From: Olivier Beyens (Olivier.Beyens_at_uantwerpen.be)
Date: Thu Jul 29 2021 - 07:11:06 CDT

Hi JC,

I meant that if using formulas 4 and 5 (from the first mail) is not formally correct for the single ligand case, how are we certain that the accuracy of Keq is not impacted by this if we apply (modifications of) formulas 4 and 5 to a single ligand case?

Kind regards,

Olivier

________________________________
Van: Gumbart, JC <gumbart_at_physics.gatech.edu>
Verzonden: donderdag 29 juli 2021 13:55:45
Aan: Olivier Beyens
CC: tutorial-l_at_ks.uiuc.edu
Onderwerp: Re: Protein:lingand standard binding free energy tutorial: theory question

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Standard binding free energy *is* concentration dependent. In the equation to convert Keq to ÄG is the standard state concentration, which is assumed to be 1 M.

Best,
JC

On Jul 29, 2021, at 3:43 AM, Olivier Beyens <Olivier.Beyens_at_uantwerpen.be<mailto:Olivier.Beyens_at_uantwerpen.be>> wrote:

Hi JC,

Thanks for all the answers and effort. But now I wonder about one more thing in your response: if the formulas 4 and 5 (from the first mail) are not formally valid for the single ligand case, how are we certain that accuracy of the end result for the value of the standard binding free energy is not impacted?

Kind regards,

Olivier
________________________________
Van: Gumbart, JC <gumbart_at_physics.gatech.edu<mailto:gumbart_at_physics.gatech.edu>>
Verzonden: woensdag 28 juli 2021 2:58:28
Aan: Olivier Beyens; tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu>
Onderwerp: Re: Protein:lingand standard binding free energy tutorial: theory question

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Well, traditionally, the equilibrium binding constant Keq does not depend on concentration and, indeed, it drops out of that equation. Most results in stat mech rely on the assumption of large numbers, without which conclusions can become unreliable.

As for this case, I have no idea if the Keq you might directly measure just by watching a single ligand bind and unbind would be the same as that from watching a lot of ligands. But I think it's irrelevant for us, anyway, as it appears what the equation is indicating is that the effect of multiple ligands is just a multiplicative factor anyway. That the result after cancelation only *happens* to reference a single ligand does not mean that it's only valid for a single ligand. In fact, a bit ironically, it may not be formally valid for the single-ligand case as the preceding cancelation could be invalid (as you suggested).

Best,
JC
________________________________
From: Olivier Beyens <Olivier.Beyens_at_uantwerpen.be<mailto:Olivier.Beyens_at_uantwerpen.be>>
Sent: Monday, July 26, 2021 6:23 AM
To: Gumbart, JC <gumbart_at_physics.gatech.edu<mailto:gumbart_at_physics.gatech.edu>>; tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu> <tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu>>
Subject: Re: Protein:lingand standard binding free energy tutorial: theory question

Hi JC,

This is why I don't understand why we are allowed to apply this formula to the problem of the tutorial, since in the tutorial the ligand is present in the same concentration as the protein. The tutorial derives an equation for 'an isolated protein in a solution of N indistinguishable ligands', and then applies the result to a periodic box containing one protein and one ligand. In order to derive the result, the assumption was made that [ligand]=N_ligand/V. However then we apply the resulting equation to a situation where the assumption '[ligand]=N_ligand/V' is (the way I understand) not valid. Why are we then allowed to apply the result to a box with periodic boundary conditions containing one ligand and one protein?

Kind regards,

Olivier

________________________________
Van: Gumbart, JC <gumbart_at_physics.gatech.edu<mailto:gumbart_at_physics.gatech.edu>>
Verzonden: zaterdag 24 juli 2021 2:35:51
Aan: Olivier Beyens; tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu>
Onderwerp: Re: Protein:lingand standard binding free energy tutorial: theory question

CAUTION: This email originated from outside of the organization. Do not click links or open attachments unless you recognize the sender and know the content is safe.

Fair point regarding the protein concentration - I think that you are right that [ligand] is supposed to be the free concentration. It is typically assumed, however, that the protein concentration is much less than the ligand concentration and, therefore, binding a few ligands minimally alters their free concentration. In fact, the tutorial basically says this: "Assuming low protein concentration, one can imagine an isolated protein in a solution of N indistinguishable ligands."

The reason I noted Prof. Luthey-Schulten's papers is merely to emphasize that when you have a very small number of ligands, you need to use a discrete, stochastic formalism rather than the concentration picture that emerges only for a relatively large number of them.

Best,
JC
________________________________
From: Olivier Beyens <Olivier.Beyens_at_uantwerpen.be<mailto:Olivier.Beyens_at_uantwerpen.be>>
Sent: Friday, July 23, 2021 2:26 PM
To: Gumbart, JC <gumbart_at_physics.gatech.edu<mailto:gumbart_at_physics.gatech.edu>>; tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu> <tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu>>
Subject: Re: Protein:lingand standard binding free energy tutorial: theory question

Hi JC,

My reasoning for using only free concentration was the following: 1) I think the thermodynamic activity would be decreased when the ligand is bound to the protein 2) For the protein the same reasoning is used in the tutorial I think , [protein] is the free protein concentration only. Maybe I should rephrase my question then, why are we allowed to use total concentration of the ligand for [ligand], while we use only the free concentration of the protein for [protein]?

(I had a very quick look at the papers from prof. Luthey-Schulten, but I don't really see how this method can be directly applied to ABF calculations, might have not adequately understood it however.)

Kind regards,

Olivier
________________________________
Van: Gumbart, JC <gumbart_at_physics.gatech.edu<mailto:gumbart_at_physics.gatech.edu>>
Verzonden: vrijdag 23 juli 2021 1:19:11
Aan: Olivier Beyens; tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu>
Onderwerp: Re: Protein:lingand standard binding free energy tutorial: theory question

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Hi Olivier,

Why would you assume [ligand] is the free concentration only? I actually don't know if this is standard or not.

That being said, I think the concentration picture breaks down anyway when you have few reactants, and you have to use a stochastic approach instead. Prof. Luthey-Schulten has done a lot of work in this area, especially with her Lattice Microbes code: http://faculty.scs.illinois.edu/schulten/Software2.0.html0a2146d4b66408d95287dabe*7C792e08fb2d544a8eaf72202548136ef6*7C0*7C0*7C637631565502710078*7CUnknown*7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0*3D*7C1000&sdata=TkZESrSN2ZcrOvjvOknQFlzYQNinsl0*2BWtd47EDR4ss*3D&reserved=0__;JSUlJSUlJSUlJSUlJSUlJSUlJQ!!DZ3fjg!sp6vXuEMS8GgwJivMWuWjtC1VC3x0Fo8NJgodJPvrOaYRyQFiulXfN7SoTJzInjKIPlB$ >

Best,
JC
________________________________
From:
owner-tutorial-l_at_ks.uiuc.edu<mailto:owner-tutorial-l_at_ks.uiuc.edu> <owner-tutorial-l_at_ks.uiuc.edu<mailto:owner-tutorial-l_at_ks.uiuc.edu>> on behalf of Olivier Beyens <Olivier.Beyens_at_uantwerpen.be<mailto:Olivier.Beyens_at_uantwerpen.be>>
Sent: Tuesday, July 20, 2021 10:25 AM
To: tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu> <tutorial-l_at_ks.uiuc.edu<mailto:tutorial-l_at_ks.uiuc.edu>>
Subject: tutorial-l: Protein:lingand standard binding free energy tutorial: theory question

Hi everyone,

I have a question regarding the section 1.1 'Theoretical underpinnings' of the tutorial on protein:ligand standard binding free energies:

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When transforming equation (4) into equation (5), they use [ligand] = N/V_bulk . However I was wondering why one can say that the concentration of the ligand at equilibrium is N/V_bulk, since at equilibrium sometimes one of the ligands is bound to the protein. Thus I would expect the equilibrium concentration of the ligand to be [ligand] = (N - p_bound)/V_bulk. Now when N is large this doesn't really matter, but we apply these equations on a system with only one ligand... So my question is, why are we allowed to say that [ligand] = N/V_bulk at equilibrium?

I have seen this equation being used in many of prof. Roux's papers, however I have always broke my head about this... I hope you can clear a few things up for me!

Kind regards,

Olivier

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