The steady states of cells affect their response to perturbation. Indeed, diagnostic markers for predicting the response to therapeutic perturbation are often based on steady state measurements. In spite of this, no method exists to systematically characterize the relationship between steady state and response. Mathematical models are established tools for studying cellular responses, but characterizing their relationship to the steady state requires that it have a parametric, or analytical, expression. For some models, this expression can be derived by the King-Altman method. However, King-Altman requires that no substrate act as an enzyme, and is therefore not applicable to most models of signal transduction. For this reason we developed py-substitution, a simple but general method for deriving analytical expressions for the steady states of mass action models. Where the King-Altman method is applicable, we show that py-substitution yields an equivalent expression, and at comparable efficiency. We use py-substitution to study the relationship between steady state and sensitivity to the anti-cancer drug candidate, dulanermin (recombinant human TRAIL). First, we use py-substitution to derive an analytical expression for the steady state of a published model of TRAIL-induced apoptosis. Next, we show that the amount of TRAIL required for cell death is sensitive to the steady state concentrations of procaspase 8 and its negative regulator, Bar, but not the other procaspase molecules. This suggests that activation of caspase 8 is a critical point in the death decision process. Finally, we show that changes in the threshold at which TRAIL results in cell death is not always equivalent to changes in the time of death, as is commonly assumed. Our work demonstrates that an analytical expression is a powerful tool for identifying steady state determinants of the cellular response to perturbation. All code is available at http://signalingsystems.ucsd.edu/models-and-code/ or as supplementary material accompanying this paper.
Diagnostic markers are derived from steady state measurements, but are used to predict the cellular response to therapy. To develop new and better diagnostics, we would like to systematically characterize the relationship between steady state and the response to a given therapeutic. Mathematical models have powerfully complemented empirical studies in this regard, but it remains challenging to employ these models to characterize the effects of steady state. To do so requires a mathematical expression for the steady state, for which no universal method has been developed. Here, we present a method for deriving a mathematical expression for the steady state of a common class of models, those that obey the Law of Mass Action. We show that our method is easy to use and scales well to large models. We then use our method to characterize the relationship between steady state and the sensitivity to the anti-cancer drug candidate, dulanermin. We find that sensitivity to the drug is strongly affected by the concentration of the signaling molecule, procaspase 8, and its inhibitor, Bar. Our work thus demonstrates the utility of analytical studies of the steady state and its relationship to drug sensitivity.
Citation: Loriaux PM, Tesler G, Hoffmann A (2013) Characterizing the Relationship between Steady State and Response Using Analytical Expressions for the Steady States of Mass Action Models. PLoS Comput Biol 9(2): e1002901. https://doi.org/10.1371/journal.pcbi.1002901
Editor: Daniel A. Beard, Medical College of Wisconsin, United States of America
Received: August 17, 2012; Accepted: December 12, 2012; Published: February 28, 2013
Copyright: © 2013 Loriaux et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The study was supported by the NIH via R01 grant GM072024 and P50 GM085763, the San Diego Center for Systems Biology. PML was supported by the Department of Energy Computational Sciences Graduate Fellowship. GT was supported by the National Science Foundation (NSF grants DMS-0718810 and CCF-1115206). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Transient activation of signaling molecules is a hallmark of the cellular response to perturbation. Far from acting as a simple relay, however, the dynamics of signaling molecules can encode information about the instigating stimulus –. Interestingly, these dynamics are affected by the steady state prior to perturbation , . Non-genetic variation in the proteome, for example, is sufficient to explain variability in the sensitivity of HeLa cells to the pro-apoptotic ligand TRAIL . Like other TNF superfamily members, TRAIL is a promising anti-cancer therapeutic . Recombinant human TRAIL, or dulanermin, as well as antibodies raised against the TRAIL receptors DR4 and DR5, are currently in clinical trials . To improve the efficacy of these and other drugs, understanding how sensitivity is affected by the cellular resting state is of great importance .
Mathematical models are powerful tools for characterizing the behavior of signaling systems in response to perturbation –. Assuming conservation of mass, these models equate the change in concentration of a molecular species with the sum of reaction velocities that produce the species, minus the sum of those that consume it. The reactions themselves are often modeled by the Law of Mass Action. This law assumes that the velocity of a reaction is proportional to the product of the concentrations of its reactants. Since many signaling reactions are bimolecular, the resulting mass balance equations are non-linear in the concentrations. A system is at steady state if no species is consumed faster than it is produced, nor produced faster than it is consumed. By this formalism, the steady state of a signaling system is equivalent to the root of a non-linear system of equations. Because of this, no universal method has been developed to identify the steady states of mass action models, despite their importance to basic and clinical research. As a result, even with the help of mathematical models, investigating the relationship between steady state and stimulus-responsiveness remains cumbersome.
Of course with any model, simulating the response to perturbation often requires the system to be at steady state prior to perturbation. To achieve this, one of several techniques is currently used. The most common technique is to assume a “trivial” steady state where every reaction velocity is zero , . While straightforward, this approach may not reflect biological reality, where tonic signaling is common ,  and can strongly influence the response to perturbation –. A second technique is to approach the steady state asymptotically via numerical integration of the mass balance equations , , . While this approach can yield non-trivial steady states, the number of integration steps required to reach the steady state may dominate the number of steps required to simulate the perturbation. Also, identifying the parameter values that result in a desired steady state is an inverse problem that requires non-linear optimization. For these reasons, numerical derivation of the steady state is impractical when characterizing its effect on the response to perturbation, and an analytical expression is required instead.
The best-known method for deriving analytical expressions for the steady states of mass action models was developed by King and Altman in 1956 . This method assumes that all molecular species can be divided into enzymes and substrates, that no enzyme is itself a substrate, and that all substrates remain constant over the time-scale of steady state formation . A number of improvements have been made to the King-Altman method over the years –. Many of these are now implemented in the Matlab application, KAPattern . The King-Altman methodology was also recently formalized using concepts from algebraic geometry , , and extended to layered signaling cascades  and post-translational modification networks . Despite these improvements, however, these methods do not extend to mass action models with arbitrary reaction structure, as is common in contemporary models of signaling systems. Furthermore, only the King-Altman method has been reduced to practice.
For these reasons we developed py-substitution, a simple, algebraic method for deriving steady state expressions for mass action models with arbitrary structure. Our method can be explained using concepts from linear algebra, and full code has been provided for all examples in this manuscript, implemented in either Matlab or Maple. A particular benefit of py-substitution is that it affords considerable flexibility when selecting independent quantities for the steady state expression. Often, this permits explicit derivation of kinetic rate constants from steady state concentration measurements. More generally, it allows independent quantities to be chosen that maximize incorporation of known or measured parameter values. This not only simplifies model fitting, but typically reduces the total number of parameters required as well. We compare py-substitution to the King-Altman method and show that, where King-Altman is applicable, the two methods yield equivalent results. Computationally, however, we find that our method is more efficient, and, because py-substitution does not require a particular reaction structure, more general than King-Altman.
Finally, we use py-substitution to derive a steady state expression for a recent model of apoptosis induced by the death-receptor ligand TRAIL . We find that incorporation of a non-trivial steady state changes the qualitative behavior of the model. Specifically, tonic signaling desensitizes the system to low doses of TRAIL, while high doses of TRAIL still result in the “snap-action” signaling dynamics indicative of cell death. We then systematically alter the steady state and show that changes in steady state affect the threshold at which TRAIL results in death. We find that the threshold is highly sensitive to the steady state abundances of procaspase 8 and its negative regulator, Bar, but not the other procaspase molecules. This suggests that the activation of caspase 8 is a critical point in the cell death decision. Finally, without recourse to a model that is tolerant to low doses of TRAIL, a common practice is to approximate the sensitivity to TRAIL by the time at which death occurs. Using our tonic signaling model, we show that these two metrics are not universally equivalent. Caution should therefore be taken when equating the dynamics of cell death with the probability that death occurs.
Materials and Methods
In this section we describe the process for deriving analytical expressions for the steady states of mass action models using py-substitution. First we describe the class of models to which py-substitution can be applied. Next, we review existing methods for deriving analytical expressions for the steady states of these models. Finally, we describe py-substitution using some formal concepts from algebra. In the results section we provide several examples, beginning with a version of the classical Michaelis-Menten model of enzyme action. All code for these examples, as well as detailed instructions for use and full transcripts of the output, are provided in Protocol S1 and on our website, http://signalingsystems.ucsd.edu/models-and-code/.
Let be the set of non-negative natural numbers and be the set of non-negative real numbers. Let be a set of species and be a set of reactions. Each reaction follows the normal definition,where is the stoichiometric coefficient of the reactant and is the stoichiometric coefficient of the product . We define to be the concentration of species and to be the velocity at which converts reactants into products. By the Law of Mass Action,(1)
The quantity is often, but not necessarily, equal to . The coefficient is called the rate constant. Assuming conservation of mass, the concentration changes according to(2)where is the first derivative of with respect to time. Any collection where the concentration of obeys Equation 2 and the velocity of obeys Equation 1 is called a mass action model. In what follows, we assume , , and are indices over the interval and is an index over . When are such that all(3)the model is said to be at steady state. If all we call the steady state trivial. In this manuscript we are concerned with symbolic, non-trivial solutions to Equation 3. A solution is symbolic if all and are left as uninterpreted variables, rather than being assigned numerical values. For a complete list of symbols and their meanings, see Table S1.
Let and be the vectors with elements = , and . Throughout this manuscript, we use to denote the element of vector and to denote the element at row , column of matrix . Let be the stoichiometric matrix, i.e., the matrix whose elements are . Using this notation, Equation 2 becomes(4)and the steady state equation becomes(5)
By convention we use the overline to denote vectors that satisfy steady state. Equation 5 often takes this form in flux balance analysis –. Here is a real-valued vector and is calculated numerically. However, prior work has shown that Equation 5 can also be used to calculate a vector of rate constants from a vector of steady state concentrations . Let be the vector with elements . Let be the diagonal matrix with elements . The vector can then be expressed as(6)
Substituting Equation 6 into Equation 5 and solving for yields the -cone  — equivalently, the left null space of the matrix product . Given a basis for this null space and a vector of steady state concentrations, a vector of rate constants can be calculated that satisfies Equation 5. While this approach is useful for deriving kinetic parameters from metabolomic measurements, it is less well suited to signaling systems where transient and low-abundance species confound accurate measurement of the concentrations.
If the velocity of every is homogeneous of degree in , then an analogous approach allows to be expressed in terms of . We call models that satisfy this condition linear models. An alternative, stoichiometric definition for a linear model is given by the following,(7)
Equation 7 requires that every reaction defines a transition from exactly one time-varying species to another. Let be the matrix with elements . If is a vector of linear reaction velocities, it can likewise be expressed as(8)
Substituting Equation 8 into Equation 5 results in the matrix product , also called the Jacobian matrix . Given a basis for the null space of the Jacobian, a vector of steady state concentrations can be calculated from a vector rate constants.
For linear models, an alternative, graphical method for deriving expressions for the steady state species concentrations was introduced by King and Altman in 1956 . Notice that Equation 7 permits a two-dimensional indexing of the rate constants,(9)
Note that is simply the Jacobian matrix for a linear model, . The general solution to Equation 12 was found in  to be the vector with elements(13)where is the minor of , formed by removing its row and column and computing its determinant. For sufficiently small systems, Equation 13 can be solved directly using modern mathematical computing software . Prior to the advent of modern computers, King and Altman realized that the minors can also be derived by graph theoretic means. Note that for a linear model, and imply a directed graph,(14)where each defines a vertex and each defines an edge between vertices and (provided and are such that ). The King-Altman method enumerates for each species the set of simple connected subgraphswhere vertex has out-degree and all other vertices have out-degree , . These are the directed spanning trees of , with all edges directed towards root . A subgraph is called a King-Altman pattern. The minor can then be expressed as(15)where is the transition rate constant between species and . For a more thorough derivation of the King-Altman method, see .
Of course, many biochemical reactions are bimolecular. By Equation 1, the velocity of a bimolecular reaction is degree in . To preserve linearity, one can assume the concentration of one reactant is so high as to be effectively constant. This concentration is incorporated into the kinetic rate constant, and the techniques described above can still be used to solve Equation 3. If this assumption fails, then Equation 2 describes a polynomial in with coefficients in . In this case the solutions to Equation 3 form an algebraic variety. Deriving an expression for the steady state of a non-linear model thus requires finding a parameterization of the variety . One way to achieve this is to calculate a Gröbner basis for the ideal generated by and eliminate variables , . Alternatively, if the model displays certain structural properties, variables can be eliminated by identifying conservation relationships. The best-known example of this is when defines a cascade of post-translational modifications. In this case, enzyme-substrate intermediates can be eliminated and the variety can be parameterized by rational functions of the free enzyme concentrations with coefficients in , . Although these methods do not require linearity, calculating a Gröbner basis can be computationally intractable, while identifying conservation relationships can be difficult for models of arbitrary reaction structure.
Py-substitution allows mass action models — a particular class of non-linear model — to be solved using simple linear algebra. We make use of the following observations: (a) is always homogeneous of degree in , and (b) is often no greater than degree in . If a subset of elements in can be found on which every has only linear dependence, then Equation 5 can be solved using linear methods.
To begin, we define sets of symbolic variables and such that and . We then relabel, or map, every element in to a unique element in so that every is linear in . By Equations 1 and 2 this requires that all are linear in . Variables that we want to remain independent, as well as variables on which has non-linear dependence, should be mapped to . As we shall see, there is considerable flexibility in choosing this map.
Let and be partitioned into disjoint (but possibly empty) subsets and . We define to be a bijective map (with a restriction given below)and extend it homomorphically over . Our linearity restriction is to consider maps of this form such that(16)for some . For , the exponent is . For , the exponent . In words, defines a change of variables such that is homogeneous of degree in . By Equation 2, becomes a homogeneous polynomial of degree in with coefficients in . We can now write(17)where is the Jacobian matrix with elements . Here and elsewhere we use the notation to mean that is the vector formed by applying the function element-wise to . Note that the trivial partition and recovers the -cone procedure described above. For the remainder of this section, we treat as an index over . Substituting Equation 17 into Equation 5 gives(18)where is called the coefficient matrix. The solution to Equation 18 is precisely the null space of . Let be a matrix whose columns form a basis for this null space. Let be the number of columns in . By the rank-nullity theorem, we have(19)where is the number of columns in . Furthermore, because is linear in and exists, the matrix must be full rank. By the properties of the rank, we can write(20)
Together, Equations 19 and 20 give(21)thus calling for the constraint . This, in conjunction with Equation 16, are the only constraints on . If we now let be some linear combination of the basis vectors,(22)then satisfies Equation 18 and steady state is achieved. In general, Equation 22 is underdetermined. Equation 22 therefore implies a partition of into independent variables (denoted ) and dependent variables (denoted ). We will now describe this partition by a second mapping function, .
Recall that a basis for the null space of can be constructed from , the reduced row echelon form of . Let be the column of . If contains a pivot position, then is a dependent variable. If does not contain a pivot, then is free, or independent. Let(23)
Let be the cardinality of . Enumerate these variables as , with . For every not containing a pivot, there is a basis vector (related by ) whose element equals and whose elements in positions are . By Equation 22, this gives an independent parameter, . Equation 22 thus defines a function . Let be the set of independent parameters. If column does contain a pivot, then depends on variables in , giving where is the specific function resulting from the row operations used to reduce to . Equation 22 can now be described in its entirety by the mapping function ,(24)
The notation indicates that for every . Note that we define as the set of all linear combinations , where and are distinct elements of . is the set of all polynomials in variables with rational numbers as coefficients. is the field of fractions of : any can be expressed as , where , .
As with , there is some flexibility in choosing how is partitioned into free variables, , and dependent variables, . A different indexing of the variables in simultaneously permutes the vector and the columns of . This leads to different reduced row echelon forms, with different partitions into free and dependent variables. The null space basis obtained by reducing to greedily classifies low-numbered columns as dependent columns when possible, or free columns when not possible. Quantities in for which good numerical estimates exist should therefore be assigned to higher indices. These quantities are favored, but not guaranteed, to be mapped to independent parameters. Quantities for which good numerical estimates do not exist should be assigned to low indices in .
Finer control over the partition of into dependent and independent parameters is possible by working directly with or . Let be the set of elements in that we want mapped to . Let be the square matrix formed by rows of . To map to requires that we find a vector such thatwhere is the vector with elements . Solving for gives(25)
Thus, for a given map , not all partitions of into and are possible, but only those for which . An example of this can be seen in the file “fum2.m” in Supporting Protocol S1, discussed below.
Next let , and . Let and be defined analogously. The composition captures the entire process of linearizing with the function , solving the linear system , and taking an arbitrary combination of solution space basis vectors:
Applying to the sets and results in a parametric description of the steady state that is typically the most useful: every element in or is mapped to an element in or , or a function in . Assigning numerical values to elements in and results in elements in taking values that satisfy the steady state equation. In some cases we may wish to reverse the substitution so that functions of variables are mapped back to functions of . To do so, let and . Let be the inverse of restricted to the independent parameters, .
The function then defines a map for whichwhere steady state velocities in are in terms of elements in and . A visual overview of the py-substitution method is given in Figure 1.
Quantities in a mass action model can be separated into kinetic rate constants (set , red) and species abundances or concentrations (set , blue). From a subset is selected on which all reaction velocities have only linear dependence. A function maps these to elements in and the remaining to elements in . A second function imposes the relations by expressing dependent variables in in terms of independent parameters . A third function, , is the inverse of restricted to the independent parameters. The composition of with results in variables in being expressed in terms of variables in , such that steady state is achieved. In the diagram, solid arrows are isomorphisms while dashed arrows are homomorphisms that replace dependent variables by equivalent expressions in independent parameters. See Table S1 for a complete listing of symbols and their meanings.
py-substitution permits flexible derivation of a steady state solution
An important goal in developing py-substitution was that it be generally applicable to any model whose reaction rates obey mass action kinetics. This requires that the independent quantities be chosen freely among the species concentrations and reaction rate constants, and that non-linear rate equations do not confound the derivation of a steady state expression. To demonstrate these capabilities we consider an open-system analog of the classical Michaelis-Menten model of enzyme kinetics (OMM, see also Figure 2). Substrate synthesis and product degradation allow this system to achieve a non-trivial steady state , which we derive here using four different substitution strategies. The set of reactions for this model is given by
Enzyme and substrate bind to form an intermediate complex, followed by catalysis and dissociation of the product. The substrate is synthesized by a zero-order reaction, , and the product is degraded by a first-order reaction, . See “omm1.m” in Protocol S1 for a complete description of the model.
By Equation 4 this results in the following system of equations,for which we now derive functions such that .
Homogeneous substitution: steady state concentrations do not uniquely determine reaction rate constants.
See “omm1.m.trace.pdf” in Protocol S1 for details of this partition and all subsequent steps. Applying to results in a reaction velocity vector that is linear in , as required by Equation 17,
From Equation 26, we observe that . Thus, of the 9 degrees of freedom in this system (5 rate constants plus 4 species concentrations), 3 will have values that are constrained by Equation 5. Since our substitution strategy only identifies 4 independent parameters, 2 additional elements mapped to must in fact be independent as well. These elements can be identified by the columns in that do not contain pivots, namely columns 2 and 5. To see this, note that Equation 26 yields the following basis for the null space of ,
Letting , Equation 22 gives(27)
To reverse the substitution, notice from Equation 27 that and , giving the following map, :
Applying this transformation to the original vector of reaction velocities yieldswhich one can verify satisfies Equation 5. An interesting implication of this trivial application of py-substitution is that, because maps every species concentration to an independent parameter, we can interpret Equation 28 to mean that any vector of steady state concentrations will be consistent with an infinite number of reaction rate constants. In this particular case, knowing all four concentrations tells us nothing about the rates of enzyme-substrate dissociation or product degradation. As we shall see, by using different substitution strategies, we have some flexibility in choosing which rate constants are constrained by the steady state concentrations, but the structure of the OMM model makes finding a unique set of rate constants impossible. In general, a unique set of reaction rate constants requires that the coefficient matrix be full rank, or(29)
In other words, a unique set of rate constants requires that the stoichiometric matrix be full rank, which is equivalent to requiring that the corresponding reaction network have no cycles. Since even a single reversible reaction represents a cycle, we conclude that in the general case, a set of steady state species concentrations does not imply a unique set of reaction rate constants.
Heterogeneous substitution: the number of independent model parameters is constant.
Often, models contain species whose concentrations are difficult to measure or reactions whose rates have been well characterized. For such models it is preferable to partition sets and so that species whose concentrations are difficult to measure are mapped to while well-characterized reaction rates are mapped to . For example, if the kinetics of the enzyme are well characterized, an attractive partitioning of the OMM model might be and . This yields a map
Again, see “omm2.m.trace.pdf” in Protocol S1 for complete details. Notice here that we have forced the enzyme kinetic parameters , , and to be independent by mapping them to elements in . The resulting coefficient matrix and null space basis arewhich yield the steady state map
Notice that even though the cardinality of differs in this example as compared to the one above (5 versus 4), the cardinality of does not (3). Let denote this cardinality. Obviously, , or equivalently,
This is simply the rank-nullity theorem again. By Equation 20, we can therefore conclude that
In other words, the final number of dependent elements in the steady state expression for a system is independent of the substitution strategy, and only depends on the structure of the reaction network.
Substitution with sublinear velocities: using py-substitution to resolve non-linearities (I).
Some reaction velocities are zero-order but well-characterized. For example, if the rate of substrate synthesis in the OMM model has been accurately measured, we may wish to partition such that . The resulting mapping function , however, fails to linearize . To compensate, we introduce a pseudospecies and let . If we now partition such that , the linearity of in is preserved and we may continue as before.
To illustrate this approach, we again let and . The remaining rate constants and one pseudospecies are partitioned into sets and , respectively, such that . See “omm3.m.trace.pdf” in Protocol S1 for details. The resulting velocity vector is linear and yields a coefficient matrix whose null space is two-dimensional,(30)
However, one of these two dimensions is constrained by the pseudospecies. We are thus not at liberty to take a general linear combination as per Equation 22 but must find such that(31)
Substitution with superlinear velocities: using py-substitution to resolve non-linearities (II).
Some reaction velocities are superlinear in their reactant concentrations. If good estimates for these concentrations do not exist, we would like to partition these species into . Analogous to the sublinear case above, doing so results in a velocity vector that is non-linear in . Fortunately, the strategy above is useful here as well: introduce a pseudospecies for each superlinearity, calculate a basis for the null space of the coefficient matrix, and identify basis vector coefficients that satisfy the constraints imposed by the pseudospecies.
Let us consider a version of the OMM model where the rate of product formation is proportional to the square of the enzyme-substrate complex, . Let us further assume that no estimate exists for the value of . We would therefore like . Since this fails to linearize the velocity, we introduce a pseudospecies and let . We now define such that
This satisfies the linearity requirement and maps and to the lowest indices in , thereby favoring these quantities to become dependent parameters. See “omm4.m.trace.pdf” in Protocol S1 for details. The resulting coefficient matrix has a null space that is spanned by the columns of
Letting maps and to and satisfies our requirement that and . As in the previous section, however, one dimension of is constrained by the pseudospecies. Specifically, we require that . by Equation 22, this requires that
This example illustrates that, using pseudospecies, a mapping function can always be found such that can be derived using linear methods. Non-linearities introduced by pseudospecies can then be resolved on a case-by-case basis, resulting in the final steady-state solution.
Py-substitution is more general, but not less efficient, than King-Altman
Some chemical reaction systems are linear in the species concentration vector, or can be rendered linear by assuming that the concentrations of certain species don't change over time. The classical model for malate synthesis is an example of the latter . Here, the enzyme fumarase binds reversibly to fumarate and hydrogen in either order, followed by reversible binding of hydroxyl and reversible formation of malate (Figure 3). The reactions for this model are
This mechanism for the conversion of fumarate to malate by the enzyme fumarase was proposed in . Fumarase binds to fumarate and hydrogen in either order, then hydroxyl, followed by formation of the product, malate. All reactions are reversible. See “fum1.m” in Protocol S1 for a complete description of the model.
By Equation 1, the corresponding reaction velocities are(32)
Notice that the submatrix formed by the first five rows of satisfies the definition for a linear model given in Equation 7. Call this submatrix and let be the vector of enzyme concentrations. If we assume that the substrate concentrations are time-invariant, the steady state equation for this model becomes(33)
Because satisfies Equation 7, we may define the following transition rate constants(34)
If we now define a matrix(36)Equation 33 becomes(37)where the elements of are given in Equation 11. The solution to Equation 37 is given by Equation 13, which we saw may be evaluated using the King-Altman method. Alternatively, we may solve Equation 33 directly using py-substitution. Given that py-substitution applies to a more general class of mass action models then King-Altman, we wondered whether this flexibility came at the cost of computational efficiency. Here we show that, for models that can be treated using the King-Altman method, py-substitution yields an equivalent result, and at no loss of efficiency.
Py-substitution and King-Altman yield equivalent steady state expressions.
In this subsection only, we use to mean the element of the vector , made to satisfy Equation 37 by the King-Altman method. The element is defined analogously for py-substitution. To solve Equation 33 by py-substitution, we partition into subsets(39)and define such that
The resulting coefficient matrix is precisely the matrix of rate constants, . The null space of is one-dimensional and spanned by a single basis vector . In our solution, the basis vector is normalized to element , which by Equations 22 and 23 yield a partition of into subsets and . After reversing the substitution we find that the steady state concentration of each enzyme likewise has the form(40)
In other words, the solutions given by KAPattern and py-substitution are not identical. The disparity arises from Equation 13, which imposes the constraint . When derived by King-Altman, the steady state expression for each enzyme is therefore a ratio of the total enzyme concentration. In contrast, py-substitution results in being expressed in terms of , the only element for which . Despite this disparity, Equations 40 to 42 can be combined to giveTherefore,
Py-substitution is not less efficient than King-Altman.
We next wondered whether the King-Altman method is computationally more efficient than direct algebraic solution of the linear steady state equation (Equation 18). The King-Altman method requires exhaustive enumeration of valid King-Altman patterns. The number of patterns depends critically on the structure of the model. A model of strongly connected species generates patterns while a simple cycle generates only . By comparison, solving Equation 18 requires Gaussian elimination on the matrix . For a fixed-precision numeric matrix, this would take at most ; however, since has symbolic entries rather than numerical ones, the sizes of the entries grow with the number of row operations. In fact, as Equation 15 shows, the number of valid King-Altman patterns is precisely the number of terms in the polynomial expansion of the minors. Thus even a few species, if highly connected, can generate thousands of terms and easily overwhelm conventional memory architectures.
To evaluate the performance of py-substitution versus KAPattern, we generated random models with six species and anywhere from 10 to 20 first-order reactions between them. Three distinct realizations were generated for each model. Models for which KAPattern failed – typically because the stoichiometric matrix described a disjoint network – were discarded. The command-line version of Matlab 2010b was used to derive the steady state concentration vector for each model using py-substitution and KAPattern, and for py-substitution the command-line version of Maple 14 was used as well. Internal memory was cleared prior to each derivation to prevent caching. The architecture used was a commodity netbook PC running Windows XP SP3 with an Intel 1.7 GHz Atom processor and 1 GB RAM. The derivation was repeated in triplicate for each realization to reduce variance introduced by the CPU scheduler. Execution times include initialization of the symbolic variables and coefficient matrix, kernel calculation, and derivation of in the case of py-substitution, and all steps prior to file writing in the case of KAPattern.
Results from the simulation are given in Figure 4. The data show that using Matlab, KAPattern provides consistently better performance and better scaling with respect to the number of reactions. This is likely because KAPattern uses Wang algebra to avoid explicit representation of the fully expanded minors in memory . In contrast, Gaussian elimination of the coefficient matrix uses MuPAD, the Matlab symbolic engine, which is memory intensive and sensitive to expression swell. Models of even modest degree exhaust physical memory and cause “thrashing”, resulting in poor runtime performance for models larger than 15 reactions. However, using Maple, direct solution of the steady state equation is typically an order of magnitude faster than KAPattern and exhibits identical scaling. This is likely because Maple's symbolic solver considers equations in increasing order of their memory footprint. This data therefore argues that the King-Altman method is not more efficient than direct solution of the steady state equation.
Given a first-order model with six species and the number of reactions indicated by the x-axis, the time required to derive an expression for the steady state of the model is indicated by the y-axis. Three random realizations were used for every model size. Every calculation was performed in triplicate, but the error in calculation time was negligible.
Py-substitution is more general than King-Altman.
To solve the steady state equation, the King-Altman method requires that the stoichiometric matrix satisfies Equation 7. As we saw above, for to satisfy Equation 7 we must be able to partition into two disjoint sets, a set of “enzymes” and a complementary set of “substrates”. The partition must be such that every reaction consumes a single species in and produces a single, different species in . All other species produced or consumed by must be in . The concentrations of these substrates are assumed to be time-invariant. As such, rows in that correspond to substrates can be removed, and the substrate concentrations can be incorporated into the kinetics of the reactions. By inspection, the only such partition for the fumarase model is Equation 39, analyzed above.
By comparison, py-substitution does not require that the stoichiometric matrix satisfies Equation 7. The substrates can therefore remain variable with respect to time and incorporated into the steady state solution, of which there are many. Without recourse to pseudospecies, the six bimolecular reaction velocities require that and be partitioned separately into sets and , or vice-versa. One such partition is
The resulting coefficient matrix has a five-dimensional null space, consistent with Equation 21 since and . A basis for this null space is given by the columns in , where
Other maps are available, however. By Equation 25, the submatrix formed by taking any 5 linearly independent rows of produces a different vector of coefficients, and thus a different partition of . For our particular choice of above, 72 partitions are possible, calculated by testing which combinations of 5 rows in are linearly independent. As an illustration, consider the case where the rate constants and are easier to measure than substrates and . Because of this, we would prefer and to be dependent variables. Equivalently, we want , , and , . Since and , any submatrix of containing rows 3 and 6 whose determinant is not zero will accomplish this. Below is the vector calculated from the matrix formed by rows 3, 5, 6, 7, and 10.This results in the desired steady state mapping,
This offers another illustration of how the choice of substitution strategy and null space basis vectors allow one to choose independent parameters flexibly among sets and when solving for steady state. See “fum2.m.trace.pdf” in Protocol S1 for details of this derivation.
Steady state establishes a threshold for drug-induced cell death
Finally, we sought to use py-substitution to characterize the relationship between steady state and the response to the cancer drug, dulanermin. Dulanermin is a recombinant human form of the endogenous ligand TRAIL, whose mechanism for triggering cell death is modeled in version 1.0 of the extrinsic apoptosis reaction model, or EARM . This model considers the biochemical events following engagement of the death receptors 4 and 5 (DR4/5), including receptor-induced cleavage of initiator caspases, positive-feedback by effector caspases, and feed-forward amplification by the mitochondrial pathway following outer membrane permeabilization, or MOMP (Figure 5). The EARM model was trained on data derived from HeLa cells co-treated with cyclohexamide, an inhibitor of protein synthesis that results in hypersensitivity to TRAIL . Accordingly, any amount of ligand in the EARM model results in cell death. The abundance of ligand still affects the time of death, defined for example by the time at which half of the caspase 3 target protein PARP has been cleaved (Figure 6A, left) . Note in this section we refer to the abundance of a species rather than its concentration, as these are the units chosen by the original authors.
Reactions new to this version include all fluxes to or from a source node, indicated by dashed lines to or from a Ø. In addition, the activation of Apaf was made reversible, as were the formation of mitochondrial pores. The complete model contains 58 species and 115 reactions. See  for a description of the original EARM model, and “xearm.mpl” in Protocol S1 for a complete description of xEARM.
(A) The dynamics of PARP cleavage are shown for EARM (left) and xEARM (right), in response to increasing doses of the TRAIL ligand (gray to blue). The abundance of cleaved PARP for each model has been normalized to the maximum observed abundance. For each model, for a particular dose of TRAIL, the time at which PARP is 50% cleaved is indicated by the dashed red lines. For xEARM, the minimum abundance of TRAIL required to observe MOMP, , is indicated on the color scale at right. (B) Changes in in response to changes in the steady state abundance of 12 primary xEARM species. Species have been sorted from left to right in order of those for which an increase in abundance results in the greatest increase in TRAIL sensitivity, to those for which an increase in abundance result in the greatest decrease in sensitivity. (C) Normalized sensitivity coefficients for , calculated using the xEARM model (blue), and , calculated using both EARM (white) and xEARM (gray), for each of the 12 primary species in (B).
In the absence of cyclohexamide, however, HeLa cells do not all die following exposure to TRAIL. Rather, a fraction of cells persist, and this resistance is a function of the proteomic state prior to stimulation . To capture this phenomenon, we extended the EARM model so that proteins continued to be synthesized and degraded following exposure to TRAIL. Specifically, we introduced 43 new synthesis and degradation fluxes as well as 2 protein inactivation reactions (see “xearm.mpl” in Protocol S1). These reactions were chosen so that every species is subject to at least one efflux. We refer to our extended model as xEARM. Because xEARM satisfies our definition of a mass action model, we use py-substitution to identify an analytical expression for its steady state. To derive this expression, a mapping function was chosen so that every non-zero parameter in EARM was mapped to an independent parameter in . As a result, we were able to preserve the snap-action dynamics of MOMP that is central to the original model (Figure 6A, right). Honoring the published parameters required that we introduce two pseudospecies, one for each the di- and tetrameric forms of Bax (variables and , respectively),(43)(44)
The coefficient matrix and null space basis matrix were calculated as before, with the latter calculation requiring less than a minute on our benchmark PC. The null space of has 17-dimensions, resulting in a matrix of basis vectors of the form
Basis vectors to preserve the steady state ratios of paired synthesis and degradation reactions. Vector , for example, ensures that a change in results in a change in , where is the abundance of Cytochrome C in the mitochondria and and are its rates of synthesis and degradation, respectively. The vector scales the steady state abundances of mitochondrial Bax and Bcl2 complexes with respect to changes in the rate of Bcl2 synthesis. Vectors and are algebraically intractable and thus defy simple biochemical interpretation. Two of these vectors, and , are constrained by the pseudospecies and . To resolve these constraints, note that Equations 43 and 44 require that(45)(46)
By our mapping function (see “xearm.mpl.trace.pdf” in Protocol S1, pp. 120–121), Equations 45 and 46 become(47)(48)where . Solving Equation 48 for gives(49)where . Substituting Equation 49 into Equation 47 and solving for gives(50)where (see “xearm.mpl.trace.pdf” in Protocol S1, pp. 121–126).
Obviously, Equation 50 identifies an explicit bistability in the xEARM model. Basis vector coefficient — and by Equation 49, — can take either of two values for any numerical realization of the model. By examination of , we find that these two coefficients affect all modified and compound species, as well as synthesis rates for proteins within and upstream of the mitochondria. Using the parameter values supplied in , however, we find that one of the solutions to Equation 50 is negative. The corresponding steady state is therefore infeasible and the solution was discarded.
In addition to parameters in , a full numerical realization of the xEARM model requires values for parameters and . All but three of these elements represent first-order degradation rate constants, to which we assigned values equivalent to a half-life of one hour. This value was based on global quantifications of protein turnover in mammalian cells, which revealed that signaling proteins tend to be short-lived . Two of the elements, and , represent first-order inactivation fluxes, which we assumed to be ten times faster than protein degradation. The final element is the steady state abundance of the mitochondrial Bax2∶Bcl2 complex, which we set to 20 molecules. Six of the elements were then modified from their initial values to better match the dynamics of caspase activation and PARP cleavage, as reported in . The complete table of parameter values required to initialize and numerically integrate the xEARM model is given in Table S2.
For comparison, Table S3 lists the steady state abundances of species in the original and extended EARM models, sorted in order of decreasing difference. As expected, every species in EARM with a non-zero abundance has precisely the same abundance in xEARM, since these are independent parameters in the steady state solution. Among species with zero abundance in EARM, the mitochondrial Bax∶Bcl2 complex exhibits the greatest disparity, with the steady state abundance in xEARM being in the low thousands of molecules. Ubiquitinated, cleaved caspase 3 and cleaved PARP are also in the low hundreds of molecules, but this represents only a small fraction of their total cellular abundance. A full 25 species with zero abundance in the EARM model have an abundance of less than 1 molecule in xEARM. This indicates that, even though the steady state reaction velocities are markedly different between EARM and xEARM, by using py-substitution we were able to engineer a steady state where the species abundances are appreciably similar between the two models.
Next we asked whether the xEARM model remained viable in the presence of low doses of TRAIL, but still exhibited MOMP when stimulated with high doses of TRAIL. To do so we created a numerical realization of the model using the parameters from Table S2, then perturbed the model from its steady state using a step increase in the abundance of TRAIL (variable ). The magnitude of the step ranged from to -fold and was followed by numerical integration of the mass balance equations out to 48 hours. As shown in Figure 6A, MOMP is only observed in xEARM when TRAIL is increased by -fold or more. We label this minimum dose of TRAIL required for MOMP . Increments less than result in a small and transient change in cleaved PARP abundance, followed by a return to the pre-stimulated steady state. By comparison, any magnitude dose of TRAIL causes MOMP in the original EARM model.
This ability of xEARM to distinguish between low and high doses of TRAIL, in conjunction with an analytical expression for its steady state, allowed us to systematically perturb the steady state and ask how these perturbations affect the sensitivity to TRAIL. To illustrate this capability we varied the steady state abundance of each major xEARM species over a 100-fold range, centered about each species' wildtype value as reported in Table S2. For each variation, we performed a binary search to identify . The results from this procedure are plotted in Figure 6B. As expected, increases in XIAP, Bcl2, FLIP, and Bar result in reduced sensitivity to TRAIL stimulation, while increases in Procaspase 8, TRAIL receptor DR4/5, Bax, and Bid result in increased sensitivity . What is interesting, however, is the following. First, TRAIL sensitivity is most affected by changes in the abundance of Procaspase 8 and Bar, an inhibitor of active caspase 8 . The ability to activate caspase 8, then, appears to be a critical determinant of TRAIL sensitivity, as previously suggested , . Second, the abundances of Procaspase 3, 6, and 9 have little effect on the sensitivity to TRAIL. This observation is in good agreement with the model-based prediction that induction of MOMP does not require positive-feedback via this caspase loop .
A common metric for describing how model parameters affect the sensitivity to TRAIL is to calculate the change in time at which death occurs in response to a small change in each parameter , , . It is conceivable, however, that changes in the time of death do not accurately reflect changes in the threshold of TRAIL at which death occurs. Therefore, to test this assumption we calculated parameter sensitivity coefficients for the ligand threshold, , and the time at which death occurs, , using the xEARM and EARM models, respectively. The numerators and were calculated by backward finite difference approximation and all sensitivities were normalized to the maximum observed sensitivity for each metric (Figure 6C). The data show good agreement for positive regulators of TRAIL sensitivity, but some disparity in the negative regulators. Specifically, while is particularly sensitive to changes in XIAP and Bcl2, is most sensitive to changes in Bar. This result argues that some caution should be taken when equating changes in the time of death with changes in TRAIL sensitivity.
We have described a simple but flexible method for deriving analytical expressions for the steady states of mass action models. Central to our method is the observation that mass action models are systems of polynomial equations that are generally no greater than degree 2. This permits a partitioning of rate constants and species concentrations into disjoint sets of quantities, and , where the reaction velocity vector is linear with respect to the variables in . If the cardinality of is greater than the rank of the stoichiometric matrix, then the steady state equation can be solved analytically using simple linear methods.
There is considerable benefit to deriving an analytical expression for the steady state of a model. An analytical expression can be used to identify network ultrasensitivity , robustness , multistationarity , and invariants . For enzyme catalytic models that have no true steady state but nevertheless satisfy the assumptions for quasi-steady state, an analytical expression can relate the rate of product formation to the initial concentrations of the substrates and enzyme . Critically, these properties do not depend on the numerical values of the parameters, which may be difficult to measure . In our companion manuscript, we show that analytical steady state expressions can be used to identify changes in the kinetic rate constants that do not alter the species concentrations. These isostatic perturbations can be used to characterize the dynamic plasticity of a system, and also how changes in the rates of protein turnover can affect the response to perturbation, independently of changes to steady state concentrations.
Even if numerical interrogation is ultimately intended and all parameters must be assigned values, deriving an analytical expression for the steady state still confers a number of benefits. First, including steady state constraints can facilitate the construction of a model . As illustrated by our treatment of the Open Michaelis-Menten model, py-substitution affords considerable flexibility in selecting which quantities are independent — thus requiring numerical values prior to simulation — and which quantities can be derived from the independent quantities. This partly transforms the problem of parameterizing a model from one of numerically fitting the rate constants to available data , to one of identifying the steady state expression that maximizes incorporation of known quantities into the independent set of parameters. Second, incorporating steady state concentration measurements can reduce the total number of parameters required. In the traditional approach to parameterization, every rate constant is assigned a value prior to simulation, as well as the abundance of any species not subject to synthesis and degradation. Using py-substitution, only independent quantities must be assigned a value. This number is equal to the total number of species and reactions, minus the rank of the stoichiometric matrix. As the stoichiometric matrix approaches full rank, this number converges to the number of species. Since most systems have more reactions than species, py-substitution often requires fewer parameters than the traditional approach. This can be observed in the xEARM model, where 119 parameters are required for simulation after deriving a steady state expression using py-substitution (100 rate constants, 18 species, and the mitochondrial volume), versus 133 parameters required for traditional parameterization (115 rate constants, 17 species, and the mitochondrial volume).
Further, in the case of the xEARM model, we have demonstrated that an analytical expression of the steady state allows systematic characterization of its effect on the response to perturbation. This was made possible in two ways. First, it allowed the model to operate at a non-trivial steady state. In the original EARM model, infinite sensitivity to TRAIL is caused by unbalanced reactions. Once the receptor is engaged, caspase cleavage and pore formation proceed deterministically to completion. As a result, for cells to be “alive” prior to stimulation, the model must assume a trivial steady state in which the abundance of TRAIL and all reaction velocities are zero. Using py-substitution, we were able to engineer a non-trivial steady state that is viable at low doses of TRAIL. Second, we were able to apply systematic changes to the steady state concentrations. By virtue of the mapping function , these resulted in compensating changes to the kinetic rate constants such that steady state was preserved. For each modification, we were then able to calculate the number of TRAIL molecules required to induce cell death, as well as the sensitivity of this threshold to changes in the steady state concentrations of different species.
Previous studies with models operating at trivial steady states employed sensitivity metrics that were with respect to the time at which death occurs, and not whether it occurs , . These studies suggested that the dynamics of TRAIL-induced cell death depend critically on Bcl-2 . Also, whether cell death proceeds to completion depends on XIAP , and whether the mitochondrial feed-forward loop is required depends on the ratio of XIAP to Procaspase 3 . In contrast, our analysis indicates that whether cell death occurs is primarily determined by the ratio of Procaspase 8 to its negative regulator, Bar. Our sensitivity analysis with respect to the threshold at which death occurs is therefore related to but distinct from analyses that consider only the timing of death, and may relate better to clinical applications since we don't assume co-treatment with cyclohexamide.
For all these reasons, an analytical expression for the steady state of a model can be of general benefit to cell systems modeling. Indeed, other methods have previously addressed the challenge of deriving analytical steady state expressions, most notably the King-Altman method. Prior to the advent of modern computers, the authors realized that for a particular class of mass action models, the laborious calculation of steady state enzyme ratios could be achieved by a conceptually simpler graphical method. As we have shown, however, this simpler approach is no longer more efficient. More significantly, the King-Altman method requires that all reactions be first- or pseudo-first order in the time-varying species. Without this stipulation, Equation 7 no longer holds and the reaction network can no longer be described by a graph. This requirement is often stated as a pair of assumptions: 1) that no enzyme is itself a substrate and 2) that all substrates remain constant over the time scale of steady state formation . The second of these can be considered common to any method that treats time-varying species as constants when solving the steady state equation. The first of these, however, is violated by any cascade of post-translational modifications, for example the well-known MAP kinase cascade .
Although recent methods relax these assumptions , , in the contemporary systems biology literature, analytical derivation of the steady state rarely, if ever, precedes numerical interrogation of a model. Since this derivation is of considerable value, we sought to develop a method that was simple, scalable, and general to mass action models. First, we described our method using only concepts from linear algebra, and we have provided complete code for all seven examples described in this manuscript, with implementations in either Matlab or Maple. Second, we show that py-substitution scales well. The xEARM model has 58 species and 115 reactions, and we were able to derive a steady state expression in less than a minute on a conventional desktop computer. Finally, we demonstrated that py-substitution can be generally applied to chemical reaction networks whose reaction velocities are modeled by mass action kinetics. This is a considerably broader class of models than can be addressed using the King-Altman and other methods, which require that the reaction network exhibit specific structural properties.
This does, however, open up an interesting avenue for further research: precisely what properties must a mass action model exhibit for its steady state to be derived using py-substitution ? How many different steady state expressions are possible, and which of these is the “best”? As we have shown with the fumarase model, even after the rate constants and species concentrations were partitioned into sets and , 72 different steady state expressions were possible. These different expressions arose from flexibility in selecting the pivot columns in the coefficient matrix, since the pivot vs. free columns partition the linear variables into dependent vs. independent variables. Equivalently, these different expressions arise from flexibility in ordering the linear variables, since different orderings permute the columns of the coefficient matrix and result in a different reduced row echelon form. Since the number of possible steady state expressions is large but finite, a combinatorial optimization strategy ought to be able to identify the best steady state expression, where the difference between any two expressions could take into account measurement uncertainty in the independent quantities, as well as computational complexity in deriving the final steady state expression.
Finally, we consider that the steady state may not be the only state of interest, but perhaps specified dynamic states as well. Essentially, this replaces the zero vector in Equation 5 with a vector of non-zero values. From linear algebra, we know that the solution to this dynamic equation can be expressed as the sum of a particular solution to the dynamic equation and an arbitrary point in the null space of the coefficient matrix. The solution is thus straightforward, raising the possibility of incorporating specific dynamic states into the parameterization of a model as well.
This zip file contains all source code required to run py-substitution, in either Matlab or Maple. Implementations for every model described in this manuscript, including a pdf trace of the steady state derivation using py-substitution, are also included. For further details please see the included README file.
This table provides a summary and description of all mathematical symbols used in this manuscript.
This table compares the non-trivial steady state abundances of all molecular species in the xEARM model with their counterparts in the EARM model, published in . All abundances are in molecules.
Conceived and designed the experiments: PML GT AH. Performed the experiments: PML GT. Analyzed the data: PML GT. Wrote the paper: PML GT AH.
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