Ammm:Umbrella sampling and WHAM - Revision history

Xy3866: /* Basic Idea */

2022-08-04T19:04:58Z

Basic Idea

← Older revision		Revision as of 19:04, 4 August 2022
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	== Basic Idea ==		== Basic Idea ==

	*The Weighted Histogram Analysis Method (WHAM) is a practical technique for the calculation of potentials of mean force (PMFs) from a group of umbrella sampling simulations. It can be applied to analyze the energy data based on not only real collective variables (have relations to the atom coordinates) but also virtual variables (temperature or interactions in the system). This technique can give out a systematic way to improve the choice of the bias potential from a set of different bias potentials so that it yields uniform sampling along the collective variables. Meanwhile, the multiple simulations can start from different points along collective variables, then the full sampling can be realized. Also, the underlying theory of this method provides us a practical way to do histogram analysis and calculate free energy under discretized language.		The Weighted Histogram Analysis Method (WHAM) is a practical technique for the calculation of potentials of mean force (PMFs) from a group of umbrella sampling simulations. It can be applied to analyze the energy data based on not only real collective variables (have relations to the atom coordinates) but also virtual variables (temperature or interactions in the system). This technique can give out a systematic way to improve the choice of the bias potential from a set of different bias potentials so that it yields uniform sampling along the collective variables. Meanwhile, the multiple simulations can start from different points along collective variables, then the full sampling can be realized. Also, the underlying theory of this method provides us with a practical way to do histogram analysis and calculate free energy under discretized language.

	== Discretization of configuration space along collective variable ==		== Discretization of configuration space along collective variable ==

Xy3866: /* Examples */

2022-08-03T18:44:17Z

Examples

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	== Ion Channel <ref>Zhifeng Jing, Joshua A. Rackers , Lawrence R. Pratt, Chengwen Liu, Susan B. Rempe and Pengyu Ren. Thermodynamics of ion binding and occupancy in potassium channels. Chem. Sci., 2021, 12, 8920-8930</ref> ==		== Ion Channel <ref>Zhifeng Jing, Joshua A. Rackers , Lawrence R. Pratt, Chengwen Liu, Susan B. Rempe and Pengyu Ren. Thermodynamics of ion binding and occupancy in potassium channels. Chem. Sci., 2021, 12, 8920-8930</ref> ==
	[[File:Image.png\|center\|thumb]]		[[File:Image.png\|center\|thumb]]

			== Tutorial of US & WHAM ==
			The Gromacs provides us with a good tutorial to show the way to perform Umbrella Sampling & WHAM post-processing: http://www.mdtutorials.com/gmx/umbrella/ <ref>Justin A. Lemkul and David R. Bevan. Assessing the Stability of Alzheimer's Amyloid Protofibrils Using Molecular Dynamics. J''. Phys. Chem. B'' 2010, 114, 4, 1652–1660</ref>

	= <span style="background-color: rgb(153,204,255)">References </span> =		= <span style="background-color: rgb(153,204,255)">References </span> =

	<references />		<references />

Xy3866: /* Derivation of WHAM[4] */

2022-08-03T18:31:34Z

Derivation of WHAM[4]

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	The most common case is that all the temperatures are the same, and the correlation factors are cancelled out. Then the equations <math>()</math> and <math>(*)</math> can be simplified into:		The most common case is that all the temperatures are the same, and the correlation factors are cancelled out. Then the equations <math>()</math> and <math>(*)</math> can be simplified into:
	:<math>\tilde \rho_{K,i}=\frac{exp(-\beta V_{K,i}) \sum_{k=1}^R N_{k,i}} {\sum_{m=1}^R n_{m} exp(-\sum_{i=1}^L \beta V_{m,i}+\beta f_{m})}\ \ (*)</math>		:<math>\tilde \rho_{K,i}=\frac{exp(-\beta V_{K,i}) \sum_{k=1}^R N_{k,i}} {\sum_{m=1}^R n_{m} exp(-\sum_{i=1}^L \beta V_{m,i}+\beta f_{m})}\ \ (*)</math>
	:<math>exp(-\~~beta_{K}~~ f_{K})=\sum_{i=1}^L \tilde \rho_{K,i}\ \ (**)</math>		:<math>exp(-\beta f_{K})=\sum_{i=1}^L \tilde \rho_{K,i}\ \ (**)</math>

	== Self-consistent algorithm ==		== Self-consistent algorithm ==

Xy3866: /* Ion Channel [7] */

2022-08-03T18:29:35Z

Ion Channel [7]

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	:<math>\tilde \rho_{K,i}=\frac{exp(-\beta_{K} V_{K,i}) \sum_{k=1}^R g_{k}^{-1} N_{k,i}} {\sum_{m=1}^R g_{m}^{-1} n_{m} exp(-\sum_{i=1}^L \beta_{m} V_{m,i}+\beta_{m} f_{m})}\ \ (*)</math>		:<math>\tilde \rho_{K,i}=\frac{exp(-\beta_{K} V_{K,i}) \sum_{k=1}^R g_{k}^{-1} N_{k,i}} {\sum_{m=1}^R g_{m}^{-1} n_{m} exp(-\sum_{i=1}^L \beta_{m} V_{m,i}+\beta_{m} f_{m})}\ \ (*)</math>
	Because <math>f_{K}</math> is related to configurational integral:		Because <math>f_{K}</math> is related to configurational integral:

			<math>exp(-\beta_{K} f_{K})=\sum_{i=1}^L \tilde \rho_{K,i}\ \ (**)</math>

			The most common case is that all the temperatures are the same, and the correlation factors are cancelled out. Then the equations <math>()</math> and <math>(*)</math> can be simplified into:
			:<math>\tilde \rho_{K,i}=\frac{exp(-\beta V_{K,i}) \sum_{k=1}^R N_{k,i}} {\sum_{m=1}^R n_{m} exp(-\sum_{i=1}^L \beta V_{m,i}+\beta f_{m})}\ \ (*)</math>
	:<math>exp(-\beta_{K} f_{K})=\sum_{i=1}^L \tilde \rho_{K,i}\ \ (**)</math>		:<math>exp(-\beta_{K} f_{K})=\sum_{i=1}^L \tilde \rho_{K,i}\ \ (**)</math>

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	[[Image:B4.JPG\|345x260px]]		[[Image:B4.JPG\|345x260px]]

	== Ion Channel <ref>~~Berneche~~,~~S and Roux~~,B~~: Energetics~~ of ion ~~conduction through the K+ channel~~, ~~nature~~,~~vol414~~,~~1 November,2001~~</ref> ==		== Ion Channel <ref>Zhifeng Jing, Joshua A. Rackers , Lawrence R. Pratt, Chengwen Liu, Susan B. Rempe and Pengyu Ren. Thermodynamics of ion binding and occupancy in potassium channels. Chem. Sci., 2021, 12, 8920-8930</ref> ==
			[[File:Image.png\|center\|thumb]]
	~~[[Image:Ion1.JPG\|259x276px]]~~

	~~<br>~~

	[[~~Image~~:~~Ion2.JPG\|319x39px]]~~

	[[Image~~:Ion3~~.~~JPG~~\|~~809x592px]]~~

	~~[[Image:Ion4.JPG~~\|~~390x378px~~]]

	= <span style="background-color: rgb(153,204,255)">References </span> =		= <span style="background-color: rgb(153,204,255)">References </span> =

	<references />		<references />

Xy3866: /* Applications */

2022-08-03T17:23:03Z

Applications

← Older revision		Revision as of 17:23, 3 August 2022
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	The true free energy change between any two bins in simulation <math>K</math> without bias potential can be calculated by:		The true free energy change between any two bins in simulation <math>K</math> without bias potential can be calculated by:
	:<math>\Delta A_{K,j \to i}=-\beta_{K}^{-1} ln(\frac{\rho_{K,i}} {\rho_{K,j}})</math>		:<math>\Delta A_{K,j \to i}=-\beta_{K}^{-1} ln(\frac{\rho_{K,i}} {\rho_{K,j}})</math>
			==Procedure of umbrella sampling==
	== ~~Applications~~ ==		*Define collective variables appropriate to describe the reaction space
			* Divide the whole reaction space into bins with proper resolution
	*~~'''<span style="font-size: larger;">US combined~~ with ~~FEP~~		*Generate different configurations at each bin by using steered MD (or directly do pulling during different simulations)
	~~</span>'''~~		*Set bias potential for each configuration (commonly used: harmonic restrain)
			*Run multiple MDs with different bias potentials and record the trajectories & energy
	~~[[Image:Ap1.JPG\|807x90px]] [[Image~~:~~Ap2.JPG\|812x527px]]~~		*Put data into the WHAM processing program (use the module from Gromacs, …)

	~~<br>~~

	~~<br>~~

	*~~'''<span style="font-size: larger;">Potential of Mean Force~~
	~~</span>'''~~

	~~[[Image:Ap3.JPG\|813x273px]]~~

	= <span style="background-color: rgb(153, 204, 255);">Examples</span> =		= <span style="background-color: rgb(153, 204, 255);">Examples</span> =

Xy3866: /* WHAM */

2022-08-03T17:19:32Z

WHAM

@@ Line 57: / Line 57: @@
 *The Weighted Histogram Analysis Method (WHAM) is a practical technique for the calculation of potentials of mean force (PMFs) from a group of umbrella sampling simulations. It can be applied to analyze the energy data based on not only real collective variables (have relations to the atom coordinates) but also virtual variables (temperature or interactions in the system). This technique can give out a systematic way to improve the choice of the bias potential from a set of different bias potentials so that it yields uniform sampling along the collective variables. Meanwhile, the multiple simulations can start from different points along collective variables, then the full sampling can be realized. Also, the underlying theory of this method provides us a practical way to do histogram analysis and calculate free energy under discretized language.
-== Derivation<ref>Roux B: Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations. Comp Phys Commu 2001, 135: 40-57.</ref>  ==
+== Discretization of configuration space along collective variable ==
-<br>
+Assume we have run <math>R</math> simulations at the same time, with each using a different or same bias potential and temperature. <math></math>
 == Applications  ==

Xy3866: /* Umbrella Sampling [2] */

2022-08-03T17:17:47Z

Umbrella Sampling [2]

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 *Ensure complete scanning of all the configuration space along the CVs under bias potential
 *Perform discretization of the probability distribution along CVs
 </span></span>
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 == Basic Idea  ==
-*The contribution of each run to a reweighting estimate should be weighed based on the magnitude of errors in their histograms. That is, runs that have greater overlap with the reweighting conditions should contribute more to the estimation of property averages.
+*The Weighted Histogram Analysis Method (WHAM) is a practical technique for the calculation of potentials of mean force (PMFs) from a group of umbrella sampling simulations. It can be applied to analyze the energy data based on not only real collective variables (have relations to the atom coordinates) but also virtual variables (temperature or interactions in the system). This technique can give out a systematic way to improve the choice of the bias potential from a set of different bias potentials so that it yields uniform sampling along the collective variables. Meanwhile, the multiple simulations can start from different points along collective variables, then the full sampling can be realized. Also, the underlying theory of this method provides us a practical way to do histogram analysis and calculate free energy under discretized language.
 == Derivation<ref>Roux B: Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations. Comp Phys Commu 2001, 135: 40-57.</ref>  ==
 <br>
 == Applications  ==

Xy3866: /* Weighted ensemble */

2022-08-03T17:12:41Z

Weighted ensemble

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	*This method demands post-processing to remove the effects of bias potential.		*This method demands post-processing to remove the effects of bias potential.
	==<span style="background-color: rgb(192, 192, 192);"> </span> Weighted ensemble ==		==<span style="background-color: rgb(192, 192, 192);"> </span> Weighted ensemble ==
	The original ensemble ~~(Equation 1)~~ can be modified by using a weight function. The weight function should be positive here.		The original ensemble can be modified by using a weight function. The weight function should be positive here.
	:<math><exp(-\beta \Delta U)>_{0}= \frac{\int\, exp(-\beta \Delta U) exp(-\beta \Delta U_{0}) dx} {\int\, exp(-\beta \Delta U_{0}) dx} </math>		:<math><exp(-\beta \Delta U)>_{0}= \frac{\int\, exp(-\beta \Delta U) exp(-\beta \Delta U_{0}) dx} {\int\, exp(-\beta \Delta U_{0}) dx} </math>
	:<math><exp(-\beta \Delta U)>_{0}= \frac{\int\, exp(-\beta \Delta U) w w^{-1} exp(-\beta \Delta U_{0}) dx} {\int\, exp(-\beta \Delta U_{0}) w w^{-1} dx}=\frac{<w^{-1} exp(-\beta \Delta U)>_{w}} {<w^{-1}>_{w}}</math>		:<math><exp(-\beta \Delta U)>_{0}= \frac{\int\, exp(-\beta \Delta U) w w^{-1} exp(-\beta \Delta U_{0}) dx} {\int\, exp(-\beta \Delta U_{0}) w w^{-1} dx}=\frac{<w^{-1} exp(-\beta \Delta U)>_{w}} {<w^{-1}>_{w}}</math>
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	To reflect the connection between bias potential and the weight function, we can represent the weight function as an exponential form: <math>w=exp(-\beta V)</math>, where <math>V</math> means the bias potential.		To reflect the connection between bias potential and the weight function, we can represent the weight function as an exponential form: <math>w=exp(-\beta V)</math>, where <math>V</math> means the bias potential.
	:<math><exp(-\beta \Delta U)>_{0}=\frac{<exp(\beta V) exp(-\beta \Delta U)>_{w}} {<exp(\beta V)>_{w}}</math>		:<math><exp(-\beta \Delta U)>_{0}=\frac{<exp(\beta V) exp(-\beta \Delta U)>_{w}} {<exp(\beta V)>_{w}}</math>

	== <span style="background-color: rgb(192, 192, 192);"> </span>Free energy calculation ==		== <span style="background-color: rgb(192, 192, 192);"> </span>Free energy calculation ==
	After defining a collective variable (CV) <math>\xi</math>, we can calculate the free energy at any point along the direction of CV by using the probability distribution function.		After defining a collective variable (CV) <math>\xi</math>, we can calculate the free energy at any point along the direction of CV by using the probability distribution function.

Xy3866: /* Umbrella Sampling(US) */

2022-08-03T17:12:17Z

Umbrella Sampling(US)

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	[[Image:P1 rc.JPG\|473x250px]]		[[Image:P1 rc.JPG\|473x250px]]

	= <span style="background-color: rgb(153, 204, 255);">Umbrella Sampling(US) </span> =		= <span style="background-color: rgb(153, 204, 255);">Umbrella Sampling (US) </span> =
	Umbrella sampling is a method in computational physics and chemistry, which can enhance the sampling of different systems when it is hard to realize ergodicity due to the energy landscape. It involves the importance sampling in statistical mechanics and was first recommended by Torrie and Valleau in 1977<ref name="stonebook">Torrie, G. M.; Valleau, J. P. (1977). "Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling". Journal of Computational Physics. 23 (2): 187–199. doi:10.1016/0021-9991(77)90121-8</ref>.In many cases, the configuration space may have multiple high-energy barriers which are hard to conquer in the limited simulation time with common molecular dynamics (MD). Therefore, it is of high probability that the trajectory is trapped in the local minimum (local state) and leaves other inaccessible important states unsampled. In order to remove the impact of the energy barriers (activation case), the researchers can chosose the umbrella sampling (US) to bridge the gap between different configuration states by adding an extra bias potential. This bias potential should cancel out the influence of these energy barriers so that the energy landscape will become more smooth, making it easier to reach other states.		Umbrella sampling is a method in computational physics and chemistry, which can enhance the sampling of different systems when it is hard to realize ergodicity due to the energy landscape. It involves the importance sampling in statistical mechanics and was first recommended by Torrie and Valleau in 1977<ref name="stonebook">Torrie, G. M.; Valleau, J. P. (1977). "Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling". Journal of Computational Physics. 23 (2): 187–199. doi:10.1016/0021-9991(77)90121-8</ref>.In many cases, the configuration space may have multiple high-energy barriers which are hard to conquer in the limited simulation time with common molecular dynamics (MD). Therefore, it is of high probability that the trajectory is trapped in the local minimum (local state) and leaves other inaccessible important states unsampled. In order to remove the impact of the energy barriers (activation case), the researchers can chosose the umbrella sampling (US) to bridge the gap between different configuration states by adding an extra bias potential. This bias potential should cancel out the influence of these energy barriers so that the energy landscape will become more smooth, making it easier to reach other states.
	== <span style="background-color: rgb(192, 192, 192);"> </span>Why do we need biased potential? ==		== <span style="background-color: rgb(192, 192, 192);"> </span>Why do we need biased potential? ==
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	The best bias potential can help remove all the possible barriers on the energy landscape, which means the free energy change under this bias potential should be 0 everywhere. So, it is obvious to know that the best bias potential must counterbalance the true free energy.		The best bias potential can help remove all the possible barriers on the energy landscape, which means the free energy change under this bias potential should be 0 everywhere. So, it is obvious to know that the best bias potential must counterbalance the true free energy.
	:<math>V(\xi)-V(\xi_{0})=-\Delta A(\xi)</math>		:<math>V(\xi)-V(\xi_{0})=-\Delta A(\xi)</math>
	=== Challenges of Umbrella Sampling ===		== <span style="background-color: rgb(192, 192, 192);"> </span>Challenges of Umbrella Sampling ==
	*Pick up the best bias potential along the CVs to realize the sufficiently flat energy landscape		*Pick up the best bias potential along the CVs to realize the sufficiently flat energy landscape
	*Ensure complete scanning of all the configuration space along the CVs under bias potential		*Ensure complete scanning of all the configuration space along the CVs under bias potential

Xy3866: /* Biased Potential */

2022-08-03T17:11:19Z

Biased Potential

← Older revision		Revision as of 17:11, 3 August 2022
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	[[Image:P1 rc.JPG\|473x250px]]		[[Image:P1 rc.JPG\|473x250px]]

	= <span style="background-color: rgb(153, 204, 255);">~~Biased Potential~~ </span> =		= <span style="background-color: rgb(153, 204, 255);">Umbrella Sampling(US) </span> =
			Umbrella sampling is a method in computational physics and chemistry, which can enhance the sampling of different systems when it is hard to realize ergodicity due to the energy landscape. It involves the importance sampling in statistical mechanics and was first recommended by Torrie and Valleau in 1977<ref name="stonebook">Torrie, G. M.; Valleau, J. P. (1977). "Nonphysical sampling distributions in Monte Carlo free-energy estimation: Umbrella sampling". Journal of Computational Physics. 23 (2): 187–199. doi:10.1016/0021-9991(77)90121-8</ref>.In many cases, the configuration space may have multiple high-energy barriers which are hard to conquer in the limited simulation time with common molecular dynamics (MD). Therefore, it is of high probability that the trajectory is trapped in the local minimum (local state) and leaves other inaccessible important states unsampled. In order to remove the impact of the energy barriers (activation case), the researchers can chosose the umbrella sampling (US) to bridge the gap between different configuration states by adding an extra bias potential. This bias potential should cancel out the influence of these energy barriers so that the energy landscape will become more smooth, making it easier to reach other states.
	== <span style="background-color: rgb(192, 192, 192);"> </span>Why do we need biased potential? ==		== <span style="background-color: rgb(192, 192, 192);"> </span>Why do we need biased potential? ==

	[[Image:Bias.JPG\|543x286px]]		[[Image:Bias.JPG\|543x286px]]

	== ~~How to determine a biased potential?~~ <~~ref name~~="~~frenkel2002">[http~~:~~//catalog.lib.utexas.edu/record=b6499358~S29 Chipot~~, ~~C.; Pohorille~~, ~~A., ''Free Energy Calculations - Theory and Applications in Chemistry and Biology''. Springer-Verlag: Berlin Heidelberg, 2007~~; ~~Vol. 86.]~~</~~ref~~> ==		== <span style="background-color: rgb(192, 192, 192);"> </span>Features of method ==
			*Pre-determined collective variables are required to describe the configuration space related to reaction.
	*~~A basic but powerful method~~ to ~~improve~~ the ~~efficiency of such computations is~~ to ~~split the interval of computation along the~~ reaction ~~coordinate into subintervals, an approach termed stratification~~.		*The target of this method is to handle activation problems in sampling, where high energy barriers can be smoothened by adding extra bias potential. (Convert activation into diffusion)
	*~~In addition,~~ in ~~each window a biasing potential can be used in order to improve the~~ sampling ~~further. In this way~~, ~~the~~ energy barriers ~~inside each window~~ can be ~~made smaller~~.		*The extra bias potential causes the loss of temporal properties.
	*~~If this biasing~~ potential is a function ~~of '''V~~(x)~~'''~~, the ~~new Hamiltonian for~~ the ~~system is '''H~~(x)+V(x)~~'''.~~		*This method supports parallel MD simulations with different bias potentials at the same time so that computational cost is reduced.
	**In general the ~~biasing~~ potential ~~needs to be guessed beforehand or~~ can ~~be gradually improved using~~ an ~~iterative refinement process~~.		*This method demands post-processing to remove the effects of bias potential.
	***Run a short simulation to estimate the free energy, A<~~sup~~>(0)</~~sup~~>(x).		==<span style="background-color: rgb(192, 192, 192);"> </span> Weighted ensemble ==
	***Use this estimate to bias the ~~system~~ using V<~~sup~~>(0)<~~/sup~~>(x)=-~~A<sup>~~(0)</~~sup~~>~~(x).~~		The original ensemble (Equation 1) can be modified by using a weight function. The weight function should be positive here.
	***With this first bias, ~~we improve the sampling and obtain a more accurate approximation of~~ the free energy A<~~sup~~>(1)</~~sup~~>(x).		:<math><exp(-\beta \Delta U)>_{0}= \frac{\int\, exp(-\beta \Delta U) exp(-\beta \Delta U_{0}) dx} {\int\, exp(-\beta \Delta U_{0}) dx} </math>
	***The biasing potential V<~~sup~~>(i)</~~sup~~>(x) can be ~~gradually improved in this fashion~~. ~~Within each window~~, ~~we obtain a relatively uniform sampling~~. ~~This leads in general~~ to ~~small statistical errors.~~		:<math><exp(-\beta \Delta U)>_{0}= \frac{\int\, exp(-\beta \Delta U) w w^{-1} exp(-\beta \Delta U_{0}) dx} {\int\, exp(-\beta \Delta U_{0}) w w^{-1} dx}=\frac{<w^{-1} exp(-\beta \Delta U)>_{w}} {<w^{-1}>_{w}}</math>
			In the equation above, the <math><...>_{w}</math> denotes the weighted ensemble average and the probability distribution function can be shown as:
			:<math>\rho_{w}(x)=\frac{w exp(-\beta \Delta U_{0})} {\int\, w exp(-\beta \Delta U_{0}) dx}</math>
			To reflect the connection between bias potential and the weight function, we can represent the weight function as an exponential form: <math>w=exp(-\beta V)</math>, where <math>V</math> means the bias potential.
			:<math><exp(-\beta \Delta U)>_{0}=\frac{<exp(\beta V) exp(-\beta \Delta U)>_{w}} {<exp(\beta V)>_{w}}</math>
			== <span style="background-color: rgb(192, 192, 192);"> </span>Free energy calculation ==
			After defining a collective variable (CV) <math>\xi</math>, we can calculate the free energy at any point along the direction of CV by using the probability distribution function.
			(The reference point is chosen at <math>\xi_{0}</math>)
			:<math>\Delta A(\xi)=-k_{B} T ln \frac {\rho_{0}(\xi)} {\rho_{0}(\xi_{0})}</math>
			When the bias potential is added, the free energy change under the weighted ensemble can be given by:
			:<math>\Delta A_{w}(\xi)=-k_{B} T ln \frac {\rho_{w}(\xi)} {\rho_{w}(\xi_{0})}</math>
			In this case, the original free energy change should be recovered by combining these two equations above: (the denominators have been offset in the ratio)
			:<math>\Delta A(\xi)=-k_{B} T ln \frac {\rho_{w}(\xi) w(\xi_{0})} {\rho_{w}(\xi_{0}) w(\xi)}</math>
			:<math>\Delta A(\xi)=\Delta A_{w}(\xi)+k_{B} T (ln w(\xi)-ln w(\xi_{0}))=\Delta A_{w}(\xi)-[V(\xi)-V(\xi_{0})]</math>
			The best bias potential can help remove all the possible barriers on the energy landscape, which means the free energy change under this bias potential should be 0 everywhere. So, it is obvious to know that the best bias potential must counterbalance the true free energy.
			:<math>V(\xi)-V(\xi_{0})=-\Delta A(\xi)</math>
			=== Challenges of Umbrella Sampling ===
			*Pick up the best bias potential along the CVs to realize the sufficiently flat energy landscape
			*Ensure complete scanning of all the configuration space along the CVs under bias potential
			*Perform discretization of the probability distribution along CVs

	= <span style="background-color: rgb(153, 204, 255);">Umbrella Sampling </span> <ref>Torrie, G.M., Valleau, J.P., Nonphysical sampling distributions in Monte Carlo free energy estimation: umbrella sampling, J.Comput.Phys.1977,23,187-199</ref> =		= <span style="background-color: rgb(153, 204, 255);">Umbrella Sampling </span> <ref>Torrie, G.M., Valleau, J.P., Nonphysical sampling distributions in Monte Carlo free energy estimation: umbrella sampling, J.Comput.Phys.1977,23,187-199</ref> =