Preface: The following question comes from an expression seen in a biophysics paper published in Nature protocols.

I'm aware that in pure mathematical notation $\delta$ is never used in the context of differential equations, however, I have an expression with the following structure;

$$\frac{\delta x}{\delta y} = G(x)$$

Where G is a function which depends on $x$.

Rather than being a differential equation, I'm assuming this is a (somewhat misleading) way of looking at non-infinitesimal change. In the legend the following text appears

"$\delta x/ \delta y$ is the rate of change of ellipticity at any time t"

But in the original paper's context this $dt$ is likely to be value equal to or greater than 10 ms - i.e. this may not hold true as $\delta t \rightarrow 0$. This suggests that treating the expression as a differential equation would be inappropriate. Any thoughts/comments would be appreciated (even if it's just a 'yup - makes sense').

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    $\begingroup$ yup - makes sense. $\endgroup$ – Stephen Montgomery-Smith Dec 27 '13 at 19:04
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    $\begingroup$ It would greatly help if you provided a link to the paper in question. $\endgroup$ – AlexR Dec 27 '13 at 19:04

Infinitesimal rate of change is always a mathematical abstraction. Even for a simple physics problem, like the motion of a free falling stone, we encounter limitation of the model as $\delta t\to 0$. At small enough scale, the position and velocity of the stone are not well-defined: the stone is not a point, its molecules move around, quantum effects become noticeable... Any observed rate of change is not instantaneous; it is the mean rate of change over some period of time $\delta t>0$. One may still say "rate of change at time $t_0$" instead of the longer "mean rate of change over some time interval of small length $\delta t$ containing $t_0$". As long as the rate of change does not change much within the $\delta t$ interval, this is a reasonable thing to say. And we do use differential equations to model physical processes, and the models perform quite well in the setting of classical mechanics.

So, I would not say that treating equation (7) as a differential equation is inappropriate just because the rate of change hasn't been measured at time intervals $\delta _t<0.01$. If one can somehow solve it as a differential equation, the result could be used to predict the behavior of $[\theta]_t$. And if the prediction diverges from reality too much, one will look for reasons why.

After all, predictions based on models are not theorems. Nobody can prove a theorem about proteins; theorems concern mathematical objects, which proteins are not.

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