User talk:Eskeptic
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I hope you enjoy editing here and being a Wikipedian! Please sign your messages on discussion pages using four tildes (~~~~); this will automatically insert your username and the date. If you need help, check out Wikipedia:Questions, ask me on my talk page, or ask your question on this page and then place {{helpme}} before the question. Again, welcome! Tim Vickers (talk) 20:43, 20 November 2009 (UTC)
- My sympathies, just forced a paper in myself after some horrible initial reviews and much "further experiments needed". Tim Vickers (talk) 21:58, 20 November 2009 (UTC)
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[edit]Please, for us math-challenged people, don't say "clearly shows", "clearly a result of" or "easily modified", as this just makes us feel worse about not following all the working! Tim Vickers (talk) 04:02, 29 November 2009 (UTC)
Hello Eskeptic. I'm new to Wikipedia, so I'm not sure if I should post this on your talk page, my talk page, or the "rate equation" article talk page. About your comments, I and my PhD supervisor read the BBA paper the argument came from and didn't get what the author was trying to say. The BBA author seemed to confuse rate and rate constant. Interchanging rate and rate constant in the 1st order kientics equation is what gives the seemingly wrong results. I think that leaving the article as it was would confuse people by undermining the validity of the 1st order kinetics equation based on rate/rate constant confusion. But I didn't want to just take out the entire section, because there is merit to thinking about the kinetics in terms of the rate rather than rate constant. This seemed like the best solution. Futurechemist1 (talk) 10:20, 27 May 2010 (UTC)
Let's see if I can clarify this. When I say instantaneous, I use it in the mathematical sense. Not to mean 0 time, but sufficiently small time so that calculus can be used on the system. If you think this distinction is vague/confusing, feel free to tweak it.
I have an Excel spreadsheet that I think clarifies my argument. However, I don't know how to get it to you through Wikipedia. Basically, my argument is this (I don't know Latex or Wikipedia equation editing, so if I could be excused for typing formulae the ugly way):
Let's take a reaction that consumes half of the reagent every second. By your argument, since half of the population decays each time interval, k should be 0.5 /s. My argument is that k is not meant to be 0.5 /s, and the exponential equation is not needed to determine the appropriate value for k.
Here's what I did in the Excel spreadsheet. I created the data that would give the amount of A left after 1s, 2s, ... (i.e. 1M, 0.5M, 0.25M, etc.) I then calculated the rate of reaction numerically by drawing a tangent line (technically a secant line). So the initial rate would be A(1s)-A(0s)]/[1s-0s] = (0.5M - 1M) / (1s - 0s) = -0.5 M/s. The rate at t=1s would be -0.25 M/s and so on. Now from the definition of rate constant, k = rate/A. In this case the initial rate is -0.5M /s and the initial concentration is 1M, giving k = -0.5 M/s / 1M = 0.5 /s. You'd get the same value of k using the data at any time. This is what you would expect it to be. But if you put k=0.5 into A=exp(-kt), it gives bad results, as you mentioned in the BBA article.
So let's go back to the raw data. Instead of determing the concentration of A every 1s, let's determine it every 0.1 s. I did this by filling the data using the equation A(t) = (1/2)^t which would work for noninteger times. (This is essentially the formula that you get at the end of your argument and I agree with it completely). Using the same analysis as the above paragraph A(0s) = 1M, A(0.1s) = 0.933M, Rate (0s) = -0.669 M/s, and k = -0.669 M/s / 1M = 0.669 /s, very different than the 0.5 /s we got before. Again, you'd get k=0.669 from the rate and concentration at any time in the data set.
If instead you use data spaced every 0.01s, then k=0.6907 /s, and if the time spacing is 0.001 s, then k = 0.6929 /s. If infinitessimally small time spacing was used, this would converge on approximately k=0.693 /s, which is exactly the rate constant you'd get from the exponential equation (technically ln(2) /s)
I think what is going wrong is that in this analysis, it assumes the rate is unchanging over the entire time interval. That is, for your argument, during the first second, rate = -0.5M/s, during the second second, rate = -0.25 M/s, etc.. But in actuality, the rate is always changing. For a very small time interval, the error in assuming an unchanging rate over the entire time interval is small. But for a large time interval (such as 1s when the rate is 50% /s), the error is very large. As such, the error has nothing to do with how Euler's constant is separated from other constants, it's an artifact of the calculus that underpins the equations.
Does this clear things up? If you'd like, I could send you the Excel spreadsheet I made for this, that might be easier to see my argument
This also explains how the rate (and rate constant) can be larger than 1. The extreme case of this is if the reaction is over before the time interval expires. In that case, the fraction of reactants that have reacted is 100%. But if the reaction was already over after only half the time interval had elapsed, the fraction that reacted would essentially be 200%
Hope that clears things up. - Larry Futurechemist1 (talk) 18:12, 27 May 2010 (UTC)
Hello again. I was just at the supermarket and I had some inspiration for another way to approach the problem, that gets rid of instantaneous times and that sort f heavy analysis. This way uses averages to estimate the rate constant. Take that same reaction where 50% is consumed every second. What's the average concentration and rate for the 1st second. The starting and ending concentrations are 1M and 0.5M. So the average concentration is 0.75M. And the average rate to get from 1M to 0.5M in 1 s is -0.5M /s. And since rate constant = -rate/concentration, that means 0.5M/s / 0.75 M = 0.67 /s. Which isn't a bad estimate of the true rate constant. To a first order approximation, the averaging method corrects for the fact that both the concentration and rate are always changing. Does that help explain why the rate constant wouldn't be 0.5 /s? - Larry Futurechemist1 (talk) 19:00, 27 May 2010 (UTC)
Kinetics
[edit]Hi Ryan. I think we can break this discussion into a part on Wikipedia and a part on your BBA paper.
For Wikipedia, the rate equation section you edited is more general than just Michaelis-Menten kinetics, it applies to all first order processes (enzyme, chemical, radioactive decay, etc.). Your comment about time scales is illustrative. Depending on the time scale of the reaction in question, the mathematical analysis will either be nearly perfect, or horribly wrong. Because it's a flaw in the specific data analysis rather than the underlying theory, I don't think it's general enough for the Wikipedia entry. In the sense that if someone miscalculated the rate constant, that doesn't affect the validity of the rate equation. My opinion is that the section on the alternative view of 1st order kinetics either be taken out entirely, or greatly shortened. Everything before the phrase "A different (but equivalent) ..." would be removed. Do you agree?
Onto the paper. I'm a physical organic chemist, so I can't comment fully on some aspects of the paper. But I have a few thoughts on the math. The same mathematical issues can appear here that I previously mentioned. If you use Excel to solve your data using improperly large time steps, you'll end up with kinetic constants that have the wrong value. But assuming that the substrate decays according to 1st order kinetics (technically, it's probably pseudo-1st order), equation 6 should be valid. That still leaves the question of whether the substrate does show 1st order behavior though. Equation 7 seems fine given those assumptions too.
Section 4. Here's where the problem starts. If you accept my argument from the previous message, then the seeming mathematical mismatch comes from inappropriate data analysis, and confusion of how k relates to r. The bits about Euler's constant don't actually apply (nor do they make sense to me). So most of this section falls apart.
Overall, it seems to me that you have a good idea. I don't see anything fundamentally wrong, other than the aforementioned stuff in Section. 4. So I don't think there's anything wrong with your steady-state decay equations (Eq. 7) and it should work. This is all assuming that the system shows proper first order behavior. A potential problem will occur if enzyme saturation causes the substrate to react at a rate significantly different than expected, or if it changes the rate in a non-obvious way as a function of time. But that's something for the biochemists to look into. I'd be interested to see how well your approach fits an array of different enzymes.
- Larry
P.S. I was looking in a recent text, "Enzyme Kinetics and Mechanism" by Cook and Cleland. They describe an alternative way to analyse the time course of M-M kinetics (the end of Chapter 3 in the text). How does their approach compare to yours? Futurechemist1 (talk) 11:46, 28 May 2010 (UTC)