# Given an EC50 value, how do I reproduce the sigmoidal curve from which this was calculated?

All-trans retinoic acid (atRA) is a potent ligand for a nuclear receptor called retinoic acid receptor alpha (RARa). The concentration of atRA at which RARa is half maximal is 19nM. The dose-response curve is sigmoidal, with concentration (log scale) on the x-axis and percentage of maximal activity on the y.

How would I use this information to reproduce the dose-response curve?

Thanks

• Depends if the EC50 is given using a constant slope and a max/min that you know. If not then you cannot reconstruct it as you are missing parameters. – cagliari2005 Apr 11 '15 at 2:29

Basic sigmoidal curve looks like that: zero at $-\inf$ and one at $+\inf$. All in between should look like an integral of gaussian distribution. Take a look on this wiki page for more information. Now, the question, why sigmoidal curve is integral of gaussian function, I will leave out for now. My understanding is that gaussian distribution tells you how much system response will increase (say, efficiency of treatment) if you increase stimulus by $dx$. E.g., how much response changes when you go from $5$ to $5+dx$ versus from $10$ to $10+dt$.

The problem, as you might realize, that gaussian distribution has two parameters: mean value and width of the function. When you construct your fit, frist one will be 19nM, but second, $\sigma$, is unknown from your question. What you don't know is how rapidly response changes when you increase concentration by little around your EC50. So all your sigmoidal plots will go through same point but with different slope, like that: Without knowing second parameter, $\sigma$, it makes little sense to plot your sigmoidal curve. However, if you want to, this is MATLAB code. See that cumsum() function effectively integrates gaussian distribution rendering sigmoidal curve.

c = 0:100; % range of concentrations you test
sigma = 1; % unknown parameter
m = 19; % your EC50 19 nM
g = 1/(sigma*sqrt(2*pi))*exp(-((c - m).^2)/(2*sigma^2)); % gaussian
gs = cumsum(g);
plot(c, gs)


If you want, you can plot it with log-scale of X, using semilogx() instead of plot().

NB here I started with Gaussian function and its integral just as a wild guess. You real function might very well be something like $f(x)=\frac{1}{1+e^{-x}}$. "Sigmoid" function is just description of general shape, not mathematical definition.

• You answer is correct for a special case of sigmoidal curves. EC50 curves do not necessarily go from 0 to 1 in real experiments. Usually there is a baseline which is non-zero and a maxima which is not 1. To reconstruct the real sigmoidal curve you need those values. – cagliari2005 Apr 11 '15 at 2:33
• I agree that the case is special, so OP misses even more parameters than I thought. I looked closer on EC50, it seems that it is defined as 50% increase after subtraction of background, that is after you accounted for zero-concentration efficency – aaaaa says reinstate Monica Apr 11 '15 at 4:44
• Yup. He can approximate it with your method though (given an estimate of the slope). That was my upvote :) – cagliari2005 Apr 11 '15 at 5:04
• even though I know that "sigmoid" shaped curves have various math definitions, I am a big proponent of link between gaussian distribution and most of sigmoidal relations in biology. Don't have any supporting data so far, unfrotunately – aaaaa says reinstate Monica Apr 11 '15 at 6:02
• Hi Andrev, this is great, thank you... By slope do you mean the derivative at the point where the function is evaluated or the slope of the straight line bit in the middle? If you mean the latter, then I can work this out from the graph I'm looking at (since my data is from a paper). Essentially I'm trying to build a mathematical model that incorporates this data, so the values of the top and bottom asymptote will be relative - meaning I have to just make them up and see if they fit. – CiaranWelsh Apr 11 '15 at 11:01