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7.3: Chemical Equilibrium—Part 1: Forward and Reverse Reactions - Biology

7.3: Chemical Equilibrium—Part 1: Forward and Reverse Reactions - Biology


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Chemical equilibrium—Part 1: forward and reverse reactions

Understanding the concept of chemical equilibrium is critical to following several of the discussions that we have in BIS2A and indeed throughout biology and the sciences. Let us, rather, begin developing our understanding of equilibrium by considering the reversible reaction below:

Hypothetical reaction #1: A hypothetical reaction involving compounds A, B and D. If we read this from left to right, we would say that A and B come together to form a larger compound: D. Reading the reaction from right to left, we would say that compound D breaks down into smaller compounds: A and B.

We first need to define what is meant by a “reversible reaction.” The term “reversible” simply means that a reaction can proceed in both directions. That is, the things on the left side of the reaction equation can react together to become the things on the right of the equation, AND the things on the right of the equation can also react together to become the things on the left side of the equation. Reactions that only proceed in one direction are called irreversible reactions.

To start our discussion of equilibrium, we begin by considering a reaction that we posit is readily reversible. In this case, it is the reaction depicted above: the imaginary formation of compound D from compounds A and B. Since it is a reversible reaction, we could also call it the decomposition of D into A and B. Let us, however, imagine an experiment in which we watch the reaction proceed from a starting point where only A and B are present.

Example #1: Left-balanced reaction

Hypothetical reaction #1: time course
Concentrationt=0t=1t=5t=10t=15t=20t=25t=30t=35t=40
[A]100908070656260606060
[B]100908070656260606060
[C]0102030453840404040

At time t = 0 (before the reaction starts), the reaction has 100 concentration units of compounds A and B and zero units of compound D. We now allow the reaction to proceed and observe the individual concentrations of the three compounds over time (t=1, 5, 10, 15, 20, 25, 30, 35, and 40 time units). As A and B react, D forms. In fact, one can see D forming from t=0 all the way to t=25. After that time, however, the concentrations of A, B and D stop changing. Once the reaction reaches the point where the concentrations of the components stop changing, we say that the reaction has reached equilibrium. Notice that the concentrations of A, B, and D are not equal at equilibrium. In fact, the reaction seems left balanced so that there is more A and B than D.

Note: Common student misconception warning

Many students fall victim to the misconception that the concentrations of a reaction’s reactants and products must be equal at equilibrium. Given that the term equilibrium sounds a lot like the word “equal,” this is not surprising. But as the experiment above tries to illustrate, this is NOT correct!

Example #2: right-balanced reaction

We can examine a second hypothetical reaction, the synthesis of compound (ce{J}) from the compounds (ce{E}) and (ce{F}).

[ ce{E +F <=> J} onumber]

Hypothetical reaction #2: A hypothetical reaction involving compounds E, F and J. If we read this from left to right, we would say that E and F come together to form a larger compound: J. Reading the reaction from right to left, we would say that compound J breaks down into smaller compounds: E and F.

The structure of hypothetical reaction #2 looks identical to that of hypothetical reaction #1, which we considered above—two things come together to make one bigger thing. We just need to assume, in this case, that E, F, and J have different properties from A, B, and D. Let’s imagine a similar experiment to the one described above and examine this data:

Hypothetical reaction #2: time course

In this case, the reaction also reaches equilibrium. This time, however, equilibrium occurs at around t=30. After that point, the concentrations of E, F, and J do not change. Note again that the concentrations of (ce{E}), (ce{F}), and (ce{J}) are not equal at equilibrium. In contrast to hypothetical reaction #1 (the ABD reaction), this time the concentration of J, the thing on the right side of the arrows, is at a higher concentration than E and F. We say that, for this reaction, equilibrium lies to the right.

Four more points need to be made at this juncture.

  • Point 1: Whether equilibrium for a reaction lies to the left or the right will be a function of the properties of the components of the reaction and the environmental conditions that the reaction is taking place in (e.g., temperature, pressure, etc.).
  • Point 2: We can also talk about equilibrium using concepts of energy, and we will do this soon, just not yet.
  • Point 3: While hypothetical reactions #1 and #2 appear to reach a point where the reaction has “stopped,” you should imagine that reactions are still happening even after equilibrium has been reached. At equilibrium the “forward” and “reverse” reactions are just happening at the same rate. That is, in example #2, at equilibrium J is forming from E and F at the same rate that it is breaking down into E and F. This explains how the concentrations of the compounds aren’t changing despite the fact that the reactions are still happening.
  • Point 4: From this description of equilibrium, we can define something we call the equilibrium constant. Typically, the constant is represented by an uppercase K and may be written as Keq. In terms of concentrations, Keq is written as the mathematical product of the reaction product concentrations (stuff on the right) divided by the mathematical product of the reactant concentrations (stuff on the left). For example, Keq,1 = [D]/[A][B], and Keq,2 = [J]/[E][F]. The square brackets "[]" indicate the “concentration of” whatever is inside the bracket.

Other Five-membered Rings with Three or more Heteroatoms, and their Fused Carbocyclic Derivatives

6.09.6.2.1 Bimolecular reactions of Δ 2 -1,2,3,4-thiatriazolines

Reaction kinetics for the interaction of 5-alkyliminothiatriazoles 52 or 58 with heterocumulenes, nitriles, ketones, imines, or other dipolarophiles ab show that the decomposition of the thiatriazole is bimolecular, and new heterocyclic five-membered rings 71 are formed ( Scheme 15 ). The term ‘masked 1,3-dipolar cycloaddition’ was used by L’abbé and co-workers for this type of reaction <1978JOC4951> , the thioimidate function being the masked 1,3-dipole. The reaction is thought to involve a thiapentalenic intermediate 70 with hypervalent sulfur. The product 71 is itself a masked dipole and often further reactions take place.

These complex cycloaddition reactions have been reviewed extensively in CHEC-II(1996) <1996CHEC-II(4)691> and practically no new data on this subject have appeared since then.

An interesting case worth mentioning is the intramolecular variant, leading to fused N, S-containing heterocycles. In this case nitriles, alkynes, or electron-poor alkenes were the dipolarophiles <1992J(P1)181, 1992T7505, 1993T4439> . A few representative examples of the reaction products 7274 are given in (Equation 6 ).


Background

NASA defines life as a self-sustaining chemical system capable of undergoing Darwinian evolution [1, 2]. Here, “self-sustaining” implies that a living system should not need continuous intervention by a higher entity (e.g. a graduate student or a god) to continue as life [1]. Another popular definition of life comes from the concept of autopoiesis: a system is said to be living if it is capable of self-sustaining owing to an inner network of reactions that regenerate all the system’s components [3]. But the phrase “self-sustaining” refers to “no-intervention” in the former definition, but “regeneration” in the latter.

In the fields that relate to essentials of life, such as biochemistry [4, 5], molecular biology [6], network autocatalysis [7–11], and non-equilibrium thermodynamics [12, 13], this phrase “self-sustaining” or “self-sustainability” is being frequently used in a vague and ambiguous manner, but mostly refers to the two different aspects mentioned above, although they are not necessarily contradictory.

In the “no-intervention” school, for example, a designed RNA enzyme system that underwent exponential amplification was said to be self-sustaining in the sense that the amplification could be continued indefinitely [4]. A chemical reaction loop that was invented to convert amines to alcohols was said to be self-sustaining in the sense that products were created, purified, and isolated without manual operations [5].

On the other hand, the “regeneration” school focuses more on the mechanism that leads to self-sustainability [8–11, 14–22]. For example, Piedrafita et al. referred “self-sustainability” to “metabolic closure” that all of the catalysts essential for the survival of an organism have to be produced internally [14, 15, 17]. In the reflexively autocatalytic and food-generated (RAF) theory, “self-sustaining” was referred to that each molecule in a chemical network can be produced starting from the food source [8, 18, 19, 23]. To be noticed, the chemical organisation theory proposed a rigorous definition of “self-sustaining” (self-maintaining, in their words) [10, 20, 21]: A set of molecules is called semi-self-maintaining if topologically all molecules that are consumed are also produced it is further called self-maintaining if the stoichiometry of the network makes the production rate of each molecule strictly nonnegative.

Although the definition in the chemical organisation theory is rigorous, it has shortcomings. Firstly, it is merely a topological description. Although the topology of the coupled network is important, the strength of the couplings (namely the reaction rates) could completely change the behaviours of the whole system [14, 24–26]. Secondly, it is defined with respect to a set of molecules, rather than a system. For a reaction system that involves N molecule types, it can be partitioned into ( + + dots + = 2^) sets of molecules, each of which may or may not be self-maintaining, based on their definition. However, different sets of molecules cannot be physically isolated as they are all involved in one system. Thirdly, this definition is too stringent: It requires all molecules that are consumed to be also produced. However, even for a living system which should be categorised as a self-sustaining system, it cannot produce the resource molecules it needs.

Another crucial point about self-sustainability is how to connect it with heredity, another essential of life. Currently, they are often considered to be independent of each other. So origins of life require one origin of self-sustainability and an independent origin of heredity, respectively. That is also why the theory stating that life began with a self-sustaining chain of chemical reactions, without the requirement for genetic information, has been heavily questioned [27–30]. But what if self-sustainability naturally guarantees heredity, or at least preliminary heredity (as we shall discuss at the end of this paper)?

The last point is how to empirically discern whether a chemical system is self-sustaining or not, which is not a trivial question at all. The definitions mentioned above are all based on the complete topological information of a chemical reaction system, including all the reactants, products, intermediates and how they are connected via reactions. However, in most real chemical experiments, only partial information is known, or even, all we know are what has been put into the system and what has been produced. The complete information is almost impossible. To get around this problem, I will base the definition of self-sustainability only on the information of what has been put into and produced from the system.

Furthermore, to connect real experiments, I will define self-sustainability in the context of a continuous-flow stirred tank reactor (CSTR) particularly, which is commonly used in chemical engineering [31–33]. Nevertheless, this definition will not lose its generality, since whether a system has the ability to self-sustain is an intrinsic property of the system itself, which we shall see later.

This paper is organised as follows. Firstly, the theoretical setup is introduced via a detailed example, followed by the formal definition of self-sustainability. And then, the general properties of a chemical system that has the potential to be self-sustaining are discussed, to give guidance for constructing or finding such systems in real biology and chemistry. After that, various self-sustaining systems that are observed in labs and real living systems are shown. In the “Discussion” section, besides some comments on the definition, two more questions are discussed: why self-sustaining systems have preliminary heredity, and why life and fire are distinct in terms of self-sustainability. The conclusions are drawn in the end.


Variations in the Form of the Equilibrium Constant Expression

Because equilibrium can be approached from either direction in a chemical reaction, the equilibrium constant expression and thus the magnitude of the equilibrium constant depend on the form in which the chemical reaction is written. For example, if we write the reaction described in Equation ( ef) in reverse, we obtain the following:

[cC+dD ightleftharpoons aA+bB label]

The corresponding equilibrium constant (K&prime) is as follows:

This expression is the inverse of the expression for the original equilibrium constant, so (K&prime = 1/K). That is, when we write a reaction in the reverse direction, the equilibrium constant expression is inverted. For instance, the equilibrium constant for the reaction (ce) is as follows:

but for the opposite reaction, (2 NO_2 ightleftharpoons N_2O_4), the equilibrium constant K&prime is given by the inverse expression:

Consider another example, the formation of water:

Because (ce

) is a good reductant and (ce) is a good oxidant, this reaction has a very large equilibrium constant ((K = 2.4 imes 10^) at 500 K). Consequently, the equilibrium constant for the reverse reaction, the decomposition of water to form (ce) and (ce), is very small: (K&prime = 1/K = 1/(2.4 imes 10^) = 4.2 imes 10^). As suggested by the very small equilibrium constant, and fortunately for life as we know it, a substantial amount of energy is indeed needed to dissociate water into (ce) and (ce). The equilibrium constant for a reaction written in reverse is the inverse of the equilibrium constant for the reaction as written originally. Writing an equation in different but chemically equivalent forms also causes both the equilibrium constant expression and the magnitude of the equilibrium constant to be different. For example, we could write the equation for the reaction with the equilibrium constant K&Prime is as follows: The values for K&prime (Equation ( ef)) and K&Prime are related as follows: In general, if all the coefficients in a balanced chemical equation were subsequently multiplied by (n), then the new equilibrium constant is the original equilibrium constant raised to the (n^) power. Example (PageIndex): The Haber Process At 745 K, K is 0.118 for the following reaction: What is the equilibrium constant for each related reaction at 745 K? Given: balanced equilibrium equation, (K) at a given temperature, and equations of related reactions Asked for: values of (K) for related reactions Write the equilibrium constant expression for the given reaction and for each related reaction. From these expressions, calculate (K) for each reaction. The equilibrium constant expression for the given reaction of (N_) with (H_) to produce (NH_) at 745 K is as follows: This reaction is the reverse of the one given, so its equilibrium constant expression is as follows: In this reaction, the stoichiometric coefficients of the given reaction are divided by 2, so the equilibrium constant is calculated as follows: At 527°C, the equilibrium constant for the reaction is (7.9 imes 10^4). Calculate the equilibrium constant for the following reaction at the same temperature: 3. Review of Modeling Strategies for BRNs

Mathematical models describe dependencies of observations on the model parameters. A general procedure for constructing mathematical models of biological systems is described in Chou and Voit (2009). The bio-reactors are mathematically described in Vargas et al. (2014), Ali et al. (2015), and Farza et al. (2016). The model building is an iterative process which is often combined with the optimum experiment design (Rodriguez-Fernandez et al., 2006b). The model structure affects the selection as well as the performance of parameter estimators. The structural identifiability and validity of multiple models together with the parameter sensitivity was considered in Jaqaman and Danuser (2006). The parameter estimation can be performed together with the discrimination among several competing models, for instance, when the model structure is only partially known. The model structure and the parameter values to achieve the desired dynamics can be obtained by the means of statistical inference (Barnes et al., 2011). The synthesis of parameter values for BRNs is also considered in ჎ška et al. (2017). The probabilistic model checking can be used to facilitate the robustness analysis of stochastic biochemical models (჎ska et al., 2014). The model checking is investigated in a number of references including Palmisano (2010), Brim et al. (2013), ჎ska et al. (2014), Mizera et al. (2014), Hussain et al. (2015), Mancini et al. (2015), ჎ška et al. (2017), and Milios et al. (2018). An iterative, feedback dependent modularization of models with the parameters identification was devised in Lang and Stelling (2016). A selection among several hierarchical models assuming Akaike information was studied in Rodriguez-Fernandez et al. (2013).

Modeling strategies of BRNs often involve the kinetics of chemical reactants which are described by the law of mass action or by the rate law (Schnoerr et al., 2017). Both these laws model the dependency of chemical reaction rates on the species concentrations. The reaction kinetics can be considered at steady state or in the transition to steady state, although the steady state may not be always achieved. There are also other kinetic models, such as the Michaelis-Menten kinetics for the enzyme-substrate reactions (Rumschinski et al., 2010), the Hill kinetics for cooperative ligand binding to macromolecules (Fey and Bullinger, 2010), the kinetics for logistic growth models in GRNs (Ghusinga et al., 2017), the kinetics for the birth-death processes (Daigle et al., 2012), and the stochastic Lotka-Volterra kinetics which are associated with the prey-predatory networks (Boys et al., 2008).

Single molecule stochastic models describe BRNs qualitatively by generating the probabilistic trajectories of species counts. A BRN can be modeled as a sequence of reactions occurring at random time instances (Amrein and Künsch, 2012). The stochastic kinetics mathematically correspond to a Markov jump process with the random state transitions between the species counts (Andreychenko et al., 2012). Alternatively, the time sequence of chemical reactions can be viewed as a hidden Markov process (Reinker et al., 2006). The Markov jump processes can be simulated exactly using the classical Gillespie algorithm, so that the competing reactions are selected assuming a Poisson process with the intensity proportional to the species counts (Golightly et al., 2012 Kügler, 2012), although, in general, the intensity can be an arbitrary function of the species counts. The random occurrences of reactions can be also described using the hazard function (Boys et al., 2008). Non-homogeneous Poisson processes can be simulated by the thinning algorithm of Lewis and Shedler (Sherlock et al., 2014).

The number of species in BRN and their molecule counts can be large, so the state space of the corresponding continuous time Markov chain (CTMC) model is huge (Angius and Horváth, 2011). The large state space can be truncated by considering only the states significantly contributing to the parameter likelihood (Singh and Hahn, 2005). The parameter likelihoods can be updated iteratively assuming the increments and decrements of the species counts (Lecca et al., 2009). The probabilistic state space representations of BRNs as dynamic systems were considered in Andreychenko et al. (2011), Gupta and Rawlings (2014), McGoff et al. (2015), and Schnoerr et al. (2017). An augmented state space representation of BRN derived from the ordinary differential equations (ODEs) is obtained in Baker et al. (2013).

More generally, mechanistic models of BRNs are obtained by assuming that biological systems are built up from the actual or perceived components which are governed by the physical laws (Hasenauer, 2013 Pullen and Morris, 2014 White et al., 2016 Fröhlich et al., 2017). It is a different strategy to empirical models which are reverse-engineered from observations (Geffen et al., 2008 Bronstein et al., 2015 Dattner, 2015). The black-box modeling can be assumed with some limitations when there is little knowledge about the underlying biological processes (Chou and Voit, 2009).

Many models containing multiple unknown parameters are often poorly constrained. Even though such models may be still fully identifiable, they are usually ill-conditioned, and often referred to as being sloppy (Toni and Stumpf, 2010 Erguler and Stumpf, 2011 White et al., 2016). The parameter estimation and experimental design for sloppy models are investigated in Mannakee et al. (2016) where it is shown that the dynamic properties of sloppy models usually depend only on several key parameters with the remaining parameters being largely unimportant. A sequence of hierarchical models with increasing complexity was proposed in White et al. (2016) to overcome the complexity and sloppiness of conventional models.

3.1. Modeling BRNs by Differential Equations

The time evolution of states with the probabilistic transitions is described by a chemical master equation (CME) (Andreychenko et al., 2011 Weber and Frey, 2017). The CME is a set of coupled first-order ODEs or partial differential equations (PDEs) (Fearnhead et al., 2014 Penas et al., 2017 Teijeiro et al., 2017) representing a continuous time approximation and describing the BRN quantitatively. The ODE model of a BRN can be also derived as a low-order moment approximation of the CME (Bogomolov et al., 2015). For the models with stochastic differential equations (SDEs), it is often difficult to find the transition probabilities (Karimi and Mcauley, 2013 Fearnhead et al., 2014 Sherlock et al., 2014). The PDE approximation can be obtained assuming a Taylor expansion of the CME (Schnoerr et al., 2017). The error bounds for the numerically obtained stationary distributions of the CME are obtained in Kuntz et al. (2017). The CME for a hierarchical BRN consisting of the dependent and independent sub-networks is solved analytically in Reis et al. (2018). A path integral form of the ODEs has been considered in Liu and Gunawan (2014) and Weber and Frey (2017). The BRN models with memory described by the delay differential equations (DDEs) are investigated in Zhan et al. (2014). The mixed-effect models assume multiple instances of the SDE based models to evaluate statistical variations between and within these models (Whitaker et al., 2017).

A comprehensive tutorial on the ODE modeling of biological systems is provided in Gratie et al. (2013). The ODE models can be solved numerically via discretization. For instance, the finite differences method (FDM) can be used to obtain difference equations (Fröhlich et al., 2016). However, the algorithms for numerically solving the deterministic ODE models or simulating the models with SDEs may not be easily parallelizable, and they may have problems with numerical stability. The ODE models are said to be stiff, if they are difficult to solve or simulate, for example, if they comprise multiple processes at largely different time scales (Sun et al., 2012 Cazzaniga et al., 2015 Kulikov and Kulikova, 2017). Alternatively, the BRN structure can be derived from its ODE representation (Fages et al., 2015). A similar strategy is assumed in Plesa et al. (2017) where the BRN is inferred from the deterministic ODE representation of the time series data.

A survey of methods for solving the CME of gene expression circuits is provided in Veerman et al. (2018). These methods involve propagators, time-scale separation, and the generating functions (Schnoerr et al., 2017). For instance, the time-scale separation can be used to robustly decompose the CME into a hierarchy of models (Radulescu et al., 2012). A reduced stochastic description of BRNs exploiting the time-scale separation is studied in Thomas et al. (2012).

If the deterministic ODEs cannot be solved analytically, one can use Langevin and Fokker-Planck equations as the stochastic diffusion approximations of the CME (Hasenauer, 2013 Schnoerr et al., 2017). The Fokker-Planck equation can be solved to obtain a deterministic time evolution of the system state distribution (Kügler, 2012 Liao et al., 2015 Schnoerr et al., 2017). The deterministic and stochastic diffusion approximations of stochastic kinetics are reviewed in Mozgunov et al. (2018). The chemical Langevin equation (CLE) is a SDE consisting of a deterministic part describing the slow macroscopic changes, and a stochastic part representing the fast microscopic changes which are dependent on the size of the deterministic part (Golightly et al., 2012 Cseke et al., 2016 Dey et al., 2018). In the limit, as the deterministic part increases, the random fluctuations can be neglected, and the deterministic kinetics described by the Langevin equation becomes the reaction rate equation (RRE) (Bronstein et al., 2015 Fröhlich et al., 2016Loos et al., 2016).

3.2. Modeling BRNs by Approximations

A popular strategy to obtain computationally efficient models is to assume approximations, such as meta-heuristics and meta-modeling (Sun et al., 2012 Cedersund et al., 2016). The quasi-steady state (QSS) and quasi-equilibrium (QE) approximations of BRNs are investigated in Radulescu et al. (2012). The modifications of QSS models are investigated in Wong et al. (2015). It is also common to approximate the system dynamics assuming continuous ODEs or SDEs (Fearnhead et al., 2014). The SDE model is preferred when the number of molecules is small, since the deterministic ODE model may be inaccurate (Gillespie and Golightly, 2012). It is generally difficult to quantify the approximation errors in the diffusion-based models. The forward-reverse stochastic diffusion with the deterministic approximation of propensities by the observed data was considered in Bayer et al. (2016).

The mass action kinetics can be used to obtain a deterministic approximation of CME. The corresponding deterministic ODEs can accurately describe the system dynamics, provided that the molecule counts of all the species are sufficiently large (Sherlock et al., 2014 Yenkie et al., 2016). Other CME approximations assume the finite state projections, the system size expansion, and the moment closure methods (Chevaliera and Samadb, 2011 Schnoerr et al., 2017). These methods are attractive, since they are easy to implement and efficient computationally. They do not require the complete statistical description, and they achieve good accuracy if the species appear in large copy numbers (Schnoerr et al., 2017). The moment closure methods leading to the coupled ODEs can approach the CME solution with a low computational complexity (Bogomolov et al., 2015 Fröhlich et al., 2016 Schilling et al., 2016). Specifically, the n-th moment of the population size depends on its (n+1) moment, and to close the model, the (n+1)-th moment is approximated by a function of the lower moments (Ruess et al., 2011 Ghusinga et al., 2017). Only the first several moments can be used to approximate the deterministic solution of CME (Schnoerr et al., 2017). The limitations of the moment closure methods are analyzed in Bronstein and Koeppl (2018). A multivariate moment closure method is developed in Lakatos et al. (2015) to describe the non-linear dynamics of stochastic kinetics. The general moment expansion method for stochastic kinetics is derived in Ale et al. (2013). The approximations of the state probabilities by their statistical moments can be used to conduct efficient simulations of stochastic kinetics (Andreychenko et al., 2015).

The leading term of the CME approximation in the system size expansion (SSE) method corresponds to a linear noise approximation (LNA). It is the first order Taylor expansion of the deterministic CME with a stochastic component where the transition probabilities are additive Gaussian noises. Other terms of the Taylor expansion can be included in order to improve the modeling accuracy (Fröhlich et al., 2016). In Sherlock et al. (2014), the LNA is used to approximate the fast chemical reactions as a continuous time Markov process (CTMP) whereas the slow reactions are represented as a Markov jump process with the time-varying hazards. There are other variants of the LNA, such as a restarting LNA model (Fearnhead et al., 2014), the LNA with time integrated observations (Folia and Rattray, 2018), and the LNA with time-scale separation (Thomas et al., 2012). The LNA for the reaction-diffusion master equation (RDME) is computed in Lötstedt (2018). The impact of parameter values on the stochastic fluctuations in a LNA of BRN is investigated in Pahle et al. (2012).

The so-called S-system model is a set of decoupled non-linear ODEs in the form of product of power-law functions (Chou et al., 2006 Meskin et al., 2011 Liu et al., 2012 Iwata et al., 2014). Such models are justified by assuming a multivariate linearization in the logarithmic coordinates. These models provide a good trade-off between the flexibility and accuracy, and offer other properties which are particularly suitable for modeling complex non-linear systems. The S-system models with additional constraints are assumed in Sun et al. (2012). The S-system modeling of biological pathways is investigated in Mansouri et al. (2015). The S-system model with weighted kinetic orders is obtained in Liu and Wang (2008a). The Bayesian inference for S-system models is investigated in Mansouri et al. (2014).

Polynomial models of biological systems are investigated in Kuepfer et al. (2007), Vrettas et al. (2011), Fey and Bullinger (2010), and Dattner (2015). Rational models as fractions of polynomial functions are examined in Fey and Bullinger (2010), Eisenberg and Hayashi (2014), and Villaverde et al. (2016). The methods for validating polynomial and rational models of BRNs are studied in Rumschinski et al. (2010). The eigenvalues are used in Hori et al. (2013) to obtain a low order linear approximation of the time series data. More generally, the models with differential-algebraic equations (DAEs) are considered in Ashyraliyev et al. (2009), Michalik et al. (2009), Rodriguez-Fernandez et al. (2013), and Deng and Tian (2014). These models have different characteristics than the ODE based models, and they are also more difficult to solve. The review of autoregressive models for parameter inferences including the stability and causality issues is presented in Michailidis and dAlchປuc (2013).

3.3. Other Models of BRNs

There are many other types of BRN models considered in the literature. The birth-death process is a special case of the CTMP having only two states (Daigle et al., 2012 Paul, 2014 Zechner, 2014). It is closely related to a telegraph process (Veerman et al., 2018). A computationally efficient tensor representation of BRNs to facilitate the parameter estimation and sensitivity analysis is devised in Liao et al. (2015). Other computational models for a qualitative description of interactions and behavioral logic in BRNs involve the Petri nets (Mazur, 2012 Sun et al., 2012 Schnoerr et al., 2017), the probabilistic Boolean networks (Liu et al., 2012 Mazur, 2012 Mizera et al., 2014), the continuous time recurrent neural networks (Berrones et al., 2016), and the agent based models (ABMs) (Hussain et al., 2015). The hardware description language (HDL) originally devised to describe the logic of electronic circuits is adopted in Rosati et al. (2018) to model spatially-dependent biological systems with the PDEs. The multi-parameter space was mapped onto a 1D manifold in Zimmer et al. (2014).

The hybrid models generally combine different modeling strategies in order to mitigate various drawbacks of specific strategies (Mikeev and Wolf, 2012 Sherlock et al., 2014 Babtie and Stumpf, 2017). For example, a hybrid model can assume deterministic description of large species populations with the stochastic variations of small populations (Mikeev and Wolf, 2012). The hybrid model consisting of the parametric and non-parametric sub-models can offer some advantages over mechanistic models (von Stosch et al., 2014).

The modeling strategies discussed in this section are summarized in Table 1 . The models are loosely categorized as physical laws, random processes, mathematical models, interaction models and the CME based models. These models are mostly quantitative except the interaction based models which are qualitative. Note that the model properties, such as sloppiness, and the model structures which may be hierarchical, modular or sequential are not distinguished in Table 1 .

Table 1

An overview of the main modeling strategies for BRNs.

StrategyMotivation and key papers
Physical lawsReaction rates in dynamic equilibrium are functions of reactant concentrations• Kinetic rate lawsJoshia et al., 2006 Chou and Voit, 2009 Engl et al., 2009 Baker et al., 2011 Villaverde et al., 2012 Voit, 2013• Mass action kineticsAngius and Horváth, 2011 Lindera and Rempala, 2015 Wong et al., 2015 Smith and Grima, 2018• Mechanistic modelsChou and Voit, 2009 Pullen and Morris, 2014 von Stosch et al., 2014 White et al., 2016Random processesProbabilistic behavioral description of chemical reactions• Markov processAndrieu et al., 2010 Goutsias and Jenkinson, 2013 Septier and Peters, 2016 Weber and Frey, 2017• Poisson processDaigle et al., 2012 Weber and Frey, 2017 Bronstein and Koeppl, 2018 Reis et al., 2018•਋irth-death processWang et al., 2010 Daigle et al., 2012 Mikelson and Khammash, 2016 Weber and Frey, 2017• Telegraph processWeber and Frey, 2017 Veerman et al., 2018Mathematical modelsAdopted models for dynamic systems• Quasi-state modelsRadulescu et al., 2012 Srivastava, 2012 Thomas et al., 2012 Wong et al., 2015 Liao, 2017 Schnoerr et al., 2017• State space representationAndrieu et al., 2010 Andreychenko et al., 2011 Brim et al., 2013 Weber and Frey, 2017• ODEs, PDEs, SDEs, DDEsJ. O. Ramsay and Cao, 2007 Jia et al., 2011 Liu and Gunawan, 2014 Fages et al., 2015 Teijeiro et al., 2017 Weber and Frey, 2017• Path integral form of ODEsWeber and Frey, 2017• Rational modelSun et al., 2012 Vanlier et al., 2013 Hussain et al., 2015 Villaverde et al., 2016•਍ifferential algebraic eqns.J. O. Ramsay and Cao, 2007 Ashyraliyev et al., 2009 Michalik et al., 2009 Deng and Tian, 2014• Tensor representationLiao et al., 2015 Wong et al., 2015 Smith and Grima, 2018• S-system modelKutalik et al., 2007 Chou and Voit, 2009 Meskin et al., 2011 Liu et al., 2012 Voit, 2013• Polynomial modelVrettas et al., 2011 ჎ška et al., 2017 Kuntz et al., 2017 Weber and Frey, 2017• Manifold mapRadulescu et al., 2012 Mannakee et al., 2016 Septier and Peters, 2016 White et al., 2016Interaction modelsQualitative modeling of chemical interactions• Petri netsChou and Voit, 2009 Liu et al., 2012 Voit, 2013•਋oolean networksChou and Voit, 2009 Emmert-Streib et al., 2012• Neural networksGoutsias and Jenkinson, 2013 von Stosch et al., 2014 Ali et al., 2015 Camacho et al., 2018•ਊgent based modelsCarmi et al., 2013 Goutsias and Jenkinson, 2013 Hussain et al., 2015 Jagiella et al., 2017CME based modelsStochastic and deterministic approximations of CME• Langevin equationThomas et al., 2012 Goutsias and Jenkinson, 2013 Septier and Peters, 2016 Schnoerr et al., 2017 Weber and Frey, 2017 Smith and Grima, 2018•ਏokker-Planck equationLiao et al., 2015 Schnoerr et al., 2017 Weber and Frey, 2017• Reaction rate equationKoeppl et al., 2012 Liu and Gunawan, 2014 Lindera and Rempala, 2015 Loos et al., 2016• Moment closureRuess et al., 2011 Andreychenko et al., 2015 Lakatos et al., 2015 Schilling et al., 2016 Schnoerr et al., 2017 Bronstein and Koeppl, 2018• Linear noise approximationGolightly et al., 2012, 2015 Thomas et al., 2012 Fearnhead et al., 2014 Schnoerr et al., 2017 Whitaker et al., 2017• System size expansionFröhlich et al., 2016 Schnoerr et al., 2017

In order to assess the level of interest in different BRN models in literature, Table S1 presents the number of occurrences for the 25 selected modeling strategies in all references cited in this review. The summary of Table S1 is reproduced in Table 2 with the inserted bar graph, and further visualized as a word cloud in Figure 2 . We observe that differential equations are the most commonly assumed models of BRNs in the literature. About half of the papers cited consider the Markov chain models or their variants, since these models naturally and accurately represent the time sequences of randomly occurring reactions in BRNs. The state space representations are assumed in over one third of the cited papers. Other more common models of BRNs include the mass action kinetics, mechanistic models, and the models involving polynomial functions.

Table 2

The coverage of modeling strategies for BRNs.

A word cloud visualizing the levels of interest in different models of BRNs.

Another viewpoint on BRN models in literature is to consider the publication years of papers. Table 3 shows the number of papers for a given modeling strategy in a given year starting from the year 2005. The dot values in tables represent zero counts to improve the readability. We can observe that the interest in some modeling strategies remain stable over the whole decade, for example, for the models involving state space representations and the models involving differential equations. The number of cited papers is the largest in years 2013 and 2014. The paper counts in Table 3 indicate that the interest in computational modeling of BRNs has been increasing steadily over the past decade.

Table 3

The number of papers concerning models of BRNs in given years.


Contents

Four isozymes of pyruvate kinase expressed in vertebrates: L (liver), R (erythrocytes), M1 (muscle and brain) and M2 (early fetal tissue and most adult tissues). The L and R isozymes are expressed by the gene PKLR, whereas the M1 and M2 isozymes are expressed by the gene PKM2. The R and L isozymes differ from M1 and M2 in that they are allosterically regulated. Kinetically, the R and L isozymes of pyruvate kinase have two distinct conformation states one with a high substrate affinity and one with a low substrate affinity. The R-state, characterized by high substrate affinity, serves as the activated form of pyruvate kinase and is stabilized by PEP and fructose 1,6-bisphosphate (FBP), promoting the glycolytic pathway. The T-state, characterized by low substrate affinity, serves as the inactivated form of pyruvate kinase, bound and stabilized by ATP and alanine, causing phosphorylation of pyruvate kinase and the inhibition of glycolysis. [3] The M2 isozyme of pyruvate kinase can form tetramers or dimers. Tetramers have a high affinity for PEP, whereas, dimers have a low affinity for PEP. Enzymatic activity can be regulated by phosphorylating highly active tetramers of PKM2 into an inactive dimers. [4]

The PKM gene consists of 12 exons and 11 introns. PKM1 and PKM2 are different splicing products of the M-gene (PKM1 contains exon 9 while PKM2 contains exon 10) and solely differ in 23 amino acids within a 56-amino acid stretch (aa 378-434) at their carboxy terminus. [5] [6] The PKM gene is regulated through heterogenous ribonucleotide proteins like hnRNPA1 and hnRNPA2. [7] Human PKM2 monomer has 531 amino acids and is a single chain divided into A, B and C domains. The difference in amino acid sequence between PKM1 and PKM2 allows PKM2 to be allosterically regulated by FBP and for it to form dimers and tetramers while PKM1 can only form tetramers. [8]

Many Enterobacteriaceae, including E. coli, have two isoforms of pyruvate kinase, PykA and PykF, which are 37% identical in E. coli (Uniprot: PykA, PykF). They catalyze the same reaction as in eukaryotes, namely the generation of ATP from ADP and PEP, the last step in glycolysis, a step that is irreversible under physiological conditions. PykF is allosterically regulated by FBP which reflects the central position of PykF in cellular metabolism. [9] PykF transcription in E. coli is regulated by the global transcriptional regulator, Cra (FruR). [10] [11] [12] PfkB was shown to be inhibited by MgATP at low concentrations of Fru-6P, and this regulation is important for gluconeogenesis. [13]

Glycolysis Edit

There are two steps in the pyruvate kinase reaction in glycolysis. First, PEP transfers a phosphate group to ADP, producing ATP and the enolate of pyruvate. Secondly, a proton must be added to the enolate of pyruvate to produce the functional form of pyruvate that the cell requires. [14] Because the substrate for pyruvate kinase is a simple phospho-sugar, and the product is an ATP, pyruvate kinase is a possible foundation enzyme for the evolution of the glycolysis cycle, and may be one of the most ancient enzymes in all earth-based life. Phosphoenolpyruvate may have been present abiotically, and has been shown to be produced in high yield in a primitive triose glycolysis pathway. [15]

In yeast cells, the interaction of yeast pyruvate kinase (YPK) with PEP and its allosteric effector Fructose 1,6-bisphosphate (FBP,) was found to be enhanced by the presence of Mg 2+ . Therefore, Mg 2+ was concluded to be an important cofactor in the catalysis of PEP into pyruvate by pyruvate kinase. Furthermore, the metal ion Mn 2+ was shown to have a similar, but stronger effect on YPK than Mg 2+ . The binding of metal ions to the metal binding sites on pyruvate kinase enhances the rate of this reaction. [16]

The reaction catalyzed by pyruvate kinase is the final step of glycolysis. It is one of three rate-limiting steps of this pathway. Rate-limiting steps are the slower, regulated steps of a pathway and thus determine the overall rate of the pathway. In glycolysis, the rate-limiting steps are coupled to either the hydrolysis of ATP or the phosphorylation of ADP, causing the pathway to be energetically favorable and essentially irreversible in cells. This final step is highly regulated and deliberately irreversible because pyruvate is a crucial intermediate building block for further metabolic pathways. [17] Once pyruvate is produced, it either enters the TCA cycle for further production of ATP under aerobic conditions, or is converted to lactic acid or ethanol under anaerobic conditions.

Gluconeogenesis: the reverse reaction Edit

Pyruvate kinase also serves as a regulatory enzyme for gluconeogenesis, a biochemical pathway in which the liver generates glucose from pyruvate and other substrates. Gluconeogenesis utilizes noncarbohydrate sources to provide glucose to the brain and red blood cells in times of starvation when direct glucose reserves are exhausted. [17] During fasting state, pyruvate kinase is inhibited, thus preventing the "leak-down" of phosphoenolpyruvate from being converted into pyruvate [17] instead, phosphoenolpyruvate is converted into glucose via a cascade of gluconeogenesis reactions. Although it utilizes similar enzymes, gluconeogenesis is not the reverse of glycolysis. It is instead a pathway that circumvents the irreversible steps of glycolysis. Furthermore, gluconeogenesis and glycolysis do not occur concurrently in the cell at any given moment as they are reciprocally regulated by cell signaling. [17] Once the gluconeogenesis pathway is complete, the glucose produced is expelled from the liver, providing energy for the vital tissues in the fasting state.

Glycolysis is highly regulated at three of its catalytic steps: the phosphorylation of glucose by hexokinase, the phosphorylation of fructose-6-phosphate by phosphofructokinase, and the transfer of phosphate from PEP to ADP by pyruvate kinase. Under wild-type conditions, all three of these reactions are irreversible, have a large negative free energy and are responsible for the regulation of this pathway. [17] Pyruvate kinase activity is most broadly regulated by allosteric effectors, covalent modifiers and hormonal control. However, the most significant pyruvate kinase regulator is fructose-1,6-bisphosphate (FBP), which serves as an allosteric effector for the enzyme.

Allosteric effectors Edit

Allosteric regulation is the binding of an effector to a site on the protein other than the active site, causing a conformational change and altering the activity of that given protein or enzyme. Pyruvate kinase has been found to be allosterically activated by FBP and allosterically inactivated by ATP and alanine. [18] Pyruvate Kinase tetramerization is promoted by FBP and Serine while tetramer dissociation is promoted by L-Cysteine. [19] [20] [21]

Fructose-1,6-bisphosphate Edit

FBP is the most significant source of regulation because it comes from within the glycolysis pathway. FBP is a glycolytic intermediate produced from the phosphorylation of fructose 6-phosphate. FBP binds to the allosteric binding site on domain C of pyruvate kinase and changes the conformation of the enzyme, causing the activation of pyruvate kinase activity. [22] As an intermediate present within the glycolytic pathway, FBP provides feedforward stimulation because the higher the concentration of FBP, the greater the allosteric activation and magnitude of pyruvate kinase activity. Pyruvate kinase is most sensitive to the effects of FBP. As a result, the remainder of the regulatory mechanisms serve as secondary modification. [9] [23]

Covalent modifiers Edit

Covalent modifiers serve as indirect regulators by controlling the phosphorylation, dephosphorylation, acetylation, succinylation and oxidation of enzymes, resulting in the activation and inhibition of enzymatic activity. [24] In the liver, glucagon and epinephrine activate protein kinase A, which serves as a covalent modifier by phosphorylating and deactivating pyruvate kinase. In contrast, the secretion of insulin in response to blood sugar elevation activates phosphoprotein phosphatase I, causing the dephosphorylation and activation of pyruvate kinase to increase glycolysis. The same covalent modification has the opposite effect on gluconeogenesis enzymes. This regulation system is responsible for the avoidance of a futile cycle through the prevention of simultaneous activation of pyruvate kinase and enzymes that catalyze gluconeogenesis. [25]

Carbohydrate response element binding protein (ChREBP) Edit

ChREBP is found to be an essential protein in gene transcription of the L isozyme of pyruvate kinase. The domains of ChREBP are target sites for regulation of pyruvate kinase by glucose and cAMP. Specifically, ChREBP is activated by a high concentration of glucose and inhibited by cAMP. Glucose and cAMP work in opposition with one another through covalent modifier regulation. While cAMP binds to Ser196 and Thr666 binding sites of ChREBP, causing the phosphorylation and inactivation of pyruvate kinase glucose binds to Ser196 and Thr666 binding sites of ChREBP, causing the dephosphorylation and activation of pyruvate kinase. As a result, cAMP and excess carbohydrates are shown to play an indirect role in pyruvate kinase regulation. [26]

Hormonal control Edit

In order to prevent a futile cycle, glycolysis and gluconeogenesis are heavily regulated in order to ensure that they are never operating in the cell at the same time. As a result, the inhibition of pyruvate kinase by glucagon, cyclic AMP and epinephrine, not only shuts down glycolysis, but also stimulates gluconeogenesis. Alternatively, insulin interferes with the effect of glucagon, cyclic AMP and epinephrine, causing pyruvate kinase to function normally and gluconeogenesis to be shut down. Furthermore, glucose was found to inhibit and disrupt gluconeogenesis, leaving pyruvate kinase activity and glycolysis unaffected. Overall, the interaction between hormones plays a key role in the functioning and regulation of glycolysis and gluconeogenesis in the cell. [27]

Inhibitory effect of metformin Edit

Metformin, or dimethylbiguanide, is the primary treatment used for type 2 diabetes. Metformin has been shown to indirectly affect pyruvate kinase through the inhibition of gluconeogenesis. Specifically, the addition of metformin is linked to a marked decrease in glucose flux and increase in lactate/pyruvate flux from various metabolic pathways. Although metformin does not directly affect pyruvate kinase activity, it causes a decrease in the concentration of ATP. Due to the allosteric inhibitory effects of ATP on pyruvate kinase, a decrease in ATP results in diminished inhibition and the subsequent stimulation of pyruvate kinase. Consequently, the increase in pyruvate kinase activity directs metabolic flux through glycolysis rather than gluconeogenesis. [28]

Gene Regulation Edit

Heterogenous ribonucleotide proteins (hnRNPs) can act on the PKM gene to regulate expression of M1 and M2 isoforms. PKM1 and PKM2 isoforms are splice variants of the PKM gene that differ by a single exon. Various types of hnRNPs such as hnRNPA1 and hnRNPA2 enter the nucleus during hypoxia conditions and modulate expression such that PKM2 is up-regulated. [29] Hormones such as insulin up-regulate expression of PKM2 while hormones like tri-iodothyronine (T3) and glucagon aid in down-regulating PKM2. [30]

Deficiency Edit

Genetic defects of this enzyme cause the disease known as pyruvate kinase deficiency. In this condition, a lack of pyruvate kinase slows down the process of glycolysis. This effect is especially devastating in cells that lack mitochondria, because these cells must use anaerobic glycolysis as their sole source of energy because the TCA cycle is not available. For example, red blood cells, which in a state of pyruvate kinase deficiency, rapidly become deficient in ATP and can undergo hemolysis. Therefore, pyruvate kinase deficiency can cause chronic nonspherocytic hemolytic anemia (CNSHA). [31]

PK-LR gene mutation Edit

Pyruvate kinase deficiency is caused by an autosomal recessive trait. Mammals have two pyruvate kinase genes, PK-LR (which encodes for pyruvate kinase isozymes L and R) and PK-M (which encodes for pyruvate kinase isozyme M1), but only PKLR encodes for the red blood isozyme which effects pyruvate kinase deficiency. Over 250 PK-LR gene mutations have been identified and associated with pyruvate kinase deficiency. DNA testing has guided the discovery of the location of PKLR on chromosome 1 and the development of direct gene sequencing tests to molecularly diagnose pyruvate kinase deficiency. [32]

Applications of pyruvate kinase inhibition Edit

Reactive Oxygen Species (ROS) Inhibition Edit

Reactive oxygen species (ROS) are chemically reactive forms of oxygen. In human lung cells, ROS has been shown to inhibit the M2 isozyme of pyruvate kinase (PKM2). ROS achieves this inhibition by oxidizing Cys358 and inactivating PKM2. As a result of PKM2 inactivation, glucose flux is no longer converted into pyruvate, but is instead utilized in the pentose phosphate pathway, resulting in the reduction and detoxification of ROS. In this manner, the harmful effects of ROS are increased and cause greater oxidative stress on the lung cells, leading to potential tumor formation. This inhibitory mechanism is important because it may suggest that the regulatory mechanisms in PKM2 are responsible for aiding cancer cell resistance to oxidative stress and enhanced tumorigenesis. [33] [34]

Phenylalanine inhibition Edit

Phenylalanine is found to function as a competitive inhibitor of pyruvate kinase in the brain. Although the degree of phenylalanine inhibitory activity is similar in both fetal and adult cells, the enzymes in the fetal brain cells are significantly more vulnerable to inhibition than those in adult brain cells. A study of PKM2 in babies with the genetic brain disease phenylketonurics (PKU), showed elevated levels of phenylalanine and decreased effectiveness of PKM2. This inhibitory mechanism provides insight into the role of pyruvate kinase in brain cell damage. [35] [36]

Pyruvate Kinase in Cancer Edit

Cancer cells have characteristically accelerated metabolic machinery and Pyruvate Kinase is believed to have a role in cancer. When compared to healthy cells, cancer cells have elevated levels of the PKM2 isoform, specifically the low activity dimer. Therefore, PKM2 serum levels are used as markers for cancer. The low activity dimer allows for build-up of phosphoenol pyruvate (PEP), leaving large concentrations of glycolytic intermediates for synthesis of biomolecules that will eventually be used by cancer cells. [8] Phosphorylation of PKM2 by Mitogen-activated protein kinase 1 (ERK2) causes conformational changes that allow PKM2 to enter the nucleus and regulate glycolytic gene expression required for tumor development. [37] Some studies state that there is a shift in expression from PKM1 to PKM2 during carcinogenesis. Tumor microenvironments like hypoxia activate transcription factors like the hypoxia-inducible factor to promote the transcription of PKM2, which forms a positive feedback loop to enhance its own transcription. [8]

A reversible enzyme with a similar function, pyruvate phosphate dikinase (PPDK), is found in some bacteria and has been transferred to a number of anaerobic eukaryote groups (for example, Streblomastix, Giardia, Entamoeba, and Trichomonas), it seems via horizontal gene transfer on two or more occasions. In some cases, the same organism will have both pyruvate kinase and PPDK. [38]


Key Concepts and Summary

Systems at equilibrium can be disturbed by changes to temperature, concentration, and, in some cases, volume and pressure volume and pressure changes will disturb equilibrium if the number of moles of gas is different on the reactant and product sides of the reaction. The system’s response to these disturbances is described by Le Châtelier’s principle: The system will respond in a way that counteracts the disturbance. Not all changes to the system result in a disturbance of the equilibrium. Adding a catalyst affects the rates of the reactions but does not alter the equilibrium, and changing pressure or volume will not significantly disturb systems with no gases or with equal numbers of moles of gas on the reactant and product side.

Disturbance Observed Change as Equilibrium is Restored Direction of Shift Effect on K
reactant added added reactant is partially consumed toward products none
product added added product is partially consumed toward reactants none
decrease in volume/increase in gas pressure pressure decreases toward side with fewer moles of gas none
increase in volume/decrease in gas pressure pressure increases toward side with more moles of gas none
temperature increase heat is absorbed toward products for endothermic, toward reactants for exothermic changes
temperature decrease heat is given off toward reactants for endothermic, toward products for exothermic changes
Table 2. Effects of Disturbances of Equilibrium and K

Chemistry End of Chapter Exercises

Under what conditions will decomposition in a closed container proceed to completion so that no CaCO3 remains?

Is an equilibrium among CH4, O2, CO2, and H2O established under these conditions? Explain your answer.

(a) Does the equilibrium constant for the reaction increase, decrease, or remain about the same as the temperature increases?


7.3: Chemical Equilibrium—Part 1: Forward and Reverse Reactions - Biology

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Activation energy is the energy needed to initiate a chemical reaction between reactants that are present. The source is often supplied by thermal energy which causes molecules to move faster and collide with one another, causing the bonds of the reactants to break.

Thus, the original reactants required free energy for the reaction to occur. Represented as the uphill part of this curve, this is the amount of activation energy for the reaction. At the peak, known as the transition state, the unbound molecules are now in unstable condition. As atoms reattach with new bonds, they release free energy into the environment, which is shown as the downhill portion of the reaction.

While humans metabolize sugar and fat for energy, if thermal energy were used to break down these molecules, so much free energy would be released as heat that the proteins in the cell would denature. Instead, substances known as catalysts are specifically added to regulate the rate of metabolism, like speeding it up. For example, a biological catalyst, an enzyme, lowers the activation energy required to break bonds, and allows reactions to occur at reasonable rates without cellular damage.

7.7: Activation Energy

Activation energy is the minimum amount of energy necessary for a chemical reaction to move forward. The higher the activation energy, the slower the rate of the reaction. However, adding heat to the reaction will increase the rate, since it causes molecules to move faster and increase the likelihood that molecules will collide. The collision and breaking of bonds represents the uphill phase of a reaction and generates the transition state. The transition state is an unstable high-energy state of the reactants. The formation of new chemical bonds and end products, and the release of free energy is the the downhill phase of the reaction. Catalysts increase the rate of a reaction by lowering the activation energy. For example, in biology, enzymes that metabolize sugar and fats increase the rate at which their breakdown happens and at the same time, prevent the overproduction of free energy that would otherwise denature proteins in the cell.

Catalysts

A catalyst is a substance that increases the rate of a reaction by lowering the activation energy, and in the process, regenerates itself. A catalyst provides an alternative pathway or mechanism for the reaction to take place and it accelerates both the forward and reverse reactions. In biology, enzymes are examples of catalysts because they lower the activation energy required for reactions in cellular metabolism.

For example, humans metabolize sugar and fat for energy. Enzymes are vital to humans for breaking down these molecules, because if thermal energy alone were to be used, the free energy released in the form of heat would cause proteins in the cell to denature. Furthermore, thermal energy would non-specifically catalyze all reactions. However, enzymes only bind to specific chemical reactants, called substrates, and lower their activation energy to catalyze selective cellular reactions.

Robinson, Peter K. &ldquoEnzymes: Principles and Biotechnological Applications.&rdquo Essays in Biochemistry 59 (November 15, 2015): 1&ndash41. [Source]


Geology and Climate

The CO2 bubbles in this photograph contain carbon that is completing (or just beginning) its several-million-year journey from the atmosphere to the ocean to marine organisms’ carbonate structures to ocean sediment to limestone subducted beneath a tectonic plate to release by magma heating and return to the surface by volcanism to begin the cycle once more. The diagram further down the page is a schematic representation of this pathway. Along the way, the carbon’s journey might have been interrupted by more rapid reactions, such as incorporation into organic molecules via photosynthesis, but as the organics decay most of the carbon on the oxygen-rich Earth ends up in its most stable oxidized form, as CO2 and carbonate.


This photo shows an ocean acidification experiment the Earth has been carrying out in a few localized areas for a very long time. The bubbles are essentially pure CO2 being emitted from the shallow floor of the Mediterranean Sea off the volcanic island of Ischia in Italy’s Bay of Naples. Unlike the mixture of hot gases and liquids emitted by the thermal vents at the juncture of tectonic plates in the deep ocean, the vents here emit the CO2 at ambient temperature. The pH of the sea in the vent area can be as low as 7.3, increasing to the usual 8.2 about 150 m from the vents. Studies of the biodiversity in this setting can help us understand the consequences of ocean acidification by increasing fossil fuel CO2 emissions.

“Venting of volcanic CO2 at a Mediterranean site off the island of Ischia provides the opportunity to observe changes in the community structure of a rocky shore ecosystem along gradients of decreasing pH close to the vents. Groups such as sea urchins, coralline algae and stony corals decline in abundance or vanish completely with decreasing pH. Sea grasses and brown algae benefit from elevated CO2 availability close to the vent by increasing their biomass. Similar high CO2/low pH conditions are on the verge of progressively developing ocean-wide through the uptake of fossil-fuel CO2 by the surface ocean.” (U. Riebesell, Nature 2008, 454, 46-47)

CO2 is stabilized by delocalization of its π electrons that make the compound about 108 kJ·mol –1 more stable than calculated from the bond enthalpy of two isolated carbon-oxygen double bonds.


7.3: Chemical Equilibrium—Part 1: Forward and Reverse Reactions - Biology

a Loschmidt Laboratories, Department of Experimental Biology and RECETOX, Faculty of Science, Masaryk University, Kamenice 5/A13, 625 00 Brno, Czech Republic
E-mail: [email protected], [email protected]

b International Clinical Research Center, St. Anne's University Hospital Brno, Pekarska 53, 656 91 Brno, Czech Republic

Abstract

Substrate inhibition is the most common deviation from Michaelis–Menten kinetics, occurring in approximately 25% of known enzymes. It is generally attributed to the formation of an unproductive enzyme–substrate complex after the simultaneous binding of two or more substrate molecules to the active site. Here, we show that a single point mutation (L177W) in the haloalkane dehalogenase LinB causes strong substrate inhibition. Surprisingly, a global kinetic analysis suggested that this inhibition is caused by binding of the substrate to the enzyme–product complex. Molecular dynamics simulations clarified the details of this unusual mechanism of substrate inhibition: Markov state models indicated that the substrate prevents the exit of the halide product by direct blockage and/or restricting conformational flexibility. The contributions of three residues forming the possible substrate inhibition site (W140A, F143L and I211L) to the observed inhibition were studied by mutagenesis. An unusual synergy giving rise to high catalytic efficiency and reduced substrate inhibition was observed between residues L177W and I211L, which are located in different access tunnels of the protein. These results show that substrate inhibition can be caused by substrate binding to the enzyme–product complex and can be controlled rationally by targeted amino acid substitutions in enzyme access tunnels.


Watch the video: CHAP 7: IONIC EQUILIBRIA: PART 1 (May 2022).