Biomod/2013/NanoUANL/Enzyme

What is an enzyme?

In biological systems, chemical transformations are typically accelerated by enzymes, macromolecules capable of turning one or more compounds into others (substrates and products). The activity is determined greatly by their three-dimensional structure. Most enzymes are proteins, although several catalytic RNA molecules have been identified. They may also need to employ organic and inorganic cofactors for the reaction to occur. The process is based upon the diminishment of the activation energy needed for a reaction, greatly increasing its rate of reaction. The rate enhancement provided by these proteins can be as high as 10^19, while maintaining high substrate specificity.

Because of this, reaction rates are millions of times faster than un-catalyzed reactions. Enzymes are not consumed by the reactions that they take part in, and they do not alter the equilibrium. Enzyme activity can be affected by a wide variety of factors. Inhibitors and activators intervene directly in the reaction rate, environmental factors like temperature, pressure, pH and substrate concentration also play a part in these kinetics. For temperature and pH, usually exist a range of values for which the enzyme works better (optimal conditions). The enzyme activity lowers dramatically as you get farther away from this range of values. As for concentration, other kind of relationship is observed. With increasing concentration, enzyme activity increases, until we reach the most optimal performance. Further increase of concentration won’t have an impact on the enzyme activity.

Being able to determine these conditions allow us to manipulate the enzyme activity, thus achieving greater control over the reaction.

Horseradish peroxidase

Enzyme information was gathered from research papers:

• Isoelectric Point: Isozymes range from 3.0 – 9.0 pH
• pH Dependence: range of 5.0 to 9.0, optimum (6.0 to 6.5)
• k₁ = 1 x 10⁷
• k_-1 = between 20 and 250
• k₂ = 200 at pH 6

Enzyme kinetics

Michaelis-Menten kinetics is one of the oldest models for describing the catalytic activity of enzymes. The reaction cycle is divided into two basic steps: the reversible binding between the enzyme and substrate to form an intermediate complex, and the irreversible catalytic step to generate the product and release the enzyme; in which the first step is affected by the constants k₁; and k_-1, whereas the irreversible step only takes into account k₂.

1.1

[math]\displaystyle{ E + S \leftrightarrow ES \rightarrow E^0 + P }[/math]

The rate of consumption can be expressed by the formation of the ES complex in the following equation:

1.2 [math]\displaystyle{ \frac{d[ES]}{dt}=k_1[E][S]-k_{-1}[ES]-k_2[ES] }[/math]

Using a steady-approximation and rearranging 1.2 we obtain:

1.3 [math]\displaystyle{ [ES]= \frac{[E][S]}{K_M+[S]} }[/math]

where K_M is the Michaelis constant defined as

1.4

[math]\displaystyle{ \operatorname{K_M}= \tfrac{k_{-1} +k_2}{k_1} }[/math]

As it was mentioned in the introduction, single-enzyme studies have proven that the "traditional" enzyme kinetics do not apply, and a new approach is needed. Enzyme concentration is meaningless in a single-molecule level, so it is more appropriate to consider the probability P_E(t) for the enzyme to find a catalytically active enzyme in a time t in the process. This is because the reaction is a stochastic event.

Therefore, the rate equations of each species are:

1.5 [math]\displaystyle{ \frac{d[E]}{dt}=-k_1[E][S]+k_{-1}[ES] }[/math]

1.6 [math]\displaystyle{ \frac{d[ES]}{dt}=k_1[E][S]-(k_{-1}+k_2)[ES] }[/math]

1.7 [math]\displaystyle{ \frac{d[E^0]}{dt}=\tfrac{d[P]}{dt}=k_2[ES] }[/math]

where t is the elapsed time, the initial conditions are [ES]=0 and [E⁰]=0 at t=0. To derive the rate equations that describe the corresponding single-molecule Michaelis-Menten kinetics, the concentrations in equations 5-7 are replaced by the probabilities P of finding the single enzyme molecule in the states E, ES, and E⁰ , leading to the equations:

1.8 [math]\displaystyle{ \frac{dP_E(t)}{dt}=-k_1^0P_E(t)+k_{-1}P_{ES}(t) }[/math]

1.9 [math]\displaystyle{ \frac{dP_{ES}(t)]}{dt}=k_1^0P_E(t)-(k_{-1}+k_2)P_{ES}(t) }[/math]

1.10 [math]\displaystyle{ \frac{dP_E^0(t)}{dt}=k_2P_{ES}(t) }[/math]

These equations must satisfy the conditions P_E(0)=1, P_ES(0)=0 and P_E⁰=0 at t=0 (start of the reaction). Also, P_E(t) + P_ES(t) + P_E⁰(t)=1. The rate constant k₁⁰ can be taken as k₁⁰=k₁[S], assuming [S] is time-independent. Given that a single enzyme is unlikely to deplete all the substrate presence, [S] can be considered constant, virtually being unaffected.

Equations 8-10 become a system of linear first-order differential equations that can be solved exactly for P_E(t), P_ES(t) and P_E⁰(t).

Knowing P_E⁰(t), the probability that a turnover occurs between t and t + Δt is f(t)Δt, the same as ΔP_E⁰(t). Taking this into account, in the limit of infinitesimal Δt:

1.11 [math]\displaystyle{ f(t)=\frac{dP_E^0(t)}{dt}=k_2P_{ES}(t) }[/math]

Solving equations 8-10, and using equation 11:

1.12 [math]\displaystyle{ f(t)=\frac{k_1k_2[S]}{2A}[exp(A+B)t-exp(B-A)t] }[/math]

in which:

1.13 [math]\displaystyle{ A=\sqrt((k_1[S]+k_{-1}+k_2)^2/4-k_1k_2[S]) }[/math]

1.14 [math]\displaystyle{ B=\frac{-(k_1[S]+k_{-1}+k_2}{2} }[/math]

when the substrate concentration dependence [S] has been shown throught the relation k₁⁰ = k₁[S].

With these equations, we have used different values for k_-1 and [S], using values found in literature.

[Descripcion y link del codigo de simulink]

[GRAFICAS Y ESO]

At the first moment of f(t), [math]\displaystyle{ \lt t\gt = \int_0^\infty dt tf(t) }[/math], which would be the mean waiting time for the reaction, <t>, and its reciprocal can be taken as the average reaction rate. Starting eqtn 12:

[math]\displaystyle{ \frac{1}{\lt t\gt }=-\frac{(A^2-B^2)^2}{2Bk_1k_2[S]} }[/math] [math]\displaystyle{ \frac{1}{\lt t\gt }=\frac{k_2[S]}{[S]+K_M} }[/math]

Randomness parameter

The probability density f(t) completely characterizes single-enzyme kinetics, with the n^th moment being given by:

1.15 [math]\displaystyle{ f(t)= \int_0^\infty dt f(t)t^n }[/math]

Although the first moment of f(t) can be described eqtn 11, higher moments of f(t) are usually calculated along with a "randomness parameter". Implying no dynamic disorder, r is given by:

1.16 [math]\displaystyle{ r=\frac{(k_1[S]+k_2+k_{-1})^2-2k_1k_2[S]}{(k1[S]+k_2+k_{-1})^2} }[/math]

As substrate concentration increases, r decreases, indicating the formation of an intermediate enzyme-substrate complex ES. At even higher concentrations, the catalytic step limits the reaction, as is the same when concentration is low and the substrate binding limits the rate.

Conclusions

We calculated the probability density of the waiting times of the enzymatic reaction, taking into account that the process is a mainly stochastic event at a single-molecule level. The reciprocal of the first moment of f(t) appears to have the same hyperbolic dependence on [S] as is described in conventional Michaelis-Menten kinetics (when assuming steady-state).