The structure of infinitesimal homeostasis in input–output networks
Homeostasis refers to a phenomenon whereby the output x o of a system is approximately constant on variation of an input I . Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to...
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| Vydáno v: | Journal of mathematical biology Ročník 82; číslo 7; s. 62 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.06.2021
Springer Nature B.V |
| Témata: | |
| ISSN: | 0303-6812, 1432-1416, 1432-1416 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Homeostasis refers to a phenomenon whereby the output
x
o
of a system is approximately constant on variation of an input
I
. Homeostasis occurs frequently in biochemical networks and in other networks of interacting elements where mathematical models are based on differential equations associated to the network. These networks can be abstracted as digraphs
G
with a distinguished input node
ι
, a different distinguished output node
o
, and a number of regulatory nodes
ρ
1
,
…
,
ρ
n
. In these models the input–output map
x
o
(
I
)
is defined by a stable equilibrium
X
0
at
I
0
. Stability implies that there is a stable equilibrium
X
(
I
)
for each
I
near
I
0
and infinitesimal homeostasis occurs at
I
0
when
(
d
x
o
/
d
I
)
(
I
0
)
=
0
. We show that there is an
(
n
+
1
)
×
(
n
+
1
)
homeostasis matrix
H
(
I
)
for which
d
x
o
/
d
I
=
0
if and only if
det
(
H
)
=
0
. We note that the entries in
H
are linearized couplings and
det
(
H
)
is a homogeneous polynomial of degree
n
+
1
in these entries. We use combinatorial matrix theory to factor the polynomial
det
(
H
)
and thereby determine a menu of different types of possible homeostasis associated with each digraph
G
. Specifically, we prove that each factor corresponds to a subnetwork of
G
. The factors divide into two combinatorially defined classes:
structural
and
appendage
. Structural factors correspond to
feedforward
motifs and appendage factors correspond to
feedback
motifs. Finally, we discover an algorithm for determining the homeostasis subnetwork motif corresponding to each factor of
det
(
H
)
without performing numerical simulations on model equations. The algorithm allows us to classify low degree factors of
det
(
H
)
. There are two types of degree 1 homeostasis (negative feedback loops and kinetic or Haldane motifs) and there are two types of degree 2 homeostasis (feedforward loops and a degree two appendage motif). |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 content type line 23 |
| ISSN: | 0303-6812 1432-1416 1432-1416 |
| DOI: | 10.1007/s00285-021-01614-1 |