A bond graph is a graphical representation of the energy flows though and between physical dynamical systems including those in the electrical, mechanical, hydraulic, thermal and chemical domains. It is used to model and analyse systems relevant to engineering and to systems biology. Because the concept of energy is common to all physical domains, the bond graph provides a unified description of all of these energy domains and can be thought of as a systematic use of the physical analogies introduced by the 19th century scientists James Clerk-Maxwell and Lord Kelvin. The mechanical–electrical analogy is one example of a physical analogy. Bond graphs use the concept of analogous power conjugate variables whose product is energy flow, or power; these variable pairs are called effort and flow and, for example, correspond to voltage and current in the electrical domain and force and velocity in the mechanical domain. These power conjugate variables are transmitted by bonds which connect bond graph components. Bond graph components are also based on analogies and, using the electrical and mechanical domains as examples, include the C-component to represent both mechanical spring and electrical capacitor, the I-component to represent both a mechanical inertia and an electrical inductor and the R-component to represent both mechanical damper and electrical resistor. The electrical circuit notions of parallel and series connections are abstracted as 0-junctions and 1-junctions in bond graph terminology and again used as connection analogues for each physical domain. The bond graph transformer (TF) and gyrator (GY) components represent energy transformation within and between domains; thus an ideal gearbox in the rotational mechanical domain is represented by the TF component and an ideal DC motor transforming electrical into mechanical energy is represented by a GY component. Non-ideal transducers with flexibility, inertia and friction are modelled by including C, I and R components. The concept of causality in the context of bond graphs is used not only to generate system equations in a number of forms including ordinary differential equation (ode), state-space and differential-algebraic equations (dae) form suitable for simulation purposes but also to investigate dynamical system properties such as invertibility and zero dynamics. Causality can also be used to guide and correct modelling choices. The bond graph use of energy flows leads to the systematic construction of hierarchical models of large multi-domain systems; thus the bond graph method provides a basis for constructing large computer models, or digital twins, of multi domain physical systems including systems relevant not only to engineering but also to systems biology and the life sciences. The bond graph approach is related to the behavioral modelling approach of Jan C Willems, and the port-Hamiltonian approach of Arjan van der Schaft and B. M. Maschke. The bond graph method was originally proposed by Henry Paynter who applied the approach to engineering systems; the use of bond graphs to model biophysical systems was introduced by Aharon Katchalsky, George Oster, and Alan Perelson in the early 1970s.
Analogies The importance of analogies between physical domains was noted by Lord Kelvin and James Clerk-Maxwell. The bond graph can be thought of as a systematic approach to analogies. This section emphasises key features of bond graph analogies; more detail appears in a number of textbooks and tutorial papers. As an introduction to the key features of bond graphs, the figure shows the bond graph of two analogous systems: one electrical and one mechanical. A brief description is given here and expanded in the following sections.
The bond graph C-component represents either the electrical capacitor C or the mechanical spring K. The bond graph I-component represents either the electrical inductor L or the mechanical mass M. The bond graph R-component represents either the electrical resistor R or the mechanical damper D. The harpoon symbol is a bond transferring energy; the harpoon direction corresponds to positive energy flow and is a sign convention. The conjugate variables are displayed on each bond for the purposes of illustration. The 1-junction represents the series connection of the electrical circuit where the same current i flows in each component and the connection of the three components in the mechanical system which share a common velocity. Thus the bond graph components share the same flow f, but the efforts are different, corresponding to the voltages and forces of the electrical and mechanical components respectively. Energy is conserved at the junction by requiring that the three efforts add to zero. The bond graph uses three classes of analogy: analogies between variables, analogies between components and analogies between component connections; these are discussed in the following sections.
Analogies between variables
The bond graph use energy flow, or power, as the basis for abstracting analogies between different physical domains. Power conjugate variables are a pair of variables whose product is power and a list of these appears in the table for various physical domains. As indicated in the table, the bond graph uses the effort/flow analogy to categorise each of the two conjugate variables; the across/through analogy is also possible but not commonly used. The conventional bond graph symbol for effort is e, and that for flow is f. As well as the two conjugate power variables e and f, the bond graph uses two integrated variables p and q where:
p = ∫ t e ( τ ) d τ and q = ∫ t f ( τ ) d τ {\displaystyle p=\int ^{t}e(\tau )d\tau ~~~{\text{and}}~~~q=\int ^{t}f(\tau )d\tau }
equivalently:
p ˙ = d p d t = e and q ˙ = d q d t = f {\displaystyle {\dot {p}}={\frac {dp}{dt}}=e~~~{\text{and}}~~~{\dot {q}}={\frac {dq}{dt}}=f}
The harpoon symbol shown in the figure represents a power bond, or bond, transferring energy; the harpoon direction corresponds to positive energy flow and corresponds to a sign convention. The conjugate variables are displayed on each bond for the purposes of illustration.
Analogies between components
The definitions relating p to e and q to f are indicated diagrammatically. Physical properties are encapsulated in constitutive equations relating the energy and power variables. In the diagram, C represents the constitutive equation relating q and e, I represents the constitutive equation relating p and f and R represents the constitutive equation relating e and f. Instead of placing e, f, p and q on the four corners of a square as in the diagram, they can alternatively be placed at the four vertices of a tetrahedron; such a diagram is called the tetrahedron of state. These three constitutive equations Φ C , Φ I and Φ R {\displaystyle \Phi _{C},\Phi _{I}~{\text{and}}~\Phi _{R}} correspond to three components each relating e and f: two dynamic components, C and I, incorporating an integrator and the R-component. The C- and I- components store, but do not dissipate energy, R dissipates, but does not store energy. It is also possible to define a memristor component linking p and q, but this component is not commonly used.
Φ C , Φ I and Φ R {\displaystyle \Phi _{C},\Phi _{I}~{\text{and}}~\Phi _{R}} may be linear or nonlinear functions relating the variables as:
q = Φ C ( e ) , p = Φ I ( f ) and e = Φ R ( f ) {\displaystyle q=\Phi _{C}(e),~~p=\Phi _{I}(f)~~{\text{and}}~~e=\Phi _{R}(f)}
in the linear case:
q = C e , p = I f , e = R f {\displaystyle q=Ce,~~p=If,~~e=Rf}
where C , I , and R {\displaystyle C,~I,~{\text{and}}~R} are scalar constants representing generalised capacitance, inertia and resistance respectively. Because e and f are conjugate variables, they are carried on a single bond and therefore the three components C, I and R are connected to a single bond and thus have a single energy port though which energy flows. The components, and impinging bond are shown in the figure. By convention, bonds point into these three components.
Notation The colon (:) notation is sometimes used to refer to the components; thus, for example I:M, C:K and R:D could be used in the bond graph of the mass-spring-damper system to emphasise the link between the bond graph components and their physical analogues.
Analogies between connections Electrical circuit diagrams have two sorts of connection: parallel and series or common voltage and common current; they distribute, but do not store or dissipate energy. The bond graph analogy of the common voltage and common current connections are the 0-junction (common effort) and 1-junction (common flow) respectively; they both distribute, but do not store or dissipate energy. All bonds impinging on a 0-junctions have the same effort. As energy is distributed, not dissipated, it follows that the sum of energy inflows (indicated by bonds pointing in) must equal the sum of energy outflows (indicated by bonds pointing out). Hence, if the common effort is e, the power flow constraint implies that the sum of the m {\displaystyle m} inflows f i i n {\displaystyle f_{i}^{in}} must equal the sum of the n {\displaystyle n} outflows f j o u t {\displaystyle f_{j}^{out}} :
∑ i = 1 m f i i n = ∑ j = 1 n f j o u t {\displaystyle \sum _{i=1}^{m}f_{i}^{in}=\sum _{j=1}^{n}f_{j}^{out}}
Using the same argument, the efforts impinging on a 1-junction are constrained by:
∑ i = 1 m e i i n = ∑ j = 1 n e j o u t {\displaystyle \sum _{i=1}^{m}e_{i}^{in}=\sum _{j=1}^{n}e_{j}^{out}}
Analogies between external connections
Systems such as that the simple electrical and mechanical systems in the figure have no connection to the environment. Such connections may include external voltages and external forces (that is efforts) and external currents and external velocities (that is flows). Similarly, external measurements of efforts and flows are also important. For the purposes of building hierarchical system, it is convenient to define energy ports though which energy can flow between systems. These five possibilities correspond to five bond graph components:
S e {\displaystyle {\text{S}}_{e}} , an effort source analogous to applying external voltages and external forces
S f {\displaystyle {\text{S}}_{f}} , a flow source analogous to applying external currents and external velocities
D e {\displaystyle {\text{D}}_{e}} , an effort sensor (detector) analogous to measuring voltage and forces
D f {\displaystyle {\text{D}}_{f}} , a flow sensor (detector) analogous to measuring currents and velocities
SS {\displaystyle {\text{SS}}} , a source/sensor component which acts both as S e {\displaystyle {\text{S}}_{e}} - D f {\displaystyle {\text{D}}_{f}} and as S f {\displaystyle {\text{S}}_{f}} - D e {\displaystyle {\text{D}}_{e}} pairs as well as an energy port for external connections. For example, the mass-spring-damper system can be augmented with an force F {\displaystyle F} applied to the mass and a measurement of the mass velocity v {\displaystyle v} using a single SS components. The colon notation has been included; the SS:io refers to the fact that the force and velocity pair could be considered as system input and output.
Analogies between energy transducers
The conjugate effort and flow variables have different units in each energy domain thus models in one domain cannot be directly be connected by bonds to a different energy domain. However, because power has the same units (J/s or W) in each domain, the two-port power transducing components TF and GY can be used to provide such connections. As the two components transmit, but do not store or dissipate power, it follows that the power associated with the conjugate variables of the left-hand and right-hand bonds must be the same:
e 2 f 2 = e 1 f 1 {\displaystyle e_{2}f_{2}=e_{1}f_{1}}
Each component has a modulus m {\displaystyle m} associated with it. In the case of the TF component:
e 2 = m e 1 and f 1 = m f 2 {\displaystyle e_{2}=me_{1}~~{\text{and}}~~f_{1}=mf_{2}}
In the case of the GY component:
e 2 = m f 1 and e 1 = m f 2 {\displaystyle e_{2}=mf_{1}~~{\text{and}}~~e_{1}=mf_{2}}
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
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 t