# Dynamic Modeling of Cell-Free Biochemical Networks Using Effective Kinetic Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Formulation and Properties of Effective Cell-Free Metabolic Models

_{1}and P

_{2}through a series of enzymatically catalyzed reactions, including a branch point at hypothetical metabolite M

_{2}. Several of these reactions involved cofactor dependence (AH or A), and various allosteric regulatory mechanisms modified the activity of pathway enzymes. Network A included feedback inhibition of the initial pathway enzyme (E

_{1}) by pathway end products P

_{1}and P

_{2}(Figure 1A). On the other hand, network B involved feedback inhibition of E

_{1}by P

_{2}and E

_{6}by P

_{1}(Figure 1B). In both networks, branch point enzymes E

_{3}and E

_{6}were subject to feed-forward activation by reduced cofactor AH. Lastly, it is known experimentally that cell-free systems have a finite operational lifespan. Loss of biosynthetic capability could be a function of many factors, e.g., cofactor or metabolite limitations. We modeled the loss of biosynthetic capability as a non-specific first-order decay of enzyme activity.

_{ij}in Equaion (10), was greater than unity, the inhibitory force was directly proportional to the cooperativity parameter, η in Equation (10). Thus, as the cooperativity parameter increased, the maximum reaction rate decreased (Figure 3B). Interestingly, our rule-based approach was unable to directly simulate competitive inhibition of enzyme activity. Taken together, the rule-based strategy captured classical regulatory patterns for both enzyme activation and inhibition. Thus, we are able to model complex kinetic phenomena such as ultrasensitivity, despite an effective description of reaction kinetics.

_{1}by P

_{1}and P

_{2}), as compared to the ON case (Figure 4A). We found this behavior was robust to the choice of underlying kinetic parameters, as we observed that same qualitative response across an ensemble of 100 randomized parameter sets, for fixed control parameters. The control ON/OFF response of network B was more subtle. In the OFF case, the behavior was qualitatively similar to network A. However, for the ON case, flux was diverted away from P

_{2}formation by feedback inhibition of E

_{6}activity at the M

_{2}branch point by P

_{1}(Figure 4B). Lower E

_{6}activity at the M

_{2}branch point allowed more flux toward P

_{1}formation, hence the yield of P

_{1}also increased (Figure 4C). Again, the control ON/OFF behavior of network B was robust to changes in kinetic parameters, as the same qualitative trend was conserved across an ensemble of 100 randomized parameters, for fixed control parameters. Taken together, these simulations suggested that the rule-based allosteric control concept could robustly capture expected feedback behavior for networks with uncertain kinetic parameters.

#### 2.2. Estimating Parameters and Effective Allosteric Regulatory Structures

_{5}and end product P

_{1}approximately every 20 min using network A. This data set is similar to published cell-free studies, both in terms of network coverage and sampling frequency [23]. We then generated an ensemble of model parameter estimates by minimizing the difference between model simulations and the synthetic data using particle swarm optimization (PSO), starting from random initial parameter guesses. The estimation of kinetic parameters was sensitive to the choice of regulatory structure (Figure 5). PSO identified an ensemble of parameters that bracketed the mean of the synthetic measurements in less than 1000 iterations when the control structure was correct (Figure 5A,B). However, with control mismatch (network B simulated with network A parameters), model simulations were not consistent with the synthetic data (Figure 5C,D). Taken together, these results suggested that we could perhaps simultaneously estimate both parameters and network control architectures, as incorrect control structures would be manifest as poor model fits.

_{2}and enzyme E

_{1}; in network A, end product P

_{2}was assumed to inhibit E

_{1}, while in network C, end product P

_{2}activated E

_{1}. Lastly, when the initial population was heavily biased towards incorrect structures (initial population seeded with 90% incorrect structures), the particle swarm misidentified the correct allosteric structure (Figure 7C). Interestingly, while each particle swarm identified parameter sets that minimized the simulation error, the estimated parameter values were not necessarily similar to the true parameters. The angle between the estimated and true parameters was not consistently small across the swarms (identical parameters would give an angle of zero). This suggested that our particle swarm approach identified a sloppy ensemble, i.e., parameter estimates that were individually incorrect but collectively exhibited the correct model behavior.

_{3}and E

_{6}control. Moreover, consistent with the correct model structure, production of end product P

_{1}was the preferred branch for all model configurations. However, there was variability in P

_{2}production flux across the population of models, especially for the uniform swarm when compared with the other cases. High P

_{1}branch flux resulted in end product inhibition of E

_{1}in both network A and network C, however in network D and E, high P

_{1}flux induced E

_{1}activation. These trends were manifested in different flux profiles, where the negatively biased population appeared more uniform across the population compared with the other swarms, and had higher E

_{1}specific activity. Interestingly, the behavior of network A and network C highlighted an artifact of our integration rule; both a positive or negative feedback connection from P

_{2}to E

_{1}were ignored because the P

_{1}inhibition of E

_{1}dominated. Thus, while theoretically distinct, network A and network C appeared operationally to the PSO algorithm to be the same network. On the other hand, networks B, D and E showed distinct behavior that was not consistent with the true network. These architectures exhibited either limited inhibition (network B) or activation (network D and E) of E

_{1}activity, resulting in significantly different metabolic flux profiles. However, the PSO was able to find low error parameter solutions, despite the mismatch in the control structures (error values similar, but not better than the best network A and network C estimates). Taken together, these results suggested that a uniform sampling approach could potentially yield an unbiassed estimate of both kinetic parameters and control structures. However, the negatively biased particle swarm results illustrated a potential shortcoming of the approach, namely convergence to a local error minimum despite a significantly incorrect control structure. This suggested that estimated model structures will need to be further evaluated, for example by generating falsifiable experimental designs which could distinguish between low error solutions.

## 3. Discussion

_{1}and P

_{2}through a series of enzymatically catalyzed reactions, including a branch point at a hypothetical metabolite M

_{2}. Each network also included the same cofactors and cofactor recycle architecture. However, while all five networks shared the same enzymatic connectivity, each had different allosteric regulatory connectivity. We found that simple effective rules, when integrated with traditional enzyme kinetic expressions, could capture complex allosteric patterns such as ultrasensitivity, or non-competitive inhibition in the absence of specific mechanistic information. Moreover, when integrated into network models, these rules captured classical regulatory patterns such as product-induced feedback inhibition. Lastly, we simultaneously estimated kinetic parameters and discriminated between competing regulatory structures, using synthetic data in combination with a modified particle swarm approach. If we considered all putative regulatory architectures to be equally likely, we were able to estimate a sloppy ensemble of models with the correct architecture and kinetic parameters. Thus, we identified parameter values that were different from their true values, but nonetheless produced reasonable model performance (low error). This suggested that we captured important parameter combinations (stiff combinations), while simultaneously missing other parameter combinations (sloppy combinations). This was similar to the earlier study of Brown and Sethna [29], which showed that reasonable model predictions were possible, despite sometimes only order of magnitude parameter estimates, if the stiff parameter combinations were well constrained.

## 4. Materials and Methods

#### 4.1. Formulation and Solution of the Model Equations

_{i}) and scaled enzyme abundance (ϵ

_{i}) in hypothetical cell-free metabolic networks:

_{j}(x, ϵ, k) denotes the rate of reaction j. Typically, reaction j is a non-linear function of metabolite and enzyme abundance, as well as unknown kinetic parameters $\mathrm{r}\phantom{\rule{0.2em}{0ex}}\left(\mathcal{K}\times 1\right)$. The quantity σ

_{ij}denotes the stoichiometric coefficient for species i in reaction j. If σ

_{ij}> 0, metabolite i is produced by reaction j. Conversely, if σ

_{ij}< 0, metabolite i is consumed by reaction j, while σ

_{ij}= 0 indicates metabolite i is not connected with reaction j. Lastly, λ

_{i}denotes the scaled enzyme degradation constant. The system material balances were subject to the initial conditions x (t

_{o}) = x

_{o}and ϵ (t

_{o}) = 1 (initially we have 100% cell-free enzyme abundance).

_{j}):

_{i}denotes the scaled enzyme activity which catalyzes reaction j, and K

_{jS}denotes the saturation constant for species s in reaction j. The product in Equaion (4) was carried out over the set of reactants for reaction j (denoted as ${m}_{j}^{-}$).

_{j}depended upon the combination of factors which influenced the activity of enzyme i. For each enzyme, we used a rule-based approach to select from competing control factors (Figure 2). If an enzyme was activated by m metabolites, we modeled this activation as:

_{ij}(Ƶ) ≤ 1 was a regulatory transfer function that calculated the influence of metabolite i on the activity of enzyme j. Conversely, if enzyme activity was inhibited by a m metabolites, we modeling this inhibition as:

^{+}and j

^{–}denoted the sets of activating and inhibitory factors for enzyme j. If an enzyme had no allosteric factors, we set v

_{j}= 1. There are many possible functional forms for 0 ≤ f

_{ij}(Ƶ) ≤ 1. However, in this study, each individual transfer function took the form:

_{j}denotes the abundance of the j factor (e.g., metabolite abundance), and κ

_{ij}and η are control parameters. The κ

_{ij}parameter represents a species gain parameter, while η is a cooperativity parameter (similar to a Hill coefficient). In the case η > 1, the allosteric interaction displays positive cooperativity. For η < 1, the interaction is negatively cooperative. Finally, if η = 1, the interaction displays no cooperativity. The effect of different values of η on reaction rate can be seen in Figure 3. The model equations were encoded using the Octave programming language and solved using the LSODE routine in Octave [54]. In some cases, metabolic fluxes (or other quantities) were scaled according to:

#### Estimation of model parameters and structures from synthetic experimental data

_{j}(τ, k) denotes the simulated value for species j at time τ, and ⍵

_{j}(τ) denotes the experimental measurement variance for species j at time τ. The outer summation is respect to time, while the inner summation is with respect to state. We approximated a realistic model identification scenario, assuming noisy experimental data, limited sampling resolution (approximately 20 min per sample) and a limited number of measurable metabolites. We assumed a constant coefficient of variation of 10% for the synthetic data set.

_{i}denotes the perturbation to the vector of parameters k

_{i}for particle i. (θ

_{1},θ

_{2},θ

_{3}) are adjustable parameters, L

_{i}denotes the best local solution found by particle i, and G denotes the best solution found over the entire population of particles. The quantities r

_{1}and r

_{2}denote uniform random vectors with the same dimension as the number of unknown model parameters (K × 1). In this study, we used (θ

_{1},θ

_{2},θ

_{3}) = (1.0,0.05564,0.02886). The quality of parameter estimates was measured using two criteria, goodness of fit (model residual) and angle between the estimated parameter vector k

_{j}and the true parameter set k*:

_{j}were perfect, the residual between the model and synthetic data and the angle between k

_{j}and the true parameter set k* would be equal to zero.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Proof-of-concept cell-free metabolic networks considered in this study. Substrate S is converted to products P

_{1}and P

_{2}through a series of chemical conversions catalyzed by enzyme(s) E

_{j}. The activity of the pathway enzymes is subject to both positive and negative allosteric regulation.

**Figure 2.**Schematic of rule-based allosteric enzyme activity control laws. Traditional enzyme kinetic expressions, e.g., Michaelis-Menten or multiple saturation kinetics, are multiplied by an enzyme activity control variable 0 ≤ v

_{j}≤ 1. Control variables are functions of many possible regulatory factors encoded by arbitrary functions of the form 0 ≤ f

_{j}(Ƶ) ≤ 1. At each simulation time step, the v

_{j}variables are calculated by evaluating integration rules such as the max or min of the set of factors f

_{1},… influencing the activity of enzyme E

_{j}.

**Figure 3.**Kinetics of simple transformations in the presence of activation and inhibition. (

**A**) The conversion of substrate S to product P by enzyme E was activated by S. For a fixed control gain parameter κ

_{control,}the reaction rate approached a step for increasing cooperativity control parameter η. For activation simulations κ

_{control}= 0.05 and η = {0.01,0.1,1,2,4,6,8,10}; (

**B**) The conversion of substrate S to product P by enzyme E with inhibitor I. For a fixed control gain parameter κ

_{control}, the reaction rate approximated non-competitive inhibition for increasing cooperativity control parameter η. For the inhibition simulations κ

_{control}= 1.5 and η = {0.01, 0.1,1, 2, 4, 6, 8,10}.

**Figure 4.**ON/OFF control simulations for Network A and Network B for an ensemble of 100 kinetic parameter sets versus time. For each case, simulations were conducted using kinetic and initial conditions generated randomly from a hypothetical true parameter set. The gray area represents ± one standard deviation surrounding the mean. Control parameters were fixed during the ensemble calculations. (

**A**) End product P

_{1}abundance versus time for Network A. The abundance of P

_{1}decreased with end product inhibition of E

_{1}activity (Control-ON) versus the no inhibition case (Control-OFF); (

**B**) End product P

_{2}abundance versus time for Network B. Inhibition of branch point E

_{6}by end product P

_{1}decreased P

_{2}abundance (Control-ON) versus the no inhibition case (Control-OFF); (

**C**) End product P

_{1}abundance versus time for Network A. Inhibition of branch point E

_{6}by end product P

_{1}decreased P

_{1}abundance (Control-ON) versus the no inhibition case (Control-OFF).

**Figure 5.**Parameter estimation from synthetic data for the same and mismatched allosteric control logic using particle swarm optimization (PSO). Synthetic experimental data was generated from a hypothetical parameter set using Network A, where substrate S, end product P

_{1}and intermediate M

_{5}were sampled approximately every 20 min. For cases (A,B) 20 particles were initialized with randomized parameters and allowed to search for 300 iterations. (A,B) PSO estimated an ensemble of 20 parameters sets consistent with the synthetic experimental data assuming the correct enzymatic and control connectivity starting from randomized initial parameters; (C,D) In the presence of control mismatch (Network B control policy simulated with Network A kinetic parameters) the ensemble of models did not describe the synthetic data. The synthetic data plotted here was unperturbed by noise. However, we assumed a constant coefficient of variation of 10% for the synthetic data during parameter estimation.

**Figure 6.**Schematic of the alternative allosteric control programs used in the structural particle swarm computation. Each network had the same enzymatic connectivity, initial conditions and kinetic parameters, but alternative feedback control structures for the first enzyme in the pathway.

**Figure 7.**Combined control and kinetic parameter search using modified particle swarm optimization (PSO). A population of 100 particles was initialized with randomized kinetic parameters and one of five possible control configurations (Network A–E). Simulation error was minimized for a synthetic data set (S, end product P

_{1}and intermediate M

_{5}sampled approximately every 20 min) generated using Network A. (

**A**) Simulation error versus parameter set angle for 100 particles biased toward the correct regulatory program (A,B,C,D,E) = (40%, 10%, 20%, 20% and 10%); (

**B**) Simulation error versus parameter set angle for 100 uniformly distributed particles (A,B,C,D,E) = (20%, 20%, 20%, 20% and 20%); (

**C**) Simulation error versus parameter set angle for 100 negatively biased particles (A,B,C,D,E) = (10%, 40%, 10%, 20% and 20%). Network A (the correct structure) was preferentially identified for positively and uniform biased particle distributions, but misidentified in the presence of a large incorrect bias.

**Figure 8.**Metabolic flux and control variables as a function of network type and particle index at t = 100 min. The particle error, the control variables governing E

_{1}, E

_{3}and E

_{6}activity (v

_{1}, v

_{3}and v

_{3}) and the scaled metabolic flux were calculated for the positively (top), uniformly (middle) and negatively (bottom) biased particle swarms (N = 100). Blue denotes a low value, while red denotes a high value for the respective quantity being plotted. The particles from each swarm were sorted based upon simulation error (low to high error). (

**A**) Model performance for the positively biased particle swarm as a function of particle index; (

**B**) Model performance for the uniformly biased particle swarm as a function of particle index; (

**C**) Model performance for the negatively biased particle swarm as a function of particle index. Models with significant control mismatch showed distinct control and flux patterns versus those models with the correct or closely related control policies. In particular, models with the correct control policy showed stronger inhibition of E

_{1}activity, leading to decreased flux from S→P

_{1}. Conversely, models with significant mismatch had increased E

_{1}activity, leading to an altered flux distribution. This is especially apparent in the negatively biased particle swarm.

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**MDPI and ACS Style**

Wayman, J.A.; Sagar, A.; Varner, J.D.
Dynamic Modeling of Cell-Free Biochemical Networks Using Effective Kinetic Models. *Processes* **2015**, *3*, 138-160.
https://doi.org/10.3390/pr3010138

**AMA Style**

Wayman JA, Sagar A, Varner JD.
Dynamic Modeling of Cell-Free Biochemical Networks Using Effective Kinetic Models. *Processes*. 2015; 3(1):138-160.
https://doi.org/10.3390/pr3010138

**Chicago/Turabian Style**

Wayman, Joseph A., Adithya Sagar, and Jeffrey D. Varner.
2015. "Dynamic Modeling of Cell-Free Biochemical Networks Using Effective Kinetic Models" *Processes* 3, no. 1: 138-160.
https://doi.org/10.3390/pr3010138