Thermodynamics #1: Metabolism's Motivation

Metabolism’s Motivation

The Game

Play a game with me. The game has three simple rules.

Rule #1 : You cannot win.

That’s alright, you say?
Ok then here’s rule two.

Rule #2 : You cannot draw.

You no longer want to play the game? This game sucks huh?
That’s too bad, here’s rule three.

Rule #3 : You cannot stay out of the game.

You think this is outrageous? You just read the first, second and third laws of thermodynamics!

What’s going on?

Before we dive into the laws of thermodynamics, there are a few terms we need to familiarize ourselves with. A thermodynamic system is a body of matter with well defined boundaries. The rest of the universe that lies outside these boundaries is known as the surroundings. Based on the properties of the boundaries of the system, it may be of the open, closed or isolated type.

• An open system is free to exchange matter and energy with its surroundings.
• A closed system can exchange energy with its surroundings but not matter.
• An isolated system can exchange neither matter nor energy with its surroundings.

Energy is a recurring term throughout this article.

Energy is the capacity to do work.

Energy transfer between two bodies of matter can happen in two modes: heat and work. The difference between these two modes of transfer becomes apparent when viewed at the atomic level. When a body of matter receives energy by the mode of work, all of its atoms move in the same direction.

Work is the mode of transfer of energy that achieves or utilizes uniform (orderly) motion in a body of matter.

On the other hand when a body of matter is heated, its molecules move in different directions.

Heat is the mode of transfer of energy that achieves or utilizes disorderly motion in a body of matter.

The total amount of energy contained in a system irrespective of how it was transfer to it is known as the internal energy of the system.

The internal energy (U) is the grand total energy of the system.

Every system has a value of internal energy however, it is not practically possible to measure this value. When a system undergoes a process or a reaction, this value of its internal energy is altered. This change in internal energy can be measured and is represented by the symbol $\Delta U$.

The first law

The first law of thermodynamics (Rule #1) is essentially a mathematical statement of the Law of Conservation of Energy. The amount of energy you put into a system is the maximum amount of work you can get out of it. You can never end up doing more work than what corresponds to the energy you put in. When you put energy into a system you increase its internal energy.
More formally :

In a thermodynamic process involving a closed system, the increment in the internal energy is equal to the difference between the heat accumulated (absorbed) by the system and the work done by it.

$\Delta U = q- w$
Where $\Delta$U is the change in internal energy, q is the heat absorbed by the system and w is the work done by the system.

When transfer of energy takes place to a system, it is usually open to the atmosphere. This means it is subjected to a constant atmospheric pressure and is not confined to a fixed volume.
During the course of energy transfer, the volume of the system changes (increases). To create this additional volume, the atmosphere needs to be pushed back. Thus, some of the energy transferred to the system is utilized to push the atmosphere out of the way. In other words, part of the supplied energy has been used to expand the system. Such work is known as expansion work. All other types of work (muscle contraction, electrical work, etc) come under non-expansion work.
$w_{total} = w_{expansion} + w_{non-expansion} \\ w_{expansion}= - P\Delta V$

The thermodynamic quantity enthalpy takes into consideration the pressure ($P$) and volume ($V$) of the system along with its internal energy. It is defined as:
$H = U + PV$
Thus, the change in enthalpy of a system, when its volume is not constant is equal to the energy supplied to the system. Change in enthalpy under constant pressure (usually atmospheric) is defined as :
$\Delta H = \Delta U + P\Delta V$
Note that enthalpy has no physical significance. Its introduction avoids the complication of taking into consideration the work of expansion.

The second law

The second law of thermodynamics introduces the concept of entropy.

Isolated systems spontaneously evolve towards thermodynamic equilibrium, the state with maximum entropy.

Energy only when in the concentrated form is useful to do work. When dispersed, work cannot be extracted from it. This dispersed, useless form of energy is called entropy. The almost depressing reality is that there are simply more ways to be disordered than ordered. Thus, due to sheer probability, it is more likely (by a large margin) that when left to itself, the universe attains a state of dispersed energy.

A more ‘disorderly’ distribution of energy and matter corresponds to a greater number of micro-states associated with the same total energy. — Atkins & de Paula (2006)

Due to this constant dispersal of energy, it is impossible to break even. You can never extract work equivalent to the energy you put into a system. It always gives out lesser than what you give in. You can never break even or draw!

The increase in entropy after a process is the amount of energy that has now become incapable of doing work. To calculate this increase (change) in entropy ($\Delta S$) we use the following equation :
$\Delta S= \frac{q_{reversible} }{ T}$
Where T is the temperature at with the energy transfer takes place in kelvin .

The rest of the article attempts to describe biochemical reactions and thus makes the following assumptions.

Conditions in a biological environment : 
a) Constant temperature
b) Constant pressure
c) Nearly constant volume
Thus, biochemical reactions are isothermal, 
isobaric and sustain negligible volume changes.

So how much work can I actually extract from a system?
Gibbs free energy ($G$), another thermodynamic quantity, tells us exactly this. Free energy is the amount of energy available for doing work. It is defined as the enthalpy (total energy of the system) minus entropy times temperature (the amount of energy work cannot be extracted from).
$G = H - T S$

Thus, after completion of a reaction, the change in Gibbs free energy ($\Delta G$) between final and initial states tells us how much usable energy we have obtained because of the reaction.
$\Delta G =\Delta H - T \Delta S \\ \Delta G =- w_{non-expansion}$
$\Delta G$ measures the total change in entropy caused by a reaction. Note that a positive $\Delta G$ corresponds to a decrease in total entropy and a negative $\Delta G$ corresponds to an increase in total entropy. Irrespective of whether the entropy of the system and the surroundings are increasing or decreasing, this total entropy of the universe is always increasing.

The third law

Entropy seems to be the bane of our existence. However, eliminating the devastating effects of entropy is not possible. Even as we lower the temperate of the reaction as close as we can to absolute zero, the best we can achieve is a constant (non - decreasing) value of entropy.
Thus, there’s no eliminating entropy; we can’t get out of the game! Entropy’s reign of terror is here to stay!

The entropy of a system approaches a constant value as its temperature approaches absolute zero.

Ok this is cool and all but why do I need to know thermo-whatever to study biochemistry?

• What drives the standard repertoire of reactions known as metabolism?
• What are the underlying principles that govern these reactions?
• In a universe that is constantly increasing in entropy, how does the human body maintain order at such an intricate, microscopic level?
  All these questions that are central to life are answered by thermodynamics.

The importance of imbalance or non-equilibrium for life (Yes, you read that right!)

The $\Delta G$ for a reaction at equilibrium is zero. For the proof refer to this Wikipedia article. Only non-equilibrium reactions produce a positive or negative $\Delta G$ value. Since, a positive $\Delta G$ corresponds to a decrease in total entropy, such reactions require energy to drive them. Thus, only non-equilibrium reactions with a negative $\Delta G$ provide free energy to perform work.

The human body needs free energy to do work such as maintenance of concentration gradients, muscle contraction and nerve impulse transmission. Thus, maintaining a state of non-equilibrium is essential for life. A non-equilibrium state provides a thermodynamic driving force. This driving force pushes the reaction towards equilibrium.

Like a sloping piece of land drives the flow of water, a thermodynamic driving force drives the flow of metabolites in a particular direction in the human body. The free energy liberated from these reactions is then stored in high energy compounds such as adenosine triphosphate (ATP) and can be utilized to maintain order in the body.

The rate of flow of metabolites in the body is known as flux.

The human body is an open system

The human body is an open system. We exchange matter and energy with our surroundings. We take in high enthalpy low entropy compounds and convert them to low enthalpy high entropy compounds that we give out. The free energy that we obtain from these compounds is used to maintain a non-equilibrium steady state within the body.

The field of irreversible thermodynamics and an introduction to the steady state.

Now that we understand the enormous importance of non-equilibrium to a living system, lets dip our toes into the field of irreversible thermodynamics. In contrast to classical thermodynamics (aka equilibrium thermodynamics) which deals with reversible process in an closed system, this field deals with irreversible processes in an open system.

The steady state in irreversible thermodynamics refers to the state in which the open system produces maximum work. It can be defined as follows :

The maintenance of a constant flux (of metabolites) in an open system (such as the human body) is known as the steady state.

It is interesting to note that the steady state abides by Le Chatelier’s principle. That is, reactions tend to compensate external changes in order to maintain steady state concentrations. Thus, steady state is analogous to equilibrium. This is clearly seen in how the excess of a metabolite often inhibits the reaction that produces it and drives the reaction that utilizes it as a substrate!

Closed system (Classical Thermodynamics) : Equilibrium :: Open system (Irreversible Thermodynamics) : Steady State

How the body maintains a steady state

Steady state is maintained by:
a) The irreversible nature of the rate determining step of the metabolic pathway.
b) Saturation of enzymes catalyzing the rate determining step.

The rate determining step
The rate determining step is an irreversible reaction that occurs earlier on in the metabolic pathway. An irreversible reaction is one whose substrate and product concentrations are very far from from equilibrium. In attempting to reach equilibrium, large losses of free energy occur, making this type of reaction essentially irreversible.
Being irreversible, the reaction commits the substrate to that pathway and forces it to traverse in only one direction. Such a step is seen earlier on in the pathway to increase efficiency by preventing unnecessary synthesis of metabolites further along the pathway.

The rate determining step is irreversible. Hence, it commits the substrate to a particular pathway

Enzymes catalyzing rate determining steps
The enzymes catalyzing rate determining steps in the pathway have low $K_m$ values in comparison to their substrate concentration. The Michaelis constant ($K_m$) is numerically equal to the substrate concentration at which the reaction rate is half of its maximum possible rate. It is a measure of enzyme affinity to the substrate. A low $K_m$value means that the enzyme has high affinity for substrate and undergoes saturation at lower substrate concentrations. Thus, for enzymes with low $K_m$ values, the enzyme is saturated and an increase in substrate concentration has no effect on reaction rate. Even during changes in substrate concentrations, a constant flux (steady state) can be maintained.

The rate of the rate determining step is determined not by substrate concentration but by the rate of enzyme activity itself.

ATP and non - spontaneous reactions the ultimate OTP

A reaction that is non-spontaneous may be driven forward by a reaction that is spontaneous by performing the two reactions simultaneously!

Consider a pair of weights. When the lighter weight is connected to a pully, it tends to move downwards. When the heavier weight is connected to a pully it also tends to move downwards. However, when both the weights are attached to the pulley on either side, the lighter weight move upwards while the heavier weight moves downwards.
Similarly endergonic reactions (analogous to the lighter weight) are forced to occur by coupling them with highly exergonic reactions (analogous to the heavier weight).

Source : Research gate
The overall reaction is spontaneous as :
$\Delta G_{endergonic}+ \Delta G_{exergonic} = \text{Negative } \Delta G$
All of life’s activities depend on coupling of this kind. The hydrolysis Adenosine triphosphate (ATP) is one such highly exergonic reaction. Thus, it drives other very useful endergonic reactions in the body and maintains order at the molecular level!

Sources

• The game : Ginsberg theorem
• Elements of Physical Chemistry by Julio de Paula and Peter Atkins
• Harper’s Illustrated Biochemistry 31st Edition
• Biochemistry by Donald Voet and Judith G. Voet
• Lehninger Principles of Biochemistry, Fourth Edition by David L. Nelson, Michael M. Cox, and Rabbi David Nelson, PhD