop} + \frac{7 P_{7} k_{off}}{2} + 2 P_{8} k_{off} + 6 P_{9} k_{off}

\frac{d P_{14}}{d t} = 12 P_{0} k_{coop} + 12 P_{0} k_{on} + \frac{P_{11} k_{coop}}{2} - P_{12} k_{coop} - 3 P_{13} k_{coop} - 4 P_{14} k_{coop} - 3 P_{14} k_{off} - 2 P_{14} k_{on} - 6 P_{15} k_{coop} - 3 P_{15} k_{on} + 2 P_{16} k_{coop} + 2 P_{18} k_{coop} - \frac{P_{19} k_{coop}}{2} - 4 P_{2} k_{coop} - 6 P_{2} k_{on} + P_{20} k_{coop} + \frac{P_{22} k_{coop}}{2} + P_{23} k_{coop} - 2 P_{24} k_{coop} + 3 P_{25} k_{off} - 2 P_{3} k_{coop} - 3 P_{3} k_{on} - 4 P_{4} k_{coop} - 6 P_{5} k_{coop} - 6 P_{5} k_{on} + \frac{3 P_{7} k_{coop}}{2} + 2 P_{8} k_{coop} + 2 P_{8} k_{on} - 2 P_{9} k_{coop}

\frac{d P_{15}}{d t} = - 2 P_{0} k_{off} - P_{14} k_{off} - 2 P_{15} k_{coop} - 4 P_{15} k_{off} - P_{15} k_{on} + 2 P_{2} k_{off} - P_{25} k_{off} + 2 P_{3} k_{off} - P_{4} k_{off} + 2 P_{5} k_{off} - P_{8} k_{off} + 2 P_{9} k_{coop} - P_{9} k_{off} + 2 P_{9} k_{on}

\frac{d P_{16}}{d t} = P_{10} k_{on} - 2 P_{16} k_{coop} - P_{16} k_{off} - 3 P_{16} k_{on} + 2 P_{24} k_{off} + P_{26} k_{off}

\frac{d P_{17}}{d t} = - 5 P_{17} k_{on} + 5 P_{22} k_{off}

\frac{d P_{18}}{d t} = \frac{3 P_{11} k_{off}}{2} + 2 P_{11} k_{on} + 2 P_{13} k_{off} - 2 P_{18} k_{coop} - 4 P_{18} k_{off} - 7 P_{18} k_{on} + \frac{7 P_{19} k_{off}}{6} - \frac{P_{20} k_{off}}{3} + \frac{3 P_{22} k_{off}}{2} + P_{23} k_{off} + 2 P_{27} k_{coop} - 2 P_{27} k_{off} - P_{28} k_{off} - \frac{3 P_{7} k_{off}}{2}

\frac{d P_{19}}{d t} = - 12 P_{0} k_{on} - 4 P_{11} k_{coop} + 6 P_{11} k_{off} + 5 P_{11} k_{on} - P_{12} k_{on} - 4 P_{13} k_{coop} - 17 P_{13} k_{off} - P_{13} k_{on} - 18 P_{14} k_{off} - 54 P_{15} k_{off} + 18 P_{16} k_{off} - 4 P_{16} k_{on} + 6 P_{18} k_{coop} + 17 P_{18} k_{off} - \frac{14 P_{19} k_{coop}}{3} - \frac{17 P_{19} k_{off}}{3} - 5 P_{19} k_{on} + 18 P_{2} k_{off} + 12 P_{2} k_{on} + \frac{4 P_{20} k_{coop}}{3} + \frac{37 P_{20} k_{off}}{12} - \frac{3 P_{21} k_{off}}{4} - 4 P_{22} k_{coop} + 6 P_{22} k_{off} - 5 P_{22} k_{on} - 2 P_{23} k_{coop} + P_{23} k_{off} - 18 P_{24} k_{off} + 4 P_{27} k_{coop} - P_{27} k_{off} + 2 P_{28} k_{coop} + 7 P_{28} k_{off} + 2 P_{29} k_{coop} + 9 P_{29} k_{off} + 36 P_{3} k_{off} - P_{30} k_{off} - 36 P_{4} k_{off} - 2 P_{4} k_{on} + 6 P_{5} k_{on} + 4 P_{7} k_{coop} + 12 P_{7} k_{off} + P_{7} k_{on} - 18 P_{8} k_{off} - 2 P_{8} k_{on} - 18 P_{9} k_{off} - 6 P_{9} k_{on}

\frac{d P_{20}}{d t} = 12 P_{0} k_{off} - 6 P_{0} k_{on} - 2 P_{11} k_{coop} - \frac{3 P_{11} k_{off}}{4} + P_{11} k_{on} - \frac{P_{12} k_{on}}{2} + 4 P_{13} k_{coop} - 25 P_{13} k_{off} + \frac{5 P_{13} k_{on}}{2} - 15 P_{14} k_{off} - 45 P_{15} k_{off} + 18 P_{16} k_{off} - 2 P_{16} k_{on} + 22 P_{18} k_{off} + \frac{5 P_{19} k_{coop}}{3} - \frac{91 P_{19} k_{off}}{12} + 3 P_{2} k_{off} + 6 P_{2} k_{on} - \frac{13 P_{20} k_{coop}}{3} + \frac{11 P_{20} k_{off}}{12} - 5 P_{20} k_{on} - \frac{3 P_{21} k_{off}}{4} - 2 P_{22} k_{coop} - \frac{3 P_{22} k_{off}}{4} - P_{22} k_{on} + 2 P_{23} k_{coop} + \frac{P_{23} k_{off}}{2} - 15 P_{24} k_{off} - 3 P_{26} k_{off} - 4 P_{27} k_{coop} + 7 P_{27} k_{off} - 2 P_{28} k_{coop} + \frac{13 P_{28} k_{off}}{2} - 5 P_{29} k_{coop} + 12 P_{29} k_{off} + 30 P_{3} k_{off} - \frac{P_{30} k_{off}}{2} - 3 P_{31} k_{coop} + 3 P_{31} k_{off} - \frac{3 P_{32} k_{off}}{2} - 27 P_{4} k_{off} - P_{4} k_{on} - 6 P_{5} k_{off} + 3 P_{5} k_{on} + 2 P_{7} k_{coop} + \frac{75 P_{7} k_{off}}{4} - P_{7} k_{on} - 12 P_{8} k_{off} - P_{8} k_{on} - 9 P_{9} k_{off} - 3 P_{9} k_{on}

\frac{d P_{21}}{d t} = 8 P_{0} k_{coop} - 132 P_{0} k_{off} + 26 P_{0} k_{on} + 28 P_{11} k_{coop} - \frac{107 P_{11} k_{off}}{4} + P_{11} k_{on} + 10 P_{12} k_{coop} + \frac{23 P_{12} k_{on}}{2} + 2 P_{13} k_{coop} + 225 P_{13} k_{off} + \frac{21 P_{13} k_{on}}{2} - 16 P_{14} k_{coop} + 153 P_{14} k_{off} - 16 P_{14} k_{on} - 48 P_{15} k_{coop} + 459 P_{15} k_{off} - 48 P_{15} k_{on} + 8 P_{16} k_{coop} - 206 P_{16} k_{off} - 2 P_{16} k_{on} - 20 P_{18} k_{coop} - 192 P_{18} k_{off} - 32 P_{18} k_{on} + \frac{7 P_{19} k_{coop}}{3} + \frac{661 P_{19} k_{off}}{12} + \frac{16 P_{19} k_{on}}{3} + 8 P_{2} k_{coop} - 21 P_{2} k_{off} - 10 P_{2} k_{on} - \frac{14 P_{20} k_{coop}}{3} - \frac{527 P_{20} k_{off}}{12} + \frac{10 P_{20} k_{on}}{3} - 5 P_{21} k_{coop} + \frac{19 P_{21} k_{off}}{4} - 3 P_{21} k_{on} + 28 P_{22} k_{coop} - \frac{107 P_{22} k_{off}}{4} + 15 P_{22} k_{on} + 2 P_{23} k_{coop} + \frac{5 P_{23} k_{off}}{2} - 16 P_{24} k_{coop} + 153 P_{24} k_{off} - 16 P_{24} k_{on} - 8 P_{26} k_{coop} + 33 P_{26} k_{off} - 8 P_{26} k_{on} - 8 P_{27} k_{coop} - 39 P_{27} k_{off} - 10 P_{28} k_{coop} - \frac{131 P_{28} k_{off}}{2} - 17 P_{29} k_{coop} - 66 P_{29} k_{off} + 32 P_{3} k_{coop} - 306 P_{3} k_{off} + 32 P_{3} k_{on} + \frac{7 P_{30} k_{off}}{2} - 3 P_{31} k_{coop} - 9 P_{31} k_{off} + 4 P_{32} k_{coop} + \frac{P_{32} k_{off}}{2} - 8 P_{33} k_{coop} - 20 P_{33} k_{off} + 4 P_{34} k_{off} - 20 P_{4} k_{coop} + 293 P_{4} k_{off} - P_{4} k_{on} - 4 P_{5} k_{coop} + 66 P_{5} k_{off} - 13 P_{5} k_{on} - 20 P_{7} k_{coop} - \frac{717 P_{7} k_{off}}{4} - 17 P_{7} k_{on} - 12 P_{8} k_{coop} + 120 P_{8} k_{off} - 9 P_{8} k_{on} - 12 P_{9} k_{coop} + 87 P_{9} k_{off} - 3 P_{9} k_{on}

\frac{d P_{22}}{d t} = P_{17} k_{on} - 2 P_{22} k_{coop} - P_{22} k_{off} - 4 P_{22} k_{on} + 2 P_{33} k_{off} + 2 P_{35} k_{off}

\frac{d P_{23}}{d t} = \frac{27 P_{0} k_{coop}}{2} - 9 P_{0} k_{off} + 12 P_{0} k_{on} - \frac{25 P_{11} k_{coop}}{8} - \frac{11 P_{11} k_{off}}{4} - P_{11} k_{on} + \frac{5 P_{12} k_{coop}}{4} + \frac{P_{12} k_{off}}{2} + P_{12} k_{on} + \frac{29 P_{13} k_{coop}}{4} + \frac{85 P_{13} k_{off}}{3} - 4 P_{13} k_{on} + 6 P_{14} k_{coop} + 18 P_{14} k_{off} + 18 P_{15} k_{coop} + 62 P_{15} k_{off} - \frac{11 P_{16} k_{coop}}{2} - \frac{79 P_{16} k_{off}}{3} + 4 P_{16} k_{on} - \frac{17 P_{18} k_{coop}}{2} - \frac{143 P_{18} k_{off}}{6} + 8 P_{18} k_{on} + \frac{31 P_{19} k_{coop}}{24} + \frac{241 P_{19} k_{off}}{36} - P_{19} k_{on} - \frac{39 P_{2} k_{coop}}{2} - 9 P_{2} k_{off} - 12 P_{2} k_{on} - \frac{4 P_{20} k_{coop}}{3} - \frac{205 P_{20} k_{off}}{72} + 2 P_{20} k_{on} + \frac{P_{21} k_{coop}}{4} + \frac{31 P_{21} k_{off}}{24} - \frac{25 P_{22} k_{coop}}{8} - \frac{11 P_{22} k_{off}}{4} - P_{22} k_{on} - \frac{17 P_{23} k_{coop}}{4} - \frac{7 P_{23} k_{off}}{3} - 3 P_{23} k_{on} + 6 P_{24} k_{coop} + 18 P_{24} k_{off} + 5 P_{26} k_{off} - \frac{9 P_{27} k_{coop}}{2} - \frac{35 P_{27} k_{off}}{6} - \frac{7 P_{28} k_{coop}}{4} - \frac{17 P_{28} k_{off}}{3} - \frac{5 P_{29} k_{coop}}{2} - \frac{67 P_{29} k_{off}}{6} - 12 P_{3} k_{coop} - 44 P_{3} k_{off} + \frac{P_{30} k_{coop}}{4} - \frac{P_{31} k_{coop}}{2} - \frac{11 P_{31} k_{off}}{3} + \frac{5 P_{32} k_{coop}}{4} + \frac{7 P_{32} k_{off}}{6} - 4 P_{33} k_{coop} - \frac{13 P_{33} k_{off}}{3} + \frac{2 P_{34} k_{off}}{3} - \frac{P_{35} k_{coop}}{2} - P_{35} k_{off} + \frac{P_{36} k_{coop}}{2} + 8 P_{37} k_{off} + \frac{73 P_{4} k_{coop}}{4} + \frac{239 P_{4} k_{off}}{6} + 2 P_{4} k_{on} - \frac{27 P_{5} k_{coop}}{4} + \frac{9 P_{5} k_{off}}{2} - 6 P_{5} k_{on} - \frac{19 P_{7} k_{coop}}{8} - \frac{283 P_{7} k_{off}}{12} + 5 P_{7} k_{on} + \frac{33 P_{8} k_{coop}}{4} + \frac{35 P_{8} k_{off}}{2} + 2 P_{8} k_{on} + \frac{51 P_{9} k_{coop}}{4} + \frac{27 P_{9} k_{off}}{2} + 6 P_{9} k_{on}

\frac{d P_{24}}{d t} = 2 P_{16} k_{coop} + 2 P_{16} k_{on} - 2 P_{24} k_{coop} - 2 P_{24} k_{off} - 2 P_{24} k_{on} + 2 P_{37} k_{off}

\frac{d P_{25}}{d t} = - 2 P_{0} k_{coop} + 2 P_{0} k_{on} - \frac{P_{11} k_{coop}}{2} + 2 P_{13} k_{coop} + 4 P_{14} k_{coop} + 2 P_{14} k_{on} + 4 P_{15} k_{coop} + 2 P_{15} k_{on} - 2 P_{16} k_{coop} - 2 P_{18} k_{coop} + \frac{P_{19} k_{coop}}{2} - 2 P_{2} k_{on} - P_{20} k_{coop} - \frac{P_{22} k_{coop}}{2} - P_{23} k_{coop} + 2 P_{24} k_{coop} - 4 P_{25} k_{off} - 2 P_{3} k_{coop} - 2 P_{3} k_{on} + 3 P_{4} k_{coop} + P_{4} k_{on} + P_{5} k_{coop} - P_{5} k_{on} - \frac{3 P_{7} k_{coop}}{2} + P_{8} k_{coop} + P_{8} k_{on} + P_{9} k_{coop} + P_{9} k_{on}

\frac{d P_{26}}{d t} = 2 P_{16} k_{on} - 4 P_{26} k_{coop} - 2 P_{26} k_{off} - 2 P_{26} k_{on} + 2 P_{37} k_{off}

\frac{d P_{27}}{d t} = 12 P_{0} k_{off} + 2 P_{11} k_{coop} + 2 P_{11} k_{off} + 2 P_{11} k_{on} + P_{12} k_{off} - P_{13} k_{off} + 4 P_{16} k_{off} - 2 P_{18} k_{coop} - 2 P_{18} k_{on} + \frac{2 P_{19} k_{off}}{3} - 12 P_{2} k_{off} + \frac{2 P_{20} k_{off}}{3} + 2 P_{22} k_{off} - 2 P_{23} k_{off} - 4 P_{27} k_{coop} - 4 P_{27} k_{off} - 8 P_{27} k_{on} + 2 P_{28} k_{off} + 2 P_{4} k_{off} - 6 P_{5} k_{off} + 2 P_{7} k_{off} + 2 P_{8} k_{off} + 6 P_{9} k_{off}

\frac{d P_{28}}{d t} = \frac{3 P_{0} k_{coop}}{2} - 18 P_{0} k_{off} - \frac{49 P_{11} k_{coop}}{8} - \frac{17 P_{11} k_{off}}{2} - \frac{7 P_{11} k_{on}}{2} + \frac{P_{12} k_{coop}}{4} + P_{12} k_{off} + \frac{25 P_{13} k_{coop}}{4} + \frac{190 P_{13} k_{off}}{3} - 4 P_{13} k_{on} + 6 P_{14} k_{coop} + 42 P_{14} k_{off} + 18 P_{15} k_{coop} + 134 P_{15} k_{off} - \frac{19 P_{16} k_{coop}}{2} - \frac{178 P_{16} k_{off}}{3} - \frac{13 P_{18} k_{coop}}{2} - \frac{169 P_{18} k_{off}}{3} + 9 P_{18} k_{on} - \frac{P_{19} k_{coop}}{24} + \frac{275 P_{19} k_{off}}{18} - \frac{11 P_{19} k_{on}}{6} - \frac{15 P_{2} k_{coop}}{2} - 24 P_{2} k_{off} - \frac{5 P_{20} k_{coop}}{3} - \frac{383 P_{20} k_{off}}{36} + \frac{5 P_{20} k_{on}}{3} + \frac{P_{21} k_{coop}}{4} + \frac{29 P_{21} k_{off}}{12} - \frac{49 P_{22} k_{coop}}{8} - \frac{17 P_{22} k_{off}}{2} - \frac{7 P_{22} k_{on}}{2} + \frac{3 P_{23} k_{coop}}{4} + \frac{2 P_{23} k_{off}}{3} + 3 P_{23} k_{on} + 6 P_{24} k_{coop} + 42 P_{24} k_{off} + 8 P_{26} k_{off} - \frac{P_{27} k_{coop}}{2} - \frac{43 P_{27} k_{off}}{3} + 4 P_{27} k_{on} - \frac{27 P_{28} k_{coop}}{4} - \frac{62 P_{28} k_{off}}{3} - 6 P_{28} k_{on} - \frac{5 P_{29} k_{coop}}{2} - \frac{77 P_{29} k_{off}}{3} - 12 P_{3} k_{coop} - 92 P_{3} k_{off} + \frac{P_{30} k_{coop}}{4} + P_{30} k_{off} + \frac{P_{31} k_{coop}}{2} - \frac{14 P_{31} k_{off}}{3} + \frac{5 P_{32} k_{coop}}{4} + \frac{14 P_{32} k_{off}}{3} - 4 P_{33} k_{coop} - \frac{34 P_{33} k_{off}}{3} + \frac{2 P_{34} k_{off}}{3} - \frac{P_{35} k_{coop}}{2} - 2 P_{35} k_{off} + \frac{P_{36} k_{coop}}{2} - 6 P_{36} k_{off} + 8 P_{37} k_{off} - 6 P_{38} k_{off} + \frac{65 P_{4} k_{coop}}{4} + \frac{277 P_{4} k_{off}}{3} - \frac{3 P_{5} k_{coop}}{4} + 9 P_{5} k_{off} - \frac{27 P_{7} k_{coop}}{8} - \frac{305 P_{7} k_{off}}{6} + \frac{7 P_{7} k_{on}}{2} + \frac{25 P_{8} k_{coop}}{4} + 39 P_{8} k_{off} + \frac{27 P_{9} k_{coop}}{4} + 33 P_{9} k_{off}

\frac{d P_{29}}{d t} = 6 P_{0} k_{coop} - \frac{P_{0} k_{off}}{2} + 6 P_{0} k_{on} + \frac{P_{11} k_{coop}}{2} + \frac{33 P_{11} k_{off}}{8} + \frac{P_{12} k_{coop}}{2} - \frac{3 P_{12} k_{off}}{4} + \frac{P_{12} k_{on}}{2} + \frac{P_{13} k_{coop}}{2} - \frac{87 P_{13} k_{off}}{4} - \frac{P_{13} k_{on}}{2} - 17 P_{14} k_{off} - 51 P_{15} k_{off} + 2 P_{16} k_{coop} + \frac{41 P_{16} k_{off}}{2} + 2 P_{16} k_{on} + \frac{43 P_{18} k_{off}}{2} + P_{18} k_{on} + \frac{5 P_{19} k_{coop}}{6} - \frac{127 P_{19} k_{off}}{24} + \frac{P_{19} k_{on}}{3} - 6 P_{2} k_{coop} + \frac{35 P_{2} k_{off}}{2} - 6 P_{2} k_{on} + \frac{P_{20} k_{coop}}{3} + \frac{29 P_{20} k_{off}}{6} + \frac{P_{20} k_{on}}{3} - \frac{3 P_{21} k_{off}}{4} + \frac{P_{22} k_{coop}}{2} + \frac{33 P_{22} k_{off}}{8} + \frac{9 P_{23} k_{off}}{4} - 17 P_{24} k_{off} - P_{26} k_{off} - 2 P_{27} k_{coop} + \frac{9 P_{27} k_{off}}{2} + \frac{25 P_{28} k_{off}}{4} - 5 P_{29} k_{coop} + \frac{11 P_{29} k_{off}}{2} - 6 P_{29} k_{on} + 34 P_{3} k_{off} - \frac{3 P_{30} k_{off}}{4} - P_{31} k_{coop} + \frac{P_{31} k_{off}}{2} - \frac{7 P_{32} k_{off}}{4} + 4 P_{33} k_{off} + \frac{3 P_{35} k_{off}}{2} + \frac{3 P_{36} k_{off}}{2} + 2 P_{38} k_{off} - P_{39} k_{off} + P_{4} k_{coop} - \frac{151 P_{4} k_{off}}{4} + P_{4} k_{on} - 3 P_{5} k_{coop} + \frac{P_{5} k_{off}}{4} - 3 P_{5} k_{on} + \frac{3 P_{7} k_{coop}}{2} + \frac{131 P_{7} k_{off}}{8} + 2 P_{7} k_{on} + P_{8} k_{coop} - \frac{67 P_{8} k_{off}}{4} + P_{8} k_{on} + 3 P_{9} k_{coop} - \frac{69 P_{9} k_{off}}{4} + 3 P_{9} k_{on}

\frac{d P_{30}}{d t} = - \frac{93 P_{0} k_{coop}}{2} + \frac{261 P_{0} k_{off}}{2} - 58 P_{0} k_{on} - \frac{193 P_{11} k_{coop}}{8} + \frac{657 P_{11} k_{off}}{8} - 21 P_{11} k_{on} - \frac{15 P_{12} k_{coop}}{4} - \frac{33 P_{12} k_{off}}{4} - 4 P_{12} k_{on} + \frac{85 P_{13} k_{coop}}{4} - \frac{1765 P_{13} k_{off}}{4} - 36 P_{13} k_{on} + 30 P_{14} k_{coop} - 324 P_{14} k_{off} - 16 P_{14} k_{on} + 90 P_{15} k_{coop} - 1032 P_{15} k_{off} - 48 P_{15} k_{on} - \frac{83 P_{16} k_{coop}}{2} + \frac{887 P_{16} k_{off}}{2} + 10 P_{16} k_{on} - \frac{23 P_{18} k_{coop}}{2} + \frac{779 P_{18} k_{off}}{2} + 53 P_{18} k_{on} - \frac{41 P_{19} k_{coop}}{24} - \frac{2327 P_{19} k_{off}}{24} - \frac{55 P_{19} k_{on}}{3} + \frac{33 P_{2} k_{coop}}{2} + \frac{387 P_{2} k_{off}}{2} + 74 P_{2} k_{on} - \frac{31 P_{20} k_{coop}}{12} + \frac{899 P_{20} k_{off}}{12} + \frac{155 P_{20} k_{on}}{12} + \frac{3 P_{21} k_{coop}}{2} - \frac{37 P_{21} k_{off}}{2} - \frac{P_{21} k_{on}}{4} - \frac{193 P_{22} k_{coop}}{8} + \frac{657 P_{22} k_{off}}{8} - 21 P_{22} k_{on} - \frac{25 P_{23} k_{coop}}{4} + \frac{73 P_{23} k_{off}}{4} + 23 P_{23} k_{on} + 30 P_{24} k_{coop} - 324 P_{24} k_{off} - 16 P_{24} k_{on} + 6 P_{26} k_{coop} - 60 P_{26} k_{off} + 10 P_{26} k_{on} + \frac{37 P_{27} k_{coop}}{2} + \frac{131 P_{27} k_{off}}{2} + 32 P_{27} k_{on} - \frac{7 P_{28} k_{coop}}{4} + \frac{431 P_{28} k_{off}}{4} - 14 P_{28} k_{on} - \frac{3 P_{29} k_{coop}}{2} + \frac{283 P_{29} k_{off}}{2} - 18 P_{29} k_{on} - 60 P_{3} k_{coop} + 708 P_{3} k_{off} + 32 P_{3} k_{on} - \frac{15 P_{30} k_{coop}}{4} - \frac{49 P_{30} k_{off}}{4} - 4 P_{30} k_{on} - \frac{9 P_{31} k_{coop}}{2} + \frac{7 P_{31} k_{off}}{2} + \frac{17 P_{32} k_{coop}}{4} - \frac{65 P_{32} k_{off}}{4} + 2 P_{33} k_{coop} + 76 P_{33} k_{off} + 10 P_{33} k_{on} - 5 P_{34} k_{off} + \frac{15 P_{35} k_{coop}}{2} + \frac{33 P_{35} k_{off}}{2} + 10 P_{35} k_{on} + \frac{17 P_{36} k_{coop}}{2} - \frac{69 P_{36} k_{off}}{2} - 60 P_{37} k_{off} + 4 P_{38} k_{coop} - 66 P_{38} k_{off} + 4 P_{39} k_{coop} + 57 P_{39} k_{off} + \frac{193 P_{4} k_{coop}}{4} - \frac{2809 P_{4} k_{off}}{4} - 55 P_{4} k_{on} + 4 P_{40} k_{coop} - 12 P_{40} k_{off} - 6 P_{41} k_{off} + \frac{93 P_{5} k_{coop}}{4} - \frac{261 P_{5} k_{off}}{4} + 29 P_{5} k_{on} - \frac{139 P_{7} k_{coop}}{8} + \frac{2891 P_{7} k_{off}}{8} + 31 P_{7} k_{on} + \frac{81 P_{8} k_{coop}}{4} - \frac{1209 P_{8} k_{off}}{4} - 29 P_{8} k_{on} + \frac{27 P_{9} k_{coop}}{4} - \frac{1035 P_{9} k_{off}}{4} - 45 P_{9} k_{on}

\frac{d P_{31}}{d t} = - 6 P_{0} k_{on} - \frac{3 P_{11} k_{off}}{4} + \frac{P_{11} k_{on}}{2} - \frac{P_{12} k_{on}}{2} + \frac{9 P_{13} k_{off}}{2} + \frac{3 P_{13} k_{on}}{2} + 3 P_{14} k_{off} + 9 P_{15} k_{off} - 4 P_{16} k_{off} - 2 P_{16} k_{on} - \frac{9 P_{18} k_{off}}{2} + P_{18} k_{on} + \frac{5 P_{19} k_{off}}{4} + \frac{P_{19} k_{on}}{2} - 3 P_{2} k_{off} + 6 P_{2} k_{on} - \frac{9 P_{20} k_{off}}{8} + \frac{P_{21} k_{off}}{8} - \frac{3 P_{22} k_{off}}{4} + \frac{P_{22} k_{on}}{2} - P_{23} k_{coop} + 3 P_{24} k_{off} - \frac{3 P_{27} k_{off}}{2} + P_{28} k_{coop} - P_{28} k_{off} + P_{29} k_{coop} - \frac{5 P_{29} k_{off}}{2} - 6 P_{3} k_{off} - 4 P_{31} k_{coop} - 5 P_{31} k_{off} - 6 P_{31} k_{on} + \frac{P_{32} k_{off}}{2} - P_{33} k_{off} + P_{36} k_{off} + 2 P_{39} k_{off} + 7 P_{4} k_{off} - P_{4} k_{on} + 3 P_{5} k_{on} - \frac{13 P_{7} k_{off}}{4} - \frac{5 P_{7} k_{on}}{2} + 3 P_{8} k_{off} - P_{8} k_{on} + 3 P_{9} k_{off} - 3 P_{9} k_{on}

\frac{d P_{32}}{d t} = 9 P_{0} k_{coop} - \frac{63 P_{0} k_{off}}{2} + 14 P_{0} k_{on} + \frac{11 P_{11} k_{coop}}{4} - \frac{87 P_{11} k_{off}}{8} + \frac{15 P_{11} k_{on}}{2} + \frac{3 P_{12} k_{coop}}{2} + \frac{3 P_{12} k_{off}}{4} + P_{12} k_{on} + \frac{51 P_{13} k_{coop}}{2} + \frac{255 P_{13} k_{off}}{4} + 5 P_{13} k_{on} + 12 P_{14} k_{coop} + 48 P_{14} k_{off} - 4 P_{14} k_{on} + 36 P_{15} k_{coop} + 156 P_{15} k_{off} - 12 P_{15} k_{on} - 15 P_{16} k_{coop} - \frac{133 P_{16} k_{off}}{2} + 6 P_{16} k_{on} - 27 P_{18} k_{coop} - \frac{105 P_{18} k_{off}}{2} - 8 P_{18} k_{on} + \frac{107 P_{19} k_{coop}}{12} + \frac{107 P_{19} k_{off}}{8} + \frac{9 P_{19} k_{on}}{2} - 21 P_{2} k_{coop} - \frac{33 P_{2} k_{off}}{2} - 10 P_{2} k_{on} - \frac{16 P_{20} k_{coop}}{3} - \frac{37 P_{20} k_{off}}{4} - \frac{3 P_{20} k_{on}}{4} + \frac{P_{21} k_{coop}}{2} + 3 P_{21} k_{off} - \frac{P_{21} k_{on}}{4} + \frac{11 P_{22} k_{coop}}{4} - \frac{87 P_{22} k_{off}}{8} + \frac{15 P_{22} k_{on}}{2} + \frac{11 P_{23} k_{coop}}{2} - \frac{15 P_{23} k_{off}}{4} + 4 P_{23} k_{on} + 12 P_{24} k_{coop} + 48 P_{24} k_{off} - 4 P_{24} k_{on} + 12 P_{26} k_{off} - 2 P_{26} k_{on} - 15 P_{27} k_{coop} - \frac{9 P_{27} k_{off}}{2} - 12 P_{27} k_{on} - \frac{25 P_{28} k_{coop}}{2} - \frac{49 P_{28} k_{off}}{4} - 4 P_{28} k_{on} - 21 P_{29} k_{coop} - \frac{25 P_{29} k_{off}}{2} - 10 P_{29} k_{on} - 24 P_{3} k_{coop} - 108 P_{3} k_{off} + 8 P_{3} k_{on} + \frac{P_{30} k_{coop}}{2} + \frac{3 P_{30} k_{off}}{4} - 8 P_{31} k_{coop} + \frac{7 P_{31} k_{off}}{2} - 4 P_{31} k_{on} - \frac{7 P_{32} k_{coop}}{2} - \frac{13 P_{32} k_{off}}{4} - 4 P_{32} k_{on} - 6 P_{33} k_{coop} - 8 P_{33} k_{off} - 2 P_{33} k_{on} + P_{34} k_{off} - 3 P_{35} k_{coop} - \frac{3 P_{35} k_{off}}{2} - 2 P_{35} k_{on} - P_{36} k_{coop} + \frac{3 P_{36} k_{off}}{2} + 12 P_{37} k_{off} - 2 P_{38} k_{coop} + 8 P_{38} k_{off} - 11 P_{39} k_{off} + \frac{63 P_{4} k_{coop}}{2} + \frac{395 P_{4} k_{off}}{4} - 3 P_{4} k_{on} + 4 P_{40} k_{off} + \frac{14 P_{41} k_{off}}{13} - \frac{8 P_{42} k_{off}}{13} - \frac{9 P_{5} k_{coop}}{2} + \frac{63 P_{5} k_{off}}{4} - 7 P_{5} k_{on} - \frac{71 P_{7} k_{coop}}{4} - \frac{445 P_{7} k_{off}}{8} - \frac{3 P_{7} k_{on}}{2} + \frac{27 P_{8} k_{coop}}{2} + \frac{171 P_{8} k_{off}}{4} - P_{8} k_{on} + \frac{33 P_{9} k_{coop}}{2} + \frac{129 P_{9} k_{off}}{4} + 3 P_{9} k_{on}

\frac{d P_{33}}{d t} = \frac{21 P_{0} k_{off}}{2} + \frac{13 P_{11} k_{off}}{8} - \frac{P_{12} k_{off}}{4} - \frac{275 P_{13} k_{off}}{12} - 13 P_{14} k_{off} - 43 P_{15} k_{off} + \frac{121 P_{16} k_{off}}{6} + \frac{115 P_{18} k_{off}}{6} - \frac{433 P_{19} k_{off}}{72} + \frac{7 P_{2} k_{off}}{2} + \frac{133 P_{20} k_{off}}{36} - \frac{5 P_{21} k_{off}}{6} + 2 P_{22} k_{coop} + \frac{13 P_{22} k_{off}}{8} + 2 P_{22} k_{on} - \frac{13 P_{23} k_{off}}{12} - 14 P_{24} k_{off} - 4 P_{26} k_{off} + \frac{37 P_{27} k_{off}}{6} + \frac{73 P_{28} k_{off}}{12} + \frac{53 P_{29} k_{off}}{6} + 29 P_{3} k_{off} - \frac{P_{30} k_{off}}{4} + \frac{11 P_{31} k_{off}}{6} - \frac{19 P_{32} k_{off}}{12} - 2 P_{33} k_{coop} + \frac{5 P_{33} k_{off}}{3} - 3 P_{33} k_{on} - \frac{P_{34} k_{off}}{3} - \frac{P_{35} k_{off}}{2} + \frac{P_{36} k_{off}}{2} - 4 P_{37} k_{off} - P_{38} k_{off} + 3 P_{39} k_{off} - \frac{347 P_{4} k_{off}}{12} - \frac{3 P_{41} k_{off}}{13} - \frac{2 P_{42} k_{off}}{13} + P_{43} k_{off} - \frac{21 P_{5} k_{off}}{4} + \frac{445 P_{7} k_{off}}{24} - \frac{45 P_{8} k_{off}}{4} - \frac{35 P_{9} k_{off}}{4}

\frac{d P_{34}}{d t} = - \frac{227 P_{0} k_{coop}}{4} + \frac{87 P_{0} k_{off}}{2} - 97 P_{0} k_{on} - \frac{395 P_{11} k_{coop}}{16} + \frac{127 P_{11} k_{off}}{8} - \frac{49 P_{11} k_{on}}{4} - \frac{81 P_{12} k_{coop}}{8} + \frac{5 P_{12} k_{off}}{4} - \frac{23 P_{12} k_{on}}{2} + \frac{379 P_{13} k_{coop}}{8} - \frac{895 P_{13} k_{off}}{12} + \frac{93 P_{13} k_{on}}{2} + 67 P_{14} k_{coop} - 45 P_{14} k_{off} + 32 P_{14} k_{on} + 177 P_{15} k_{coop} - 179 P_{15} k_{off} + 84 P_{15} k_{on} - \frac{321 P_{16} k_{coop}}{4} + \frac{515 P_{16} k_{off}}{6} - 65 P_{16} k_{on} - \frac{177 P_{18} k_{coop}}{4} + \frac{160 P_{18} k_{off}}{3} - 31 P_{18} k_{on} + \frac{157 P_{19} k_{coop}}{48} - \frac{839 P_{19} k_{off}}{72} + \frac{71 P_{19} k_{on}}{12} + \frac{79 P_{2} k_{coop}}{4} - \frac{7 P_{2} k_{off}}{2} + 89 P_{2} k_{on} - \frac{115 P_{20} k_{coop}}{24} + \frac{109 P_{20} k_{off}}{72} - \frac{29 P_{20} k_{on}}{6} + \frac{7 P_{21} k_{coop}}{2} - \frac{109 P_{21} k_{off}}{24} + 2 P_{21} k_{on} - \frac{395 P_{22} k_{coop}}{16} + \frac{127 P_{22} k_{off}}{8} - \frac{49 P_{22} k_{on}}{4} + \frac{13 P_{23} k_{coop}}{8} - \frac{209 P_{23} k_{off}}{12} + \frac{51 P_{23} k_{on}}{2} + 37 P_{24} k_{coop} - 40 P_{24} k_{off} + 8 P_{24} k_{on} + 12 P_{25} k_{off} - 7 P_{26} k_{coop} - 20 P_{26} k_{off} + 4 P_{26} k_{on} + \frac{91 P_{27} k_{coop}}{4} + \frac{25 P_{27} k_{off}}{3} + P_{27} k_{on} - \frac{145 P_{28} k_{coop}}{8} + \frac{227 P_{28} k_{off}}{12} - 30 P_{28} k_{on} - \frac{61 P_{29} k_{coop}}{4} + \frac{59 P_{29} k_{off}}{3} - 27 P_{29} k_{on} - 140 P_{3} k_{coop} + 139 P_{3} k_{off} - 76 P_{3} k_{on} + \frac{11 P_{30} k_{coop}}{8} + \frac{5 P_{30} k_{off}}{4} - \frac{17 P_{31} k_{coop}}{4} + \frac{73 P_{31} k_{off}}{6} - 6 P_{31} k_{on} - \frac{21 P_{32} k_{coop}}{8} + \frac{97 P_{32} k_{off}}{12} - P_{32} k_{on} + \frac{55 P_{33} k_{off}}{3} - 6 P_{33} k_{on} - 4 P_{34} k_{coop} - \frac{26 P_{34} k_{off}}{3} - 2 P_{34} k_{on} - \frac{39 P_{35} k_{coop}}{4} + \frac{5 P_{35} k_{off}}{2} - P_{35} k_{on} + \frac{7 P_{36} k_{coop}}{4} - \frac{59 P_{36} k_{off}}{2} - 24 P_{37} k_{coop} - 44 P_{37} k_{off} - 12 P_{37} k_{on} - P_{38} k_{coop} - 32 P_{38} k_{off} + 6 P_{39} k_{off} + \frac{711 P_{4} k_{coop}}{8} - \frac{1249 P_{4} k_{off}}{12} + \frac{49 P_{4} k_{on}}{2} + 2 P_{40} k_{off} - \frac{8 P_{41} k_{off}}{13} - \frac{14 P_{42} k_{off}}{13} - 6 P_{43} k_{coop} + 5 P_{43} k_{off} - 12 P_{44} k_{off} + \frac{227 P_{5} k_{coop}}{8} - \frac{87 P_{5} k_{off}}{4} + \frac{97 P_{5} k_{on}}{2} - \frac{889 P_{7} k_{coop}}{16} + \frac{1679 P_{7} k_{off}}{24} - \frac{211 P_{7} k_{on}}{4} + \frac{415 P_{8} k_{coop}}{8} - \frac{183 P_{8} k_{off}}{4} + \frac{21 P_{8} k_{on}}{2} + \frac{69 P_{9} k_{coop}}{8} - \frac{73 P_{9} k_{off}}{4} - \frac{81 P_{9} k_{on}}{2}

\frac{d P_{35}}{d t} = 21 P_{0} k_{off} + \frac{13 P_{11} k_{off}}{4} - \frac{P_{12} k_{off}}{2} - \frac{275 P_{13} k_{off}}{6} - 26 P_{14} k_{off} - 86 P_{15} k_{off} + \frac{121 P_{16} k_{off}}{3} + \frac{115 P_{18} k_{off}}{3} - \frac{433 P_{19} k_{off}}{36} + 7 P_{2} k_{off} + \frac{133 P_{20} k_{off}}{18} - \frac{5 P_{21} k_{off}}{3} + \frac{13 P_{22} k_{off}}{4} + 2 P_{22} k_{on} - \frac{13 P_{23} k_{off}}{6} - 28 P_{24} k_{off} - 8 P_{26} k_{off} + \frac{37 P_{27} k_{off}}{3} + \frac{73 P_{28} k_{off}}{6} + \frac{53 P_{29} k_{off}}{3} + 58 P_{3} k_{off} - \frac{P_{30} k_{off}}{2} + \frac{11 P_{31} k_{off}}{3} - \frac{19 P_{32} k_{off}}{6} + \frac{22 P_{33} k_{off}}{3} - \frac{2 P_{34} k_{off}}{3} - 4 P_{35} k_{coop} - 3 P_{35} k_{off} - 3 P_{35} k_{on} + P_{36} k_{off} - 8 P_{37} k_{off} - 2 P_{38} k_{off} + 6 P_{39} k_{off} - \frac{347 P_{4} k_{off}}{6} - \frac{6 P_{41} k_{off}}{13} - \frac{4 P_{42} k_{off}}{13} - P_{43} k_{off} - \frac{21 P_{5} k_{off}}{2} + \frac{445 P_{7} k_{off}}{12} - \frac{45 P_{8} k_{off}}{2} - \frac{35 P_{9} k_{off}}{2}

\frac{d P_{36}}{d t} = - \frac{37 P_{0} k_{coop}}{4} + 3 P_{0} k_{off} - 19 P_{0} k_{on} + \frac{75 P_{11} k_{coop}}{16} - \frac{15 P_{11} k_{off}}{4} + \frac{P_{11} k_{on}}{4} - \frac{7 P_{12} k_{coop}}{8} + \frac{P_{12} k_{off}}{2} - \frac{P_{12} k_{on}}{2} - \frac{83 P_{13} k_{coop}}{8} + 10 P_{13} k_{off} + \frac{13 P_{13} k_{on}}{2} - 13 P_{14} k_{coop} + 9 P_{14} k_{off} + 2 P_{14} k_{on} - 39 P_{15} k_{coop} + 27 P_{15} k_{off} + 6 P_{15} k_{on} + \frac{49 P_{16} k_{coop}}{4} - 11 P_{16} k_{off} - 7 P_{16} k_{on} + \frac{49 P_{18} k_{coop}}{4} - \frac{21 P_{18} k_{off}}{2} - 2 P_{18} k_{on} - \frac{31 P_{19} k_{coop}}{16} + \frac{19 P_{19} k_{off}}{12} + \frac{13 P_{19} k_{on}}{12} + \frac{89 P_{2} k_{coop}}{4} - 11 P_{2} k_{off} + 17 P_{2} k_{on} + \frac{25 P_{20} k_{coop}}{8} - \frac{49 P_{20} k_{off}}{24} - \frac{5 P_{20} k_{on}}{12} - \frac{P_{21} k_{coop}}{2} + \frac{3 P_{21} k_{off}}{8} + \frac{P_{21} k_{on}}{4} + \frac{75 P_{22} k_{coop}}{16} - \frac{15 P_{22} k_{off}}{4} + \frac{P_{22} k_{on}}{4} + \frac{35 P_{23} k_{coop}}{8} - P_{23} k_{off} + \frac{3 P_{23} k_{on}}{2} - 13 P_{24} k_{coop} + 8 P_{24} k_{off} + 2 P_{24} k_{on} + P_{26} k_{coop} + 4 P_{26} k_{on} - \frac{3 P_{27} k_{coop}}{4} - \frac{3 P_{27} k_{off}}{2} + \frac{9 P_{28} k_{coop}}{8} - \frac{7 P_{28} k_{off}}{2} - 2 P_{28} k_{on} + \frac{5 P_{29} k_{coop}}{4} - \frac{11 P_{29} k_{off}}{2} + 26 P_{3} k_{coop} - 19 P_{3} k_{off} - 4 P_{3} k_{on} - \frac{3 P_{30} k_{coop}}{8} + \frac{P_{30} k_{off}}{2} + \frac{P_{31} k_{coop}}{4} - 4 P_{31} k_{on} - \frac{3 P_{32} k_{coop}}{8} + P_{32} k_{off} + 2 P_{33} k_{coop} - 3 P_{33} k_{off} + \frac{7 P_{35} k_{coop}}{4} - 2 P_{35} k_{off} + P_{35} k_{on} - \frac{15 P_{36} k_{coop}}{4} - 7 P_{36} k_{off} - 4 P_{36} k_{on} + 3 P_{38} k_{coop} - 8 P_{38} k_{off} - 2 P_{39} k_{coop} + 10 P_{39} k_{off} - \frac{239 P_{4} k_{coop}}{8} + \frac{41 P_{4} k_{off}}{2} - \frac{P_{4} k_{on}}{2} - 2 P_{40} k_{coop} - \frac{6 P_{41} k_{off}}{13} - \frac{4 P_{42} k_{off}}{13} - P_{43} k_{off} + \frac{37 P_{5} k_{coop}}{8} - \frac{3 P_{5} k_{off}}{2} + \frac{19 P_{5} k_{on}}{2} + \frac{121 P_{7} k_{coop}}{16} - \frac{29 P_{7} k_{off}}{4} - \frac{29 P_{7} k_{on}}{4} - \frac{119 P_{8} k_{coop}}{8} + \frac{19 P_{8} k_{off}}{2} - \frac{5 P_{8} k_{on}}{2} - \frac{141 P_{9} k_{coop}}{8} + \frac{19 P_{9} k_{off}}{2} - \frac{15 P_{9} k_{on}}{2}

\frac{d P_{37}}{d t} = 2 P_{24} k_{coop} + 2 P_{24} k_{on} + 2 P_{26} k_{coop} + P_{26} k_{on} - 2 P_{37} k_{coop} - 3 P_{37} k_{off} - P_{37} k_{on} + P_{44} k_{off}

\frac{d P_{38}}{d t} = 3 P_{0} k_{coop} + \frac{21 P_{0} k_{off}}{2} + 12 P_{0} k_{on} - \frac{19 P_{11} k_{coop}}{4} + \frac{17 P_{11} k_{off}}{8} + \frac{P_{11} k_{on}}{2} + \frac{P_{12} k_{coop}}{2} - \frac{P_{12} k_{off}}{4} + \frac{43 P_{13} k_{coop}}{2} - \frac{293 P_{13} k_{off}}{12} + \frac{P_{13} k_{on}}{2} + 18 P_{14} k_{coop} - 15 P_{14} k_{off} + 54 P_{15} k_{coop} - 49 P_{15} k_{off} - 21 P_{16} k_{coop} + \frac{133 P_{16} k_{off}}{6} + 2 P_{16} k_{on} - 22 P_{18} k_{coop} + \frac{62 P_{18} k_{off}}{3} - \frac{7 P_{18} k_{on}}{2} + \frac{65 P_{19} k_{coop}}{12} - \frac{457 P_{19} k_{off}}{72} + \frac{4 P_{19} k_{on}}{3} - 21 P_{2} k_{coop} + \frac{11 P_{2} k_{off}}{2} - 12 P_{2} k_{on} - \frac{67 P_{20} k_{coop}}{12} + \frac{287 P_{20} k_{off}}{72} - \frac{25 P_{20} k_{on}}{24} + \frac{3 P_{21} k_{coop}}{4} - \frac{23 P_{21} k_{off}}{24} - \frac{P_{21} k_{on}}{8} - \frac{19 P_{22} k_{coop}}{4} + \frac{17 P_{22} k_{off}}{8} + \frac{P_{22} k_{on}}{2} - \frac{5 P_{23} k_{coop}}{2} - \frac{7 P_{23} k_{off}}{12} + \frac{P_{23} k_{on}}{2} + 18 P_{24} k_{coop} - 16 P_{24} k_{off} - 4 P_{26} k_{off} - 3 P_{26} k_{on} - 4 P_{27} k_{coop} + \frac{17 P_{27} k_{off}}{3} - \frac{7 P_{27} k_{on}}{2} - \frac{11 P_{28} k_{coop}}{2} + \frac{73 P_{28} k_{off}}{12} - P_{28} k_{on} - 8 P_{29} k_{coop} + \frac{25 P_{29} k_{off}}{3} - \frac{9 P_{29} k_{on}}{2} - 36 P_{3} k_{coop} + 33 P_{3} k_{off} + \frac{P_{30} k_{coop}}{2} - \frac{P_{30} k_{off}}{4} - 3 P_{31} k_{coop} + \frac{11 P_{31} k_{off}}{6} + 2 P_{31} k_{on} + \frac{3 P_{32} k_{coop}}{2} - \frac{13 P_{32} k_{off}}{12} + \frac{P_{32} k_{on}}{2} - 4 P_{33} k_{coop} + \frac{11 P_{33} k_{off}}{3} - P_{33} k_{on} - \frac{P_{34} k_{off}}{3} - P_{35} k_{coop} - \frac{P_{35} k_{off}}{2} - 3 P_{36} k_{coop} - \frac{P_{36} k_{off}}{2} - P_{36} k_{on} - 4 P_{37} k_{off} - 10 P_{38} k_{coop} - 5 P_{38} k_{on} + 2 P_{39} k_{coop} - 9 P_{39} k_{off} + \frac{81 P_{4} k_{coop}}{2} - \frac{395 P_{4} k_{off}}{12} + 4 P_{4} k_{on} + 2 P_{40} k_{coop} + \frac{4 P_{41} k_{off}}{13} - \frac{6 P_{42} k_{off}}{13} - P_{43} k_{off} + P_{45} k_{off} - \frac{3 P_{5} k_{coop}}{2} - \frac{21 P_{5} k_{off}}{4} - 6 P_{5} k_{on} - \frac{65 P_{7} k_{coop}}{4} + \frac{481 P_{7} k_{off}}{24} + \frac{3 P_{7} k_{on}}{2} + \frac{37 P_{8} k_{coop}}{2} - \frac{53 P_{8} k_{off}}{4} + 3 P_{8} k_{on} + \frac{39 P_{9} k_{coop}}{2} - \frac{43 P_{9} k_{off}}{4} + 6 P_{9} k_{on}

\frac{d P_{39}}{d t} = \frac{3 P_{0} k_{coop}}{2} - \frac{21 P_{0} k_{off}}{2} + \frac{13 P_{0} k_{on}}{2} - \frac{13 P_{11} k_{coop}}{8} - \frac{17 P_{11} k_{off}}{8} + \frac{21 P_{11} k_{on}}{8} + \frac{P_{12} k_{coop}}{4} + \frac{P_{12} k_{off}}{4} - \frac{P_{12} k_{on}}{4} + \frac{33 P_{13} k_{coop}}{4} + \frac{293 P_{13} k_{off}}{12} - \frac{53 P_{13} k_{on}}{4} + 6 P_{14} k_{coop} + 15 P_{14} k_{off} - 10 P_{14} k_{on} + 18 P_{15} k_{coop} + 49 P_{15} k_{off} - 30 P_{15} k_{on} - \frac{15 P_{16} k_{coop}}{2} - \frac{133 P_{16} k_{off}}{6} + \frac{27 P_{16} k_{on}}{2} - \frac{17 P_{18} k_{coop}}{2} - \frac{62 P_{18} k_{off}}{3} + \frac{23 P_{18} k_{on}}{2} + \frac{17 P_{19} k_{coop}}{8} + \frac{457 P_{19} k_{off}}{72} - \frac{67 P_{19} k_{on}}{24} - \frac{15 P_{2} k_{coop}}{2} - \frac{11 P_{2} k_{off}}{2} + \frac{7 P_{2} k_{on}}{2} - 2 P_{20} k_{coop} - \frac{287 P_{20} k_{off}}{72} + \frac{31 P_{20} k_{on}}{12} + \frac{P_{21} k_{coop}}{4} + \frac{23 P_{21} k_{off}}{24} - \frac{P_{21} k_{on}}{2} - \frac{13 P_{22} k_{coop}}{8} - \frac{17 P_{22} k_{off}}{8} + \frac{21 P_{22} k_{on}}{8} - \frac{P_{23} k_{coop}}{4} + \frac{7 P_{23} k_{off}}{12} + \frac{5 P_{23} k_{on}}{4} + 6 P_{24} k_{coop} + 16 P_{24} k_{off} - 10 P_{24} k_{on} + 4 P_{26} k_{off} - 2 P_{26} k_{on} - \frac{5 P_{27} k_{coop}}{2} - \frac{17 P_{27} k_{off}}{3} + \frac{3 P_{27} k_{on}}{2} - \frac{11 P_{28} k_{coop}}{4} - \frac{73 P_{28} k_{off}}{12} + \frac{11 P_{28} k_{on}}{4} - \frac{9 P_{29} k_{coop}}{2} - \frac{25 P_{29} k_{off}}{3} + \frac{7 P_{29} k_{on}}{2} - 12 P_{3} k_{coop} - 33 P_{3} k_{off} + 20 P_{3} k_{on} + \frac{P_{30} k_{coop}}{4} + \frac{P_{30} k_{off}}{4} - \frac{P_{30} k_{on}}{4} - \frac{3 P_{31} k_{coop}}{2} - \frac{11 P_{31} k_{off}}{6} + \frac{7 P_{31} k_{on}}{2} + \frac{5 P_{32} k_{coop}}{4} + \frac{13 P_{32} k_{off}}{12} - \frac{P_{32} k_{on}}{4} - 2 P_{33} k_{coop} - \frac{11 P_{33} k_{off}}{3} + 2 P_{33} k_{on} + \frac{P_{34} k_{off}}{3} - \frac{P_{35} k_{coop}}{2} + \frac{P_{35} k_{off}}{2} + \frac{P_{35} k_{on}}{2} - \frac{3 P_{36} k_{coop}}{2} + \frac{P_{36} k_{off}}{2} - \frac{P_{36} k_{on}}{2} + 4 P_{37} k_{off} - 2 P_{38} k_{coop} + 4 P_{38} k_{off} - 6 P_{39} k_{coop} - 11 P_{39} k_{off} - 5 P_{39} k_{on} + \frac{57 P_{4} k_{coop}}{4} + \frac{395 P_{4} k_{off}}{12} - \frac{81 P_{4} k_{on}}{4} + \frac{4 P_{41} k_{off}}{13} - \frac{6 P_{42} k_{off}}{13} + P_{43} k_{off} - \frac{3 P_{5} k_{coop}}{4} + \frac{21 P_{5} k_{off}}{4} - \frac{13 P_{5} k_{on}}{4} - \frac{47 P_{7} k_{coop}}{8} - \frac{481 P_{7} k_{off}}{24} + \frac{87 P_{7} k_{on}}{8} + \frac{25 P_{8} k_{coop}}{4} + \frac{53 P_{8} k_{off}}{4} - \frac{33 P_{8} k_{on}}{4} + \frac{27 P_{9} k_{coop}}{4} + \frac{43 P_{9} k_{off}}{4} - \frac{27 P_{9} k_{on}}{4}

\frac{d P_{40}}{d t} = - \frac{27 P_{0} k_{coop}}{4} + \frac{3 P_{0} k_{off}}{2} - 6 P_{0} k_{on} + \frac{P_{11} k_{coop}}{16} - \frac{21 P_{11} k_{off}}{8} - \frac{5 P_{11} k_{on}}{4} - \frac{5 P_{12} k_{coop}}{8} + \frac{P_{12} k_{off}}{4} - \frac{P_{12} k_{on}}{2} - \frac{5 P_{13} k_{coop}}{8} + \frac{11 P_{13} k_{off}}{4} + \frac{5 P_{13} k_{on}}{2} - 3 P_{14} k_{coop} + 3 P_{14} k_{off} - 9 P_{15} k_{coop} + 9 P_{15} k_{off} + \frac{3 P_{16} k_{coop}}{4} - \frac{9 P_{16} k_{off}}{2} - 3 P_{16} k_{on} + \frac{9 P_{18} k_{coop}}{4} - 3 P_{18} k_{off} - P_{18} k_{on} + \frac{P_{19} k_{coop}}{48} - \frac{3 P_{19} k_{off}}{8} + \frac{7 P_{19} k_{on}}{12} + \frac{39 P_{2} k_{coop}}{4} - \frac{7 P_{2} k_{off}}{2} + 6 P_{2} k_{on} - \frac{P_{20} k_{coop}}{6} - \frac{3 P_{20} k_{off}}{8} - \frac{7 P_{20} k_{on}}{6} - \frac{P_{21} k_{coop}}{8} + \frac{P_{21} k_{off}}{8} + \frac{P_{22} k_{coop}}{16} - \frac{21 P_{22} k_{off}}{8} - \frac{5 P_{22} k_{on}}{4} + \frac{5 P_{23} k_{coop}}{8} + \frac{P_{23} k_{off}}{4} - \frac{P_{23} k_{on}}{2} - 3 P_{24} k_{coop} + 2 P_{24} k_{off} - \frac{3 P_{27} k_{coop}}{4} + \frac{3 P_{28} k_{coop}}{8} - \frac{7 P_{28} k_{off}}{4} + \frac{P_{29} k_{coop}}{4} - 2 P_{29} k_{off} + 6 P_{3} k_{coop} - 7 P_{3} k_{off} - \frac{P_{30} k_{coop}}{8} + \frac{P_{30} k_{off}}{4} - \frac{P_{31} k_{coop}}{4} + \frac{P_{31} k_{off}}{2} - \frac{5 P_{32} k_{coop}}{8} + \frac{3 P_{32} k_{off}}{4} - 2 P_{33} k_{off} - P_{33} k_{on} + \frac{P_{35} k_{coop}}{4} - \frac{3 P_{35} k_{off}}{2} - \frac{P_{36} k_{coop}}{4} - \frac{P_{36} k_{off}}{2} + P_{38} k_{off} - 3 P_{39} k_{off} - \frac{57 P_{4} k_{coop}}{8} + \frac{29 P_{4} k_{off}}{4} - 4 P_{40} k_{coop} - 4 P_{40} k_{off} - 4 P_{40} k_{on} + \frac{3 P_{41} k_{off}}{13} + \frac{2 P_{42} k_{off}}{13} - P_{43} k_{off} + \frac{27 P_{5} k_{coop}}{8} - \frac{3 P_{5} k_{off}}{4} + 3 P_{5} k_{on} + \frac{11 P_{7} k_{coop}}{16} - \frac{15 P_{7} k_{off}}{8} - \frac{7 P_{7} k_{on}}{4} - \frac{33 P_{8} k_{coop}}{8} + \frac{13 P_{8} k_{off}}{4} - P_{8} k_{on} - \frac{51 P_{9} k_{coop}}{8} + \frac{11 P_{9} k_{off}}{4} - 3 P_{9} k_{on}

\frac{d P_{41}}{d t} = - 3 P_{0} k_{coop} - 198 P_{0} k_{off} - 29 P_{0} k_{on} + \frac{61 P_{11} k_{coop}}{4} - \frac{103 P_{11} k_{off}}{2} + \frac{51 P_{11} k_{on}}{4} - \frac{P_{12} k_{coop}}{2} + 9 P_{12} k_{off} - \frac{3 P_{12} k_{on}}{2} - \frac{113 P_{13} k_{coop}}{2} + \frac{1912 P_{13} k_{off}}{3} - 100 P_{13} k_{on} - 60 P_{14} k_{coop} + 370 P_{14} k_{off} - 74 P_{14} k_{on} - 180 P_{15} k_{coop} + 1194 P_{15} k_{off} - 222 P_{15} k_{on} + 63 P_{16} k_{coop} - \frac{1654 P_{16} k_{off}}{3} + 83 P_{16} k_{on} + 57 P_{18} k_{coop} - \frac{1687 P_{18} k_{off}}{3} + \frac{213 P_{18} k_{on}}{2} - \frac{163 P_{19} k_{coop}}{12} + \frac{3113 P_{19} k_{off}}{18} - \frac{121 P_{19} k_{on}}{4} + 63 P_{2} k_{coop} - 198 P_{2} k_{off} + 103 P_{2} k_{on} + \frac{50 P_{20} k_{coop}}{3} - \frac{4031 P_{20} k_{off}}{36} + \frac{199 P_{20} k_{on}}{8} - \frac{5 P_{21} k_{coop}}{2} + \frac{269 P_{21} k_{off}}{12} - \frac{23 P_{21} k_{on}}{8} + \frac{61 P_{22} k_{coop}}{4} - \frac{103 P_{22} k_{off}}{2} + \frac{51 P_{22} k_{on}}{4} + \frac{33 P_{23} k_{coop}}{2} + \frac{98 P_{23} k_{off}}{3} + 10 P_{23} k_{on} - 60 P_{24} k_{coop} + 396 P_{24} k_{off} - 74 P_{24} k_{on} + 84 P_{26} k_{off} + 5 P_{26} k_{on} - 3 P_{27} k_{coop} - \frac{577 P_{27} k_{off}}{3} + \frac{65 P_{27} k_{on}}{2} + \frac{11 P_{28} k_{coop}}{2} - \frac{557 P_{28} k_{off}}{3} + \frac{43 P_{28} k_{on}}{2} + P_{29} k_{coop} - \frac{845 P_{29} k_{off}}{3} + \frac{99 P_{29} k_{on}}{2} + 120 P_{3} k_{coop} - 798 P_{3} k_{off} + 148 P_{3} k_{on} - \frac{P_{30} k_{coop}}{2} + 9 P_{30} k_{off} - \frac{3 P_{30} k_{on}}{2} + 3 P_{31} k_{coop} - \frac{248 P_{31} k_{off}}{3} + 37 P_{31} k_{on} + \frac{31 P_{32} k_{coop}}{2} + \frac{122 P_{32} k_{off}}{3} + 8 P_{32} k_{on} + 4 P_{33} k_{coop} - \frac{346 P_{33} k_{off}}{3} + 17 P_{33} k_{on} + \frac{26 P_{34} k_{off}}{3} + P_{35} k_{coop} + 8 P_{35} k_{off} + 3 P_{35} k_{on} + 7 P_{36} k_{coop} - 22 P_{36} k_{off} + 84 P_{37} k_{off} + 24 P_{38} k_{coop} - 14 P_{38} k_{off} + 19 P_{38} k_{on} - 42 P_{39} k_{coop} - 26 P_{39} k_{off} - 37 P_{39} k_{on} - \frac{249 P_{4} k_{coop}}{2} + \frac{2545 P_{4} k_{off}}{3} - \frac{343 P_{4} k_{on}}{2} - 16 P_{40} k_{coop} - 20 P_{40} k_{off} + 2 P_{40} k_{on} - \frac{48 P_{41} k_{coop}}{13} - 4 P_{41} k_{off} - \frac{24 P_{41} k_{on}}{13} + \frac{20 P_{42} k_{coop}}{13} + 4 P_{42} k_{off} + \frac{10 P_{42} k_{on}}{13} + 26 P_{43} k_{off} - 7 P_{45} k_{off} - 20 P_{46} k_{off} + \frac{3 P_{5} k_{coop}}{2} + 99 P_{5} k_{off} + \frac{29 P_{5} k_{on}}{2} + \frac{191 P_{7} k_{coop}}{4} - \frac{2999 P_{7} k_{off}}{6} + \frac{281 P_{7} k_{on}}{4} - \frac{121 P_{8} k_{coop}}{2} + 337 P_{8} k_{off} - \frac{161 P_{8} k_{on}}{2} - \frac{123 P_{9} k_{coop}}{2} + 297 P_{9} k_{off} - \frac{177 P_{9} k_{on}}{2}

\frac{d P_{42}}{d t} = \frac{9 P_{0} k_{coop}}{2} - 15 P_{0} k_{off} + \frac{31 P_{0} k_{on}}{4} - \frac{27 P_{11} k_{coop}}{8} - \frac{3 P_{11} k_{off}}{4} - \frac{33 P_{11} k_{on}}{16} + \frac{3 P_{12} k_{coop}}{4} - \frac{P_{12} k_{off}}{2} + \frac{5 P_{12} k_{on}}{8} + \frac{79 P_{13} k_{coop}}{4} - \frac{P_{13} k_{off}}{2} + \frac{121 P_{13} k_{on}}{8} + 12 P_{14} k_{coop} + 4 P_{14} k_{off} + 7 P_{14} k_{on} + 36 P_{15} k_{coop} + 16 P_{15} k_{off} + 21 P_{15} k_{on} - \frac{33 P_{16} k_{coop}}{2} - 5 P_{16} k_{off} - \frac{43 P_{16} k_{on}}{4} - \frac{41 P_{18} k_{coop}}{2} + 5 P_{18} k_{off} - \frac{67 P_{18} k_{on}}{4} + \frac{125 P_{19} k_{coop}}{24} - \frac{19 P_{19} k_{off}}{12} + \frac{215 P_{19} k_{on}}{48} - \frac{33 P_{2} k_{coop}}{2} + 11 P_{2} k_{off} - \frac{59 P_{2} k_{on}}{4} - \frac{53 P_{20} k_{coop}}{12} + \frac{5 P_{20} k_{off}}{3} - \frac{83 P_{20} k_{on}}{24} + \frac{P_{21} k_{coop}}{2} + \frac{P_{21} k_{off}}{2} + \frac{P_{21} k_{on}}{4} - \frac{27 P_{22} k_{coop}}{8} - \frac{3 P_{22} k_{off}}{4} - \frac{33 P_{22} k_{on}}{16} + \frac{5 P_{23} k_{coop}}{4} - \frac{7 P_{23} k_{off}}{2} + \frac{23 P_{23} k_{on}}{8} + 12 P_{24} k_{coop} + 4 P_{24} k_{off} + 7 P_{24} k_{on} + 4 P_{26} k_{off} - P_{26} k_{on} - \frac{17 P_{27} k_{coop}}{2} + 9 P_{27} k_{off} - \frac{39 P_{27} k_{on}}{4} - \frac{33 P_{28} k_{coop}}{4} + \frac{11 P_{28} k_{off}}{2} - \frac{63 P_{28} k_{on}}{8} - \frac{29 P_{29} k_{coop}}{2} + 13 P_{29} k_{off} - \frac{63 P_{29} k_{on}}{4} - 24 P_{3} k_{coop} - 12 P_{3} k_{off} - 14 P_{3} k_{on} + \frac{3 P_{30} k_{coop}}{4} - \frac{P_{30} k_{off}}{2} + \frac{5 P_{30} k_{on}}{8} - \frac{9 P_{31} k_{coop}}{2} + 7 P_{31} k_{off} - \frac{27 P_{31} k_{on}}{4} + \frac{11 P_{32} k_{coop}}{4} - \frac{5 P_{32} k_{off}}{2} + \frac{21 P_{32} k_{on}}{8} - 6 P_{33} k_{coop} + 4 P_{33} k_{off} - 6 P_{33} k_{on} - \frac{3 P_{35} k_{coop}}{2} + P_{35} k_{off} - \frac{5 P_{35} k_{on}}{4} - \frac{21 P_{36} k_{coop}}{2} + 7 P_{36} k_{off} - \frac{39 P_{36} k_{on}}{4} + 4 P_{37} k_{off} - 10 P_{38} k_{coop} + 8 P_{38} k_{off} - 9 P_{38} k_{on} - 2 P_{39} k_{coop} - 3 P_{39} k_{on} + \frac{123 P_{4} k_{coop}}{4} + \frac{3 P_{4} k_{off}}{2} + \frac{173 P_{4} k_{on}}{8} - 2 P_{40} k_{coop} + 4 P_{40} k_{off} - 3 P_{40} k_{on} - \frac{6 P_{41} k_{coop}}{13} - \frac{3 P_{41} k_{on}}{13} - \frac{82 P_{42} k_{coop}}{13} - 10 P_{42} k_{off} - \frac{41 P_{42} k_{on}}{13} + 4 P_{45} k_{off} + 4 P_{46} k_{off} - \frac{9 P_{5} k_{coop}}{4} + \frac{15 P_{5} k_{off}}{2} - \frac{31 P_{5} k_{on}}{8} - \frac{105 P_{7} k_{coop}}{8} - \frac{17 P_{7} k_{off}}{4} - \frac{139 P_{7} k_{on}}{16} + \frac{51 P_{8} k_{coop}}{4} + \frac{3 P_{8} k_{off}}{2} + \frac{69 P_{8} k_{on}}{8} + \frac{57 P_{9} k_{coop}}{4} - \frac{7 P_{9} k_{off}}{2} + \frac{87 P_{9} k_{on}}{8}

\frac{d P_{43}}{d t} = 2 P_{33} k_{coop} + 2 P_{33} k_{on} + 2 P_{35} k_{coop} + P_{35} k_{on} - 2 P_{43} k_{coop} - 3 P_{43} k_{off} - 2 P_{43} k_{on} + 2 P_{46} k_{off}

\frac{d P_{44}}{d t} = 8 P_{37} k_{coop} + 4 P_{37} k_{on} - 4 P_{44} k_{off}

\frac{d P_{45}}{d t} = - 45 P_{0} k_{coop} - \frac{27 P_{0} k_{off}}{2} - \frac{43 P_{0} k_{on}}{2} - \frac{21 P_{11} k_{coop}}{4} - \frac{31 P_{11} k_{off}}{8} + \frac{5 P_{11} k_{on}}{8} + \frac{P_{12} k_{coop}}{2} + \frac{3 P_{12} k_{off}}{4} - \frac{P_{12} k_{on}}{4} + \frac{487 P_{13} k_{coop}}{6} + \frac{589 P_{13} k_{off}}{12} + \frac{271 P_{13} k_{on}}{12} + 48 P_{14} k_{coop} + 28 P_{14} k_{off} + 10 P_{14} k_{on} + 160 P_{15} k_{coop} + 90 P_{15} k_{off} + 38 P_{15} k_{on} - \frac{221 P_{16} k_{coop}}{3} - \frac{251 P_{16} k_{off}}{6} - \frac{119 P_{16} k_{on}}{6} - \frac{197 P_{18} k_{coop}}{3} - \frac{263 P_{18} k_{off}}{6} - \frac{46 P_{18} k_{on}}{3} + \frac{761 P_{19} k_{coop}}{36} + \frac{971 P_{19} k_{off}}{72} + \frac{443 P_{19} k_{on}}{72} - 7 P_{2} k_{coop} - \frac{33 P_{2} k_{off}}{2} + \frac{19 P_{2} k_{on}}{2} - \frac{215 P_{20} k_{coop}}{18} - \frac{317 P_{20} k_{off}}{36} - \frac{145 P_{20} k_{on}}{72} + \frac{10 P_{21} k_{coop}}{3} + \frac{5 P_{21} k_{off}}{3} + \frac{25 P_{21} k_{on}}{24} - \frac{21 P_{22} k_{coop}}{4} - \frac{31 P_{22} k_{off}}{8} + \frac{5 P_{22} k_{on}}{8} + \frac{17 P_{23} k_{coop}}{6} + \frac{35 P_{23} k_{off}}{12} + \frac{41 P_{23} k_{on}}{12} + 52 P_{24} k_{coop} + 30 P_{24} k_{off} + 12 P_{24} k_{on} + 16 P_{26} k_{coop} + 6 P_{26} k_{off} + 7 P_{26} k_{on} - \frac{53 P_{27} k_{coop}}{3} - \frac{95 P_{27} k_{off}}{6} - \frac{16 P_{27} k_{on}}{3} - \frac{113 P_{28} k_{coop}}{6} - \frac{179 P_{28} k_{off}}{12} - \frac{59 P_{28} k_{on}}{12} - \frac{73 P_{29} k_{coop}}{3} - \frac{139 P_{29} k_{off}}{6} - \frac{17 P_{29} k_{on}}{3} - 108 P_{3} k_{coop} - 60 P_{3} k_{off} - 26 P_{3} k_{on} + \frac{P_{30} k_{coop}}{2} + \frac{3 P_{30} k_{off}}{4} - \frac{P_{30} k_{on}}{4} - \frac{13 P_{31} k_{coop}}{3} - \frac{43 P_{31} k_{off}}{6} - \frac{7 P_{31} k_{on}}{6} + \frac{23 P_{32} k_{coop}}{6} + \frac{41 P_{32} k_{off}}{12} + \frac{11 P_{32} k_{on}}{12} - \frac{32 P_{33} k_{coop}}{3} - \frac{28 P_{33} k_{off}}{3} - \frac{7 P_{33} k_{on}}{3} + \frac{4 P_{34} k_{coop}}{3} + \frac{2 P_{34} k_{off}}{3} + \frac{2 P_{34} k_{on}}{3} + 3 P_{35} k_{coop} + \frac{P_{35} k_{off}}{2} + \frac{5 P_{35} k_{on}}{2} + P_{36} k_{coop} - \frac{5 P_{36} k_{off}}{2} + \frac{P_{36} k_{on}}{2} + 16 P_{37} k_{coop} + 6 P_{37} k_{off} + 8 P_{37} k_{on} + 8 P_{38} k_{coop} - 2 P_{38} k_{off} + 5 P_{38} k_{on} - 14 P_{39} k_{coop} - 2 P_{39} k_{off} - 9 P_{39} k_{on} + \frac{619 P_{4} k_{coop}}{6} + \frac{781 P_{4} k_{off}}{12} + \frac{253 P_{4} k_{on}}{12} - 2 P_{40} k_{off} + \frac{12 P_{41} k_{coop}}{13} + \frac{2 P_{41} k_{off}}{13} + \frac{6 P_{41} k_{on}}{13} + \frac{8 P_{42} k_{coop}}{13} + \frac{10 P_{42} k_{off}}{13} + \frac{4 P_{42} k_{on}}{13} + 4 P_{43} k_{coop} + 2 P_{43} k_{off} + 2 P_{43} k_{on} - 8 P_{45} k_{coop} - 7 P_{45} k_{off} - 4 P_{45} k_{on} - 2 P_{46} k_{off} + \frac{45 P_{5} k_{coop}}{2} + \frac{27 P_{5} k_{off}}{4} + \frac{43 P_{5} k_{on}}{4} - \frac{821 P_{7} k_{coop}}{12} - \frac{911 P_{7} k_{off}}{24} - \frac{491 P_{7} k_{on}}{24} + \frac{81 P_{8} k_{coop}}{2} + \frac{103 P_{8} k_{off}}{4} + \frac{27 P_{8} k_{on}}{4} + \frac{59 P_{9} k_{coop}}{2} + \frac{93 P_{9} k_{off}}{4} + \frac{5 P_{9} k_{on}}{4}

\frac{d P_{46}}{d t} = 42 P_{0} k_{coop} + 21 P_{0} k_{on} + \frac{13 P_{11} k_{coop}}{2} + \frac{13 P_{11} k_{on}}{4} - P_{12} k_{coop} - \frac{P_{12} k_{on}}{2} - \frac{275 P_{13} k_{coop}}{3} - \frac{275 P_{13} k_{on}}{6} - 52 P_{14} k_{coop} - 26 P_{14} k_{on} - 172 P_{15} k_{coop} - 86 P_{15} k_{on} + \frac{242 P_{16} k_{coop}}{3} + \frac{121 P_{16} k_{on}}{3} + \frac{230 P_{18} k_{coop}}{3} + \frac{115 P_{18} k_{on}}{3} - \frac{433 P_{19} k_{coop}}{18} - \frac{433 P_{19} k_{on}}{36} + 14 P_{2} k_{coop} + 7 P_{2} k_{on} + \frac{133 P_{20} k_{coop}}{9} + \frac{133 P_{20} k_{on}}{18} - \frac{10 P_{21} k_{coop}}{3} - \frac{5 P_{21} k_{on}}{3} + \frac{13 P_{22} k_{coop}}{2} + \frac{13 P_{22} k_{on}}{4} - \frac{13 P_{23} k_{coop}}{3} - \frac{13 P_{23} k_{on}}{6} - 56 P_{24} k_{coop} - 28 P_{24} k_{on} - 16 P_{26} k_{coop} - 8 P_{26} k_{on} + \frac{74 P_{27} k_{coop}}{3} + \frac{37 P_{27} k_{on}}{3} + \frac{73 P_{28} k_{coop}}{3} + \frac{73 P_{28} k_{on}}{6} + \frac{106 P_{29} k_{coop}}{3} + \frac{53 P_{29} k_{on}}{3} + 116 P_{3} k_{coop} + 58 P_{3} k_{on} - P_{30} k_{coop} - \frac{P_{30} k_{on}}{2} + \frac{22 P_{31} k_{coop}}{3} + \frac{11 P_{31} k_{on}}{3} - \frac{19 P_{32} k_{coop}}{3} - \frac{19 P_{32} k_{on}}{6} + \frac{44 P_{33} k_{coop}}{3} + \frac{22 P_{33} k_{on}}{3} - \frac{4 P_{34} k_{coop}}{3} - \frac{2 P_{34} k_{on}}{3} - 2 P_{35} k_{coop} - P_{35} k_{on} + 2 P_{36} k_{coop} + P_{36} k_{on} - 16 P_{37} k_{coop} - 8 P_{37} k_{on} - 4 P_{38} k_{coop} - 2 P_{38} k_{on} + 12 P_{39} k_{coop} + 6 P_{39} k_{on} - \frac{347 P_{4} k_{coop}}{3} - \frac{347 P_{4} k_{on}}{6} - \frac{12 P_{41} k_{coop}}{13} - \frac{6 P_{41} k_{on}}{13} - \frac{8 P_{42} k_{coop}}{13} - \frac{4 P_{42} k_{on}}{13} - 2 P_{43} k_{coop} - 2 P_{46} k_{coop} - 4 P_{46} k_{off} - P_{46} k_{on} + P_{47} k_{off} - 21 P_{5} k_{coop} - \frac{21 P_{5} k_{on}}{2} + \frac{445 P_{7} k_{coop}}{6} + \frac{445 P_{7} k_{on}}{12} - 45 P_{8} k_{coop} - \frac{45 P_{8} k_{on}}{2} - 35 P_{9} k_{coop} - \frac{35 P_{9} k_{on}}{2}

\frac{d P_{47}}{d t} = 10 P_{46} k_{coop} + 5 P_{46} k_{on} - 5 P_{47} k_{off}

Equations as text

48 coordinates, rank 47

The system retains one redundant coordinate on every allowed topology.

Count identity

52260*P0 + 7605*P11 - 1170*P12 - 107250*P13 - 59280*P14 - 202800*P15 + 94380*P16 + 936*P17 + 89700*P18 - 28145*P19 + 13260*P2 + 17290*P20 - 3900*P21 + 12285*P22 - 5070*P23 - 65520*P24 + 1560*P25 - 18720*P26 + 28860*P27 + 28470*P28 + 41340*P29 + 132600*P3 - 1170*P30 + 8580*P31 - 7410*P32 + 21840*P33 - 1560*P34 + 2340*P35 + 2340*P36 - 18720*P37 - 4680*P38 + 14040*P39 - 133770*P4 - 1080*P41 - 720*P42 + 4680*P46 + 936*P47 - 27690*P5 - 1560*P6 + 86775*P7 - 51090*P8 - 44070*P9 = 0

Polynomial jump closure (pJC) and EMGF

Finite event model. Removing one redundant coordinate leaves 47 independent counts. Rule applications on all 83 topology/occupancy classes generate 247 coordinate-jump classes.

pJCCLOSEDcomplete finite language

Independent lift4748 generated coordinates minus one identity

Function algebra8383 state signatures

Operator span / order182 / 1nested commutators / EMGF

D = \sum_{e} k_{e} (e^{λ δ_{e}} - 1) a_{e} (\partial_{λ})

Derived event classes · grouped view

The 247 exact event classes are grouped by rule and rate.

Rule / rate	Classes	Distinct δ	Max order	Example class IDs
`basal_bind` basal_bind	91	91	1	`basal_bind-jump-1, basal_bind-jump-2, …`
`cooperative_bind` cooperative_bind	65	65	1	`cooperative_bind-jump-1, cooperative_bind-jump-2, …`
`unbind` unbind	91	91	1	`unbind-jump-1, unbind-jump-2, …`

5 · Initial states

The same ODE is used in all three experiments. Only the initial vector changes: yᵢ(0) = evaluation of the exported count expression on the chosen initial graph. All eight slots are free in all three examples.

Initial count	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0

Every unlisted coordinate is initially zero. Coordinates distinguish component topologies through their graph-count expressions. Thus P0(0)=0 and P1(0)=8 alone do not specify the complete ODE initial condition.

Complete three 48-dimensional initial vectors

Coordinate	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0

A probability distribution over allowed graphs gives the weighted average of their vectors. A 4+4 graph is one allowed two-ring state; it is not a probability mixture of two alternative four-node systems.

6 · Dynamics

Numerical solutions of the exported ODE. Adjust every reaction rate independently. Each change reintegrates the saved ODE rather than rescaling a stored curve.

Rate program Initial experiment Compare all three starts

Rates from t=0Basal binding · k_on 0.08Cooperative binding · k_coop 1.2Unbinding · k_off 0.4

Time horizon 30

All ODE coordinates for the selected scenario

Rates are illustrative. Curves use numerical matrix exponentials at the selected rates.

Reference plots and parameters

Neighbour-enhanced binding

Static reference: k_on = 0.08, k_coop = 1.2, k_off = 0.4. The independent sliders control the interactive curves. All eight slots initially free in each topology.

In the static reference, occupancy starts with slope k_on × 8 = 0.64, then the curves separate as neighbour correlations develop differently. The interactive curves instead use your selected rates; their initial slope is 8 k_on. Rules never change topology. The reference replay agrees with the independent P0 samples within 2.13e-14.

Initial count	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0

Coordinate	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0

Initial count	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0

Coordinate	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0

Initial count	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0

Coordinate	8	4+4	3+5
P0	0	0	0
P1	8	8	8
P2	0	0	0
P3	0	0	0
P4	0	0	0
P5	0	0	0
P6	0	0	6
P7	0	0	0
P8	0	0	0
P9	0	0	0
P10	0	16	0
P11	0	0	0
P12	0	0	0
P13	0	0	0
P14	0	0	0
P15	0	0	0
P16	0	0	0
P17	0	0	10
P18	0	0	0
P19	0	0	0
P20	0	0	0
P21	0	0	0
P22	0	0	0
P23	0	0	0
P24	0	0	0
P25	0	0	0
P26	0	0	0
P27	0	0	0
P28	0	0	0
P29	0	0	0
P30	0	0	0
P31	0	0	0
P32	0	0	0
P33	0	0	0
P34	0	0	0
P35	0	0	0
P36	0	0	0
P37	0	0	0
P38	0	0	0
P39	0	0	0
P40	0	0	0
P41	0	0	0
P42	0	0	0
P43	0	0	0
P44	0	0	0
P45	0	0	0
P46	0	0	0
P47	0	0	0