14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bond graph | 4/11 | https://en.wikipedia.org/wiki/Bond_graph | reference | science, encyclopedia | 2026-05-05T14:13:40.582713+00:00 | kb-cron |
v
=
κ
(
exp
μ
A
R
T
−
exp
μ
B
R
T
)
=
κ
(
K
A
x
A
−
K
B
x
B
)
{\displaystyle v=\kappa \left(\exp {\frac {\mu _{A}}{RT}}-\exp {\frac {\mu _{B}}{RT}}\right)=\kappa \left(K_{A}x_{A}-K_{B}x_{B}\right)}
where the subscripts correspond to the species. This is the simple mass-action equation:
v
=
k
+
x
A
−
k
−
x
B
{\displaystyle v=k^{+}x_{A}-k^{-}x_{B}}
where
k
+
=
κ
K
A
and
k
−
=
κ
K
B
{\displaystyle k^{+}=\kappa K_{A}{\text{ and }}k^{-}=\kappa K_{B}}
.
==== Enzyme-catalysed reaction ====
As discussed in section 1.4 of Keener & Sneyd, an enzyme-catalysed reaction reversibly transforming species
A
{\displaystyle {\ce {A}}}
to species
B
{\displaystyle {\ce {B}}}
via enzyme
E
{\displaystyle {\ce {E}}}
and enzyme complex
C
{\displaystyle {\ce {C}}}
can be written as the pair of reactions:
A
+
E
↽
−
−
⇀
C
↽
−
−
⇀
B
+
E
{\displaystyle {\ce {A + E <=> C <=> B + E}}}
The enzyme complex
C
{\displaystyle {\ce {C}}}
is formed from
A
+
E
{\displaystyle {\ce {A + E}}}
and decomposes into the species
B
{\displaystyle {\ce {B}}}
and releases enzyme
E
{\displaystyle {\ce {E}}}
. The bond graph shown in the figure shows how the enzyme is recycled. The bond graph can be used to derive the properties of these reactions which are of generalised Michaelis-Menten form.
=== Energy transduction ===
The bond graph TF (transformer) component represents energy transduction either within or between energy domains. (Note that the TF component has been called the TD (transduction) component - TF is more widely used.) This section focuses on transduction between the chemical domain with effort
μ
{\displaystyle \mu }
(J/mol) and flow
v
{\displaystyle v}
(mol/s) and a generic domain with effort
e
{\displaystyle e}
and flow
f
{\displaystyle f}
. The key feature of the TF component is that it transmits energy without dissipation; hence, with reference to the figure:
e
f
=
μ
v
{\displaystyle ef=\mu v}
The transformer has a modulus m (with appropriate units) so that:
f
=
m
v
{\displaystyle f=mv}
the energy formula then implies that:
μ
=
m
e
{\displaystyle \mu =me}
==== Stoichiometry ====
The stoichiometry of a chemical reaction determines how many of each chemical species occurs. Thus, for example, the reaction
A
↽
−
−
⇀
m
B
{\displaystyle {\ce {A <=> m B}}}
converts one mol of species
A
{\displaystyle {\ce {A}}}
to m mol of species
B
{\displaystyle {\ce {B}}}
. The case where
m
=
1
{\displaystyle m=1}
corresponds to the simple reaction of the first example above. Using the same approach for general
m
{\displaystyle m}
, the reaction flow is:
v
=
κ
(
exp
μ
A
R
T
−
exp
m
μ
B
R
T
)
=
κ
(
K
A
x
A
−
(
K
B
x
B
)
m
)
{\displaystyle v=\kappa \left(\exp {\frac {\mu _{A}}{RT}}-\exp {\frac {m\mu _{B}}{RT}}\right)=\kappa \left(K_{A}x_{A}-(K_{B}x_{B})^{m}\right)}
==== Chemoelectrical transduction ==== This section looks at the case where the generic domain is the electrical domain so that effort is (electrical) voltage
V
{\displaystyle {\mathcal {V}}}
(
e
=
V
{\displaystyle e={\mathcal {V}}}
) and the flow is current (
f
=
i
{\displaystyle f=i}
). Consider the flow
v
{\displaystyle v}
of charged ions where the charge on the molecule is
z
ϵ
{\displaystyle z\epsilon }
(Coulomb) where
ϵ
{\displaystyle \epsilon }
is the charge on the electron measured in Coulomb; the charge associated with a mole of ions is thus
z
ϵ
N
A
{\displaystyle z\epsilon N_{A}}
where
N
A
{\displaystyle N_{A}}
is the Avogadro constant. The equivalent current is then
i
=
z
ϵ
N
A
v
=
z
F
v
{\displaystyle i=z\epsilon N_{A}v=z{\mathcal {F}}v}
where
F
=
ϵ
N
A
{\displaystyle {\mathcal {F}}=\epsilon N_{A}}
is the Faraday constant; thus the corresponding TF modulus is:
m
=
z
F
{\displaystyle m=z{\mathcal {F}}}
(C/mol) Again, it follows that
μ
=
m
e
=
z
F
V
{\displaystyle \mu =me=z{\mathcal {F}}{\mathcal {V}}}
In this context, the bond graph TF component can be used to model energy flows associated with action potential, membrane transporters, cardiac action potential, and the mitochondrial electron transport chain.
==== Chemomechanical transduction ==== Consider a long rigid molecule such as actin where a sub unit of length
δ
{\displaystyle \delta }
(m) is added at a rate of
v
{\displaystyle v}
(mol/sec). Then the tip velocity
V
{\displaystyle V}
is given by:
V
=
δ
N
A
v
{\displaystyle V=\delta N_{A}v}
where
N
A
{\displaystyle N_{A}}
is the Avogadro constant. Thus the modulus
m
=
δ
N
A
{\displaystyle m=\delta N_{A}}
(m/mol) and
μ
=
m
F
=
δ
N
A
F
{\displaystyle \mu =mF=\delta N_{A}F}
where
F
{\displaystyle F}
is the corresponding force at the tip. These formulae have been used to generate force/velocity curves for actin filaments. The approach provides a useful alternative to the Brownian Ratchet approach as the bond graph TF component can be potentially used with modular bond graph models of cellular systems.
=== Applications === A number of systems relevant to systems biology have been modelled using bond graphs. These include:
Enterocyte homeostasis Glucose transport Cardiac Cellular Electrophysiological Modeling Cerebral Circulation Gene regulatory networks Actin filament polymerization Biochemical oscillators Photosynthesis Blood circulation Simplified E. coli Mitochondrial Electron Transport Chain Action potential
== Causality ==
Causality is a word with many uses and connotations. In the context of bond graphs, however, it has a limited, precise but important meaning and allows the bond graph model of a system to be converted to various other forms including a (nonlinear) state-space representation. The causality concept can also be used to examine structural properties, including inversion, of the system represented by a bond graph as well as to expose modelling errors.
=== R, C and I components ===