12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bond graph | 3/11 | https://en.wikipedia.org/wiki/Bond_graph | reference | science, encyclopedia | 2026-05-05T14:13:40.582713+00:00 | kb-cron |
For example, a frictionless, massless piston of area
A
{\displaystyle A}
converts hydraulic power to mechanical power so that the hydraulic pressure
P
{\displaystyle P}
is related to mechanical force
F
{\displaystyle F}
by:
F
=
A
P
{\displaystyle F=AP}
and hydraulic flow
V
{\displaystyle V}
to piston velocity
v
{\displaystyle v}
by:
V
=
A
v
{\displaystyle V=Av}
Thus the bond graph analogy is the TF component with modulus
m
=
A
{\displaystyle m=A}
where
e
1
=
P
,
f
1
=
V
{\displaystyle e_{1}=P,~~f_{1}=V}
and
e
2
=
F
,
f
2
=
v
{\displaystyle e_{2}=F,~~f_{2}=v}
. For example, an ideal DC motor converts electrical power into rotational mechanical power so that the mechanical torque
T
{\displaystyle T}
is related to the electrical current
i
{\displaystyle i}
by:
T
=
k
i
{\displaystyle T=ki}
and back EMF (voltage)
E
{\displaystyle E}
to angular velocity
Ω
{\displaystyle \Omega }
by
E
=
k
Ω
{\displaystyle E=k\Omega }
Thus the bond graph analogy is the GY component with modulus
m
=
k
{\displaystyle m=k}
where
e
1
=
E
,
f
1
=
i
{\displaystyle e_{1}=E,~~f_{1}=i}
and
e
2
=
T
,
f
2
=
Ω
{\displaystyle e_{2}=T,~~f_{2}=\Omega }
. In both cases, non-ideal transduction behaviour can be modelled by including C ,I and R bond graph components in the model.
== Bond graphs in systems biology == Bond graphs have been used to model systems relevant to the life sciences, including physiology and biology. In particular, the use of bond graphs to model biophysical systems was introduced by Aharon Katchalsky, George Oster, and Alan Perelson in the early 1970s. More recently, these ideas were used in the context of Systems Biology to provide an energy-based approach to modelling the biochemical reaction systems of cellular biology and to modelling the entire physiome. The bond graph approach has a number of features which make it a good basis for building large computational models of the physiome.
It is energy based, which implies that: the models are physically-plausible detailed balance (Wegscheider's conditions) for reaction kinetics are automatically satisfied energy flow, usage and dissipation can be directly considered It is modular: bond graph components can themselves be bond graphs Energy transduction between physical domains is simply represented - see below Symbolic code, which may be used for simulation, can be automatically generated
=== Variables === The bond graph variables for biochemical systems are:
Displacement: quantity of chemical species measured in moles, symbol
x
{\displaystyle x}
(mol) Flow: rate of change of chemical species, symbol
v
{\displaystyle v}
(mol/s) Effort: chemical potential, or Gibbs energy, per mole of a chemical species, symbol
μ
{\displaystyle \mu }
(J/mol) Note that the product of effort and flow (μv) is, as always in the bond graph formulation, power (J/s).
=== Components === As detailed below, the main features of the components used to model biochemical systems are: the R and C components are nonlinear, there is no I component required and the R component is replaced by a two-port Re component.
==== Junction components ==== The bond graph zero (0) and one (1) components are no different in this context.
==== C component ==== The C component integrates the flow
v
{\displaystyle v}
to give the amount
x
{\displaystyle x}
of species:
x
(
t
)
=
∫
t
v
(
τ
)
d
τ
{\displaystyle x(t)=\int ^{t}v(\tau )d\tau }
The effort, chemical potential
μ
{\displaystyle \mu }
, is given by the formula:
μ
=
μ
0
+
R
T
ln
x
x
0
{\displaystyle \mu =\mu ^{0}+RT\ln {\frac {x}{x^{0}}}}
where
μ
0
{\displaystyle \mu ^{0}}
is the chemical potential corresponding to
x
=
x
0
{\displaystyle x=x^{0}}
,
R
{\displaystyle R}
is the gas constant and
T
{\displaystyle T}
is the absolute temperature in degrees Kelvin. The formula for
μ
{\displaystyle \mu }
can be rewritten in a simplified form as:
μ
=
R
T
ln
K
x
{\displaystyle \mu =RT\ln Kx}
where
K
=
1
x
0
exp
μ
0
R
T
{\displaystyle K={\frac {1}{x^{0}}}\exp {\frac {\mu ^{0}}{RT}}}
Because of the special form of this particular C component it is sometimes given a special name Ce analogously to the special Re component.
==== Re component ==== The Re (reaction) component has two energy ports corresponding to the left (forward) and right (reverse) sides of a chemical reaction. The forward
A
f
{\displaystyle A^{f}}
and reverse
A
r
{\displaystyle A^{r}}
affinities are defined as the net chemical potential due to the species on the left and right sides of the reaction respectively. The Re component then gives the reaction flow
v
{\displaystyle v}
as:
v
=
κ
(
exp
A
f
R
T
−
exp
A
r
R
T
)
{\displaystyle v=\kappa \left(\exp {\frac {A^{f}}{RT}}-\exp {\frac {A^{r}}{RT}}\right)}
where
κ
{\displaystyle \kappa }
(mol/s) is a rate constant. Note that it is not possible to use the usual R component with 1-junction formulation as the flow depends on both the forward
A
f
{\displaystyle A^{f}}
and reverse
A
r
{\displaystyle A^{r}}
affinities rather than the difference
A
f
−
A
r
{\displaystyle A^{f}-A^{r}}
.
=== Modelling simple reactions ===
==== Reaction ====
A
↽
−
−
⇀
B
{\displaystyle {\ce {A <=> B}}}
Sources:
The simple reaction
A
↽
−
−
⇀
B
{\displaystyle {\ce {A <=> B}}}
is represented by three components:
C:A represents the species A with chemical potential
μ
A
{\displaystyle \mu _{A}}
; the flow is
−
v
{\displaystyle -v}
. C:B represents the species B with chemical potential
μ
B
{\displaystyle \mu _{B}}
; the flow is
v
{\displaystyle v}
. Re_r1 represents the reaction with flow
v
{\displaystyle v}
, forward affinity
A
f
=
μ
A
{\displaystyle A^{f}=\mu _{A}}
and reverse affinity
A
r
=
μ
B
{\displaystyle A^{r}=\mu _{B}}
. The bonds and junctions transfer chemical energy with effort and flow variables indicated. Using the above equations, the flow
v
{\displaystyle v}
is given by