1.4. Descripción de los Servicios que Brindará el ISP
1.4.4. Servicio de Voz sobre IP
In order to analyze the influence of external forces on the activity of enzymes single enzymes will have to be analyzed for example with the AFM. Therefore, the following chapter will introduce the basic concepts for the analysis of single enzyme data and give a short overview of the information, which can be gained from single enzyme measurements.
At the single molecule level the concentration of an enzyme is no longer relevant. The enzymatic reaction is considered to be a stochastic event. Therefore, for single molecule experiments the probabilities for the enzyme to be in one of the possible states along the reaction pathway are considered. To describe one complete enzymatic turnover again the following reaction scheme is used.
k1 k2
E + S
ES
E + P
k-1The following equations can be used to describe the time dependent probabilities for the enzyme to be either in its free state
E
or to be bound in the enzyme substrate complexES
: ! dPE( )
t dt ="k1[ ]
S PE( )
t +k"1PES( )
t +k2PES(t) (eq. 7.6) ! dPES( )
t dt =k1[ ]
S PE( )
t "k"1PES( )
t "k2PES(t) (eq. 7.7)The different states cannot be measured in a single enzyme experiment. The parameter, which can be measured, is the time the enzyme molecule needs for one complete turnover cycle. This time is also called the “waiting time”
τ
. If a huge number of turnover cycles is measured a waiting time distribution is obtained, which is the basis for the analysis of the dynamic behavior of the enzyme.Relating the waiting time distribution to the differential equations and solving them for the initial conditions
P
E(0)
= 1 andP
ES(0)
= 0 at t = 0 and the constraintP
E(t)
+P
ES(t)
= 1 one obtains the following relationship between the mean waiting timeτ
and the substrate concentration[S]
(Kou et al., 2005):!
1
"
=
k
2[ ]S
S
[ ]+K
M (eq. 7.8)Comparing this equation to the classical Michaelis-Menten equation it is evident, that the reciprocal of the mean waiting time is related to the enzymatic velocity measured in an ensemble measurement. This relation originates from the
Molecular force sensors for the manipulation of enzyme activity
equivalence between time averaging over a long time trace and ensemble averaging over a large number of identical molecules (English et al., 2006).
The above relation shows that experiments with single enzymes can yield the same result as ensemble measurements. However, this is not always the case. Enzymes are dynamic entities exhibiting distributions and fluctuations of catalytic rate constants. These effects can be analyzed in single molecule experiments only. The most important features of enzymes, which are hidden in ensemble measurements, are static and dynamic disorder (Xie and Lu, 1999). Static disorder is the result of differences in the activity of individual enzyme molecules. The existence of static disorder was shown for lactate dehydrogenase first (Xue and Yeung, 1995). A solution of the enzyme and its substrate was filled in a narrow capillary. The enzyme concentration was adjusted to ensure that single enzyme molecules are separated by a relatively large distance. The accumulation of product was measured after an incubation time of one hour. The amount of the product varied by a factor of 4. This result could be reproduced with the same enzyme molecules showing that the enzymes indeed possess different activities. Dynamic disorder refers to fluctuations of the rate constants of the reaction caused by transitions among different enzyme conformers. Dynamic disorder has been shown for different enzymes. The first experiment proving the existence of dynamic disorder was carried out with cholesterol oxidase (Lu et al., 1998). Later it was shown for staphyloccal nuclease (Ha et al., 1999), horseradish peroxidase (Edman et al., 1999), bacteriophage λ exonuclease (van Oijen et al., 2003), lysozyme (Lu,
2004), the lipase B of Candida antarctica (Velonia et al., 2005) and β-galactosidase
(English et al., 2006). It is considered to be a general property of enzymes (Xie, 2001).
The fluctuations resulting in dynamic disorder can occur on a time scale comparable to or longer than the time scale of the enzymatic reaction, so that the rate of the product formation is no longer governed by a single rate constant but effectively by a distribution of rate constants. The simplest case is shown in the reaction scheme below where
E
a shows very high catalytic activity andE
b shows low catalytic activity: k1a k2aE
a+ S
E
aS
E
a+ P
k-1a k1b k2bE
b+ S
E
bS
E
b+ P
k-1bMolecular force sensors for the manipulation of enzyme activity
Depending on the rates of the enzymatic reaction and the rates for the interconversion between the different conformations of the enzyme two general cases can be distinguished. If
k
2a is slow compared to the interconversion rates,k
2a is rate limiting and no dynamic disorder will be observed. However, if the interconversion rates are slower thank
2a the enzyme can remain in the highly active conformation for several turnovers and then convert into the less active conformation and remain in this conformation for a while. The effect that long waiting times are followed by long waiting times and short waiting times are followed by short waiting times is a non-Markovian behavior. Therefore, dependent on the combination of the rate constants memory effects can be observed. Interestingly, if the constants of the basic Michaelis-Menten equation are replaced by the weighted averages of the distributions of the conformers the equation is still valid. This has been shown theoretically and experimentally for the enzyme β-galactosidase(English et al., 2006). Information about dynamic disorder can be obtained from the time trajectories of the enzymatic turnover events. The rate constants can be obtained from a multiexponential fit to the probability density function of the waiting times. Alternatively the autocorrelation function of the waiting times shows the range of the timescales of the fluctuations (Xie, 2001; Kou et al., 2005; Min et al., 2005; English et al., 2006).