Tutorial on Excitation Transfer

In this part of the tutorial, you will learn about the transfer of excitation energy between pairs of bacteriochlorophyl molecules (BChls) in light harvesting complex II (LH-II). The tutorial consists of two sections. In the first section, you will use VMD to investigate the structure of LH-II in order to see how proteins and pigments are organized in the complex. Then, in the second section you will calculate the rate of excitation energy transfer between pairs of BChls in LH-II using VMD and Mathematica.

File Setup

The working directory for this tutorial is at

photo-tutorial-files/1-excitation

It contains five files and one directory.

1. lh2.pdb is a pdb file for LH-II.

2. TwoDimer.vmd is a VMD state file for two LH-II structural units.

3. EightDimer.vmd is a VMD state file for LH-II ring.

4. lh2-dipole.vmd is a VMD state file for showing direction of transition dipole moments of BChls in LH-II.

5. transfer.nb is a Mathematica file for excitation transfer rate calculation.

6. example-output is a directory containing example output files e.g. OneDimer.vmd is a VMD state file for an LH-II structural unit.

Architecture of Light Harvesting Complex II (LH-II)

In this section, you are going to view and investigate the three dimensional structure of the light harvesting complex II from Rhodospirillum (Rh.) molischianum using VMD.

Your objective in this section is to see how proteins and pigments (BChls and carotenoids) are arranged in LH-II by using VMD. It should take you about 45 minutes to complete.

LH-II from Rh. molischianum is a membrane protein complex which absorbs light and transfers its energy to a reaction center in a photosynthetic unit. It consists of eight copies of a basic structural unit (building block) containing a heterodimer of two protein subunits ($\alpha$-apoprotein and $\beta$-apoprotein), three bacteriochlorophyls (BChls named B850a, B850b and B800, according to their absorption wavelength in nm) and one carotenoid. Eight of these units are assembled into the LH-II ring as shown in Figure 1. In total, there are 16 protein segments, 24 BChls (each contains a Mg$^{2+}$ ion at its center) and 8 carotenoids.

You will start from viewing a building block of LH-II. From there, you will then examine how two heterodimers fit together in LH-II. Finally, you will investigate the complete LH-II ring which is made of eight of these heterodimers.

Figure 1: Top and side view of a light harvesting complex II from Rh. molischianum.

Building block of LH-II

In this subsection, you will examine the building block of LH-II as shown in Figure 2 and investigate how the two proteins and the four pigments are arranged.

In the pdb file of LH-II (lh2.pdb), there are eight copies of the $\alpha$-apoprotein with segment names ALP1, ALP2,.. .., ALP8 and eight copies of the $\beta$-apoprotein with segment names BET1, BET2,.. .., BET8. The corresponding eight groups of BChls have segment names BCA1, BCA2,.. .., BCA8 and each consists of three individual BChls with residue identification number (ResID) 57, 58 and 59 corresponding to BChls B850b, B800 and B850a respectively. Finally there are eight carotenoids with segment names LYC1, LYC2,.. .., LYC8.

Figure 2: Building Block of LH-II.

You will first render a heterodimer of the two proteins made of the segment ALP1 and BET1 in Cartoon representation and color the $\alpha$- and $\beta$-apoproteins in blue and purple respectively according to their segment names. You will then render the four pigments contained in the heterodimer in Licorice representation using various ways of coloring methods which allow you to color the carotenoid in yellow and color BChl B850a, B850b and B800 (resid 59,57 and 58) in red, green and pink, respectively, to show their locations relative to each other. Finally, you will render the two proteins in Surf representation and color them so that you can learn how they pack with each other and with the four pigments. Here is one way how to do this all.

1

Open VMD and load the LH-II coordinate file lh2.pdb. Open a terminal and change directory to the Excitation Transfer tutorial directory.

tbss> cd $\sim$/tbss.work/photo-tutorial-files/1-excitation

Open a VMD session and load the LH-II coordinate file lh2.pdb.

tbss> vmd lh2.pdb

2

Select the $\alpha$-apoprotein with segment name ALP1 to be displayed in the VMD OpenGL Window. In the Selected Atoms text entry of the Graphical Representation Window, delete the word all and type segname ALP1 and then press Enter.

3

Render the $\alpha$-protein in Cartoon representation and color it in blue according to its segment name. In the Draw style tab of the Graphical Representation Window, select Cartoon as Drawing Method and Segname as Coloring Method. The $\alpha$-apoprotein should be colored in blue by default.

4

Reduce the distortion of the displayed protein due to Perspective viewing mode by changing it to Orthographic. In the VMD Main Menu Window, click the third pull-down menu Display and click Orthographic in the second box. Orthographic mode renders three dimensional structure of the proteins by projecting them normally onto the screen.

5

Make a new representation and select the $\beta$-apoprotein with segment name BET1. In the Graphical Representation Window, click Create Rep to create a new representation. In the Selected Atoms text entry, delete ALP1 and type BET1 and then press Enter. Make sure that the $\beta$-apoprotein is drawn in Cartoon representation.

6

Color the $\beta$-apoprotein in purple according to its segment name. In the VMD Main Window, click the third pull down menu Graphics and select the second option Colors to open a Color Controls Window. In the first column Categories of the Color Controls Window, scroll down and click Segname. In the second column Names, click BET1. In the third column Colors, select 11 purple.

Q1: What is the main secondary structure of $\alpha$- and $\beta$-apoproteins?

7

Change the drawing method for the two proteins to Surf representation to see how both proteins are packed with each other. NOTE: Surface rendering may take a-little while.

Q2: Where do $\alpha$- and $\beta$-apoproteins make contact with each other?

8

Change the coloring method for the two proteins to ResType to see how polar and non-polar residues are distributed on the two proteins.

Q3: Can you tell that $\alpha$- or $\beta$-apoproteins are transmembrane proteins?

9

Change the Drawing Method for the two proteins to Trace representation and use Bonds labelling to measure the distance between the transmembrane helices on the $\alpha$- and $\beta$-apoproteins. To use Bonds Llabelling, click Mouse in the VMD Main Window. Then click and hold Label and select Bonds 2. The mouse arrow should change to a cross sign +. This will provide a separation distance of any two atoms you click. A quick and easy way to use this Bonds Labelling is to press the button 2 on your keyboard.

Q4: What is the approximate distance between the transmembrane helices of $\alpha$- and $\beta$-apoproteins?

Q5: Can you show that the N-terminals of both proteins are on the same side of the membrane while their C-terminals are on the other side.

You can see that $\alpha$- and $\beta$-apoproteins are transmembrane proteins which consist mainly of $\alpha$-helices. You also might have noticed that there is sufficient space in the middle of the heterodimer to accommodate 3 BChls and 1 carotenoid molecule. Now, let's have a look at how pigments are arranged in the heterodimer.

NOTE: To delete the labels of atoms and distances on the VMD OpenGL Display, Click Graphics in the VMD Main Window and select Labels. Select Atoms to see a list of atom labels. Select atom names, then click Delete. Select Bonds to see a list of distance labels. Select pairs of atom names, then click Delete.

10

Change the Drawing Method for the two proteins to Cartoon representation and color them according to SegName

11

Create a new representation and select the BChl B850a with segment name BCA1 and Residue ID 59 to be displayed. In the Graphical Representation Window, click Create Rep to create a new representation. In the Selected Atoms text entry, delete all the words and type segname BCA1 and resid 59. Press Enter.

12

Render the BChl B850a in Licorice representation and color it in red according to its Chain ID. In the Draw style tab, select Licorice as Drawing Method and Chain as Coloring Method. In the first column Categories of the Color Controls Window, scroll down and click on Chain. In the second column Names, click on X. In the third column Colors, select 1 Red.

13

Create a new representation and select the BChl B850b with segment name BCA1 and residue ID 57 to be displayed.

14

Render the BChl B850b in Licorice representation and color it in green according to its Residue Type. In the Draw style tab, select Licorice as Drawing Method and ResType as Coloring Method. In the first column Categories of the Color Controls Window, scroll down and click on ResType. In the second column Names, click on Unassigned. In the third column Colors, select 7 green.

15

Create a new representation and select the BChls B800 with segment name BCA1 and Residue ID 58 to be displayed.

16

Render the BChl B800 in Licorice representation and color it in pink according to its Residue Name. In the Draw style tab, select Licorice as Drawing Method and ResName as Coloring Method. In the first column Categories of the Color Controls Window, scroll down and click on Resname. In the second column Names, click on BCA. In the third column Colors, select 9 pink.

17

Create a new representation and select the Carotenoid with segment name LYC1 to be displayed.

18

Render the Carotenoid in Licorice representation and color it in yellow according to its Residue Name. In the Draw style tab, select Licorice as Drawing Method and ResName as Coloring Method. In the first column Categories of the Color Controls Window, scroll down and click on Resname. In the second column Names, click on LYC. In the third column Colors, select 4 yellow.

19

Create another representation and select the three magnesium ions in the segment name BCA1 to be displayed. In the Graphical Representation Window, click Create Rep to create a new representation. In the Selected Atoms text entry, delete all the words in the entry and type name MG and segname BCA1 and then press Enter.

20

Render the three magnesium ions on the BChls in VDW representation and color them in white according to atom name. In the Draw style tab, select VDW as Drawing Method and Name as Coloring Method. In the first column Categories of the Color Controls Window, scroll down and click on Name. In the second column Names, click on M. In the third column Colors select 8 white.

Your LH-II structural unit should look like what is shown in Figure 2 except that your BChls are rendered in their full chemical structure.

21

Use the Bonds Labelling to measure the distances between BChls.

Q6: What are the distances between (1) BChls B850a and B850b; (2) BChls B850a and B800; (3) BChls B850b and B800?

22

Change the Drawing Method for the two proteins to VDW representation and color them according to Residue Type.

Q7: What are the polar protein residues which make close contact with the three BChls?

You can see that the BChls B850a (red), B850b (green) and the carotenoid (yellow) are packed in the space between the $\alpha$- and $\beta$-apoproteins (blue and purple) while BChl B800 make a contact to the N-terminal helix of the $\alpha$-apoprotein. It can be seen that there are two histidine side chains: one ($\alpha$His34) contacta BChl B850a and the other ($\beta$His35) contacts BChl B850b. BChl 800 (pink) is coordinated by an aspartate ($\alpha$Asp6). A very important is how two of these building blocks assemble together. Experimentally, it has been shown that these proteins actively self-assemble into LH-II.

23

Before you proceed to the next subsection, save your work as a VMD state my-lh2-unit.vmd and delete the LH-II unit system. From the VMD Main Window, click File and then select Save State. Type my-lh2-unit.vmd and click OK. In the VMD Main Window, highlight lh2.pdb. Click Molecule and then select Delete Molecule.

Packing of two LH-II structural units

In this section, you will study two LH-II structural units from a VMD state prepared by us. You will then investigate the packing of proteins and pigments between them.

Figure 3: Packing of two LH-II structural units.

24

Load the VMD state TwoDimer.vmd of two LH-II structural units. In the VMD Main Window, click File and then select Load State to open a file selection window. Click TwoDimer.vmd and then click OK. You should see two copies of the LH-II structural unit which you rendered in the previous section (see Figure 3). All the tails and branches of the BChls have been removed to make it easier for viewing the system.

25

Change the drawing method for all the proteins to Surf representation. Take a look at the system particularly at the interface between two proteins to see how they pack.

Q1: How well do the two $\alpha$-apoproteins pack with each other? How about the packing between the two $\beta$-apoproteins?

26

Now render all the proteins in Trace representation.

Q2: What is the distance in Å (1) between two $\alpha$-apoproteins; (2) between two $\beta$-apoproteins?

27

Hide all the proteins and carotenoids.

Q3: What is the distance in Å between (1) two BChls B850a (red); (2) two BChls B850b (green); (3) BChls B850a (red, BCA1) and B850b'(green, BCA2) ; (4) B800 (pink, BCA1) and B850b'; (5) B800 and B850a' (red, BCA2)?

It can be seen that the transmembrane helices of the $\alpha$-apoproteins are packed closely while the transmembrane helices of the $\beta$-apoproteins make very little contact with each other. The termini of both proteins also make extensive contact with each other. Can you imagine how a ring of eight of these structural units looks like?

LH-II ring

In this subsection, you will investigate an LH-II as a ring of 8 heterodimers containing 24 BChls and 8 carotenoids.

28

Delete the previous system and load the VMD state EightDimer.vmd. The structure of the LH-II will be loaded as 8 copies of the LH-II structural unit. One can readily recognize that this system forms a ring. (see Figure 1)

29

Have a look at how proteins are organized in LH-II by hiding all the pigments and vice versa.

Q1: What geometrical shape do the proteins and the pigments in LH-II form? What are the advantages of having such an arrangement?

30

Draw the protein in Trace representation and hide all the pigments.

Q2: What are the diameters of the rings of $\alpha$- and $\beta$-apoproteins in LH-II?

31

Create a new representation. Make a selection of amino acids Tyrosine (TYR) and Tryptophan (TRP) on all the proteins in LH-II. Render them in VDW according to ResType. These two amino acids are usually used to identify the positions at which a membrane protein is aligned with a lipid bilayer. They are usually located at the interface between the fatty acids and the lipid head groups in the lipid bilayer.

Q3: Can you estimate the thickness of the membrane (in Å) in which the LH-II is embedded?

32

Hide all the proteins and show all the pigments. See how they are arranged relative to one another.

Q4: What are the diameters of the B850 ring and the B800 ring?

Q5: What are the distances between BChls B850a (red) and its two adjacent BChls B850b (green)? Are they equal?

An LH-II is made up of eight heterodimers. They are assembled into a circular ring with eight $\alpha$-apoproteins (blue) forming an inner ring and eight $\beta$-apoprotein (purple) forming an outer ring. Pigments are embedded within the proteins and are placed roughly in the middle of the transmembrane part of the protein away from the bulk water at the interfaces of the membrane. Sixteen BChls (B850a and B850b) and eight carotenoids are packed between the rings of $\alpha$- and $\beta$-apoproteins forming a ring of 16 BChls. Eight BChls B800 are held between $\beta$-apoproteins and supported by short helices from $\alpha$-apoproteins. They form a larger ring of 8 BChls.

In LH-II, light is absorbed by pigment molecules such as BChls and carotenoids while proteins function as a scaffolding structure that hold all the pigments in their places. In the next section you will do some calculations to see how fast excitation energy is transfered between pairs of BChls when their separation is larger than their molecular sizes.

Excitation Transfer between two Bacteriochlorophyls in LH-II

In this section, you are going to determine the rate of excitation transfer between pairs of BChls within LH-II.

Your objective in this section is to learn how the rate of excitation transfer between a pair of BChls is calculated. This should take about half an hour to complete.

To do this, you will need a few mathematical formulae that are used to calculate the excitation transfer rate. You will start by determining the orientations of the transition dipole moments of BChls in LH-II and represent them as cones using VMD. Then you will use this structural information to compute the excitation transfer rate between a pair of BChls using Mathematica.

Figure 4: BChl and its transition dipole moment

Figure 4 shows the ring part of a BChl in LH-II. Magnesium ions (Mg$^{2+}$) are shown as white circles at the center. The direction of the transition dipole moment is shown as a cyan arrow. It points from the nitrogen atom of ring I (NB) to that of ring III (ND).

Figure 5: Top and side view of 24 BChls in an LH-II from Rhodospirillum molischianum.

Figure 5 shows the top and side view of 24 BChls in LH-II forming two rings of BChls. The inner ring consists of sixteen BChls (eight BChls B850a (red) and eight BChls B850b (green) arranged alternatively). The outer ring consists of eight BChls B800 colored in pink. Direction of the transition dipole moments are represented by cones connecting two nitrogen atoms NB and ND (see Figure 5). First, you will learn how to compute an excitation transfer rate.

The rate $k_{ij}$ of excitation energy transfer from BChl i to BChl j can be determined via

$$k_{ij} = \frac {2 \pi} {\hbar} \left\vert U_{ij} \right\vert^2 J_{ij}$$

where $U_{ij}$ is the Coulomb coupling between the excited state of BChl i and the ground state of BChl j; $J_{ij}$ is an overlap integral between the emission spectrum of BChl i and the absorption spectrum of BChl j which can be determined spectroscopically. For a pair of BChls B850a $J_{ij} = 1.05 \times 10^{-3}$ cm; $\pi$ = 3.1416 and $\hbar$ = 5.31 ps cm$^{-1}$.

When the distance between the two BChls is larger than their molecular sizes, the Coulomb interaction term $U_{ij}$ can be approximated as an induced dipole-induced dipole interaction involving the transition dipole moments $\vec{d}_i$ and $\vec{d}_j$ of the two BChls whose central Mg$^{2+}$ ions are separated by a distance $r_{ij}$.

$$U_{ij} = C \frac { \hat{d}_i \cdot \hat{d}_j - 3 (\hat{d}_i \cdot \hat{r}_{ij})(\hat{d}_j \cdot \hat{r}_{ij})} {r_{ij}^3}$$

where $\hat{d}_i$, $\hat{d}_j$ and $\hat{r}_{ij}$ are unit vectors along $\vec{d}_i$, $\vec{d}_j$ and $\vec{r}_{ij}$ respectively; and C = 505,644 Å$^3$.cm$^{-1}$ is a constant.

Let's ask a more specific question. What is the excitation energy transfer rate between a pair of BChls B850a (say segment names BCA1 and BCA2)? Let's call them the first and the third BChls in the ring of 16 BChls. To be able to compute the excitation energy transfer $k_{13}$ between the two BChls, we need to determine the transition dipole moment $\vec{d}_1$ and $\vec{d}_3$ of the BChls and the relative position vector $\vec{r}_{13}$. These vectors can be determined from the positions of nitrogen atoms NB and ND and Mg$^{2+}$ ions (see Figure 4).

Orientation of BChl Transition Dipole Moments in LH-II

In this subsection, you will use VMD to measure the positions of the nitrogen atoms NB, ND and the Mg$^{2+}$ ions named MG of the BChls B850a from the segments BCA1 and BCA2 in the pdb file lh2.pdb. Their positions will be used to determine the directions of the transition dipole moments of the two BChls and the relative position vectors between them. You will then draw the transition dipole moments as cones to show how they are oriented in LH-II.

1

From the previous VMD session, hide all the proteins, the carotenoids and the Mg$^{2+}$ ions.

2

Locate the positions of nitrogen atoms NB, ND and magnesium ions MG on a pair of two BChls B850a (red) from segments BCA1 and BCA2. Read out their coordinates and copy them into a file. In the VMD Main Menu, click on the pull down menu Mouse, then click Label and hold the button and then move your mouse to select Atoms 1. The mouse arrow should change to a cross sign +. This will provide a label and coordinates of any atom you click (Coordinates will be shown in the VMD Console Window). Now use the figure of a BChl (Fig. 4) to help you identify the two nitrogen atoms NB and ND and the Mg$^{2+}$ ions on the pair of BChls B850a (red, ResID 59) in segments BCA1 and BCA2. Click on them, see the labels and the VMD console output to check that they are the atoms you are looking for. Then read out their (x, y, z) coordinates from the VMD Console Window as rNB1 = (xNB1, yNB1, zNB1), rND1 = (xND1, yND1, zND1), rMG1 = (xMG1, yMG1, zMG1) and rNB3 = (xNB3, yNB3, zNB3), rND3 = (xND3, yND3, zND3), rMG3 = (xMG3, yMG3, zMG3). Copy these numbers into a text file using nedit.

3

Set the drawing color to red.At the prompt command in the VMD Console Window type draw color red and press Enter.

4

Draw a cone pointing from atom NB to atom NA on each BChl. Read the coordinates of atoms NB and ND from your text file. Then type draw cone {xNB1 yNB1 zNB1} {xND1 yND1 zND1} resolution 20 where xNB1, yNB1, zNB1 and xND1, yND1, zND1 are the actual coordinates (e.g. (1.23, 4.56, 7.89)) of the two nitrogen atom NB and ND on the BChl 850a with segment BCA1 and resid 59. A red cone pointing from atom NB to atom ND will be drawn on the VMD OpenGL Window. Repeat the process by drawing another cone on the other BChl using the same command. Type draw cone {xNB3 yNB3 zNB3} {xND3 yND3 zND3} resolution 20 where (xNB3, yNB3, zNB3) and (xND3, yND3, zND3) are the actual coordinates (e.g. (1.23, 4.56, 7.89)) of the two nitrogen atom NB and ND on the BChl 850a with segment BCA2 and resid 59.

5

Load a VMD state lh2-dipole.vmd to view the directions of all BChl transition dipole moments as cones in LH-II. Look at how the transition dipole moments of BChls in LH-II are oriented. To save your time, we have already drawn all the rest of the BChls as cones and save it as a VMD state for you to investigate. You should see all the BChls drawn in Licorice representation with red, green and pink cones on BChls B850a, B850b and B800 respectively. (see Figure 5

Q1: How is the transition dipole moment of a BChl in the LH-II oriented relative to that of its neighboring BChls.?

It can be seen that the BChls in LH-II have eight-fold symmetry. The transition dipole moment of BChl B850a (red) points roughly opposite to its neighboring BChl B850b (green) and it points in the same direction to its adjacent BChl B800. Knowing the positions of nitrogen atoms NB and ND and Mg$^{2+}$ ions, you can start to calculate the excitation transfer rate in the next subsection.

Calculation of Excitation transfer rate

In this final part, you will use the atomic positions obtained from the previous subsection to calculate the excitation transfer rate $k_{13}$ between two nearest BChls B850a (the first and the third BChls in the ring of 16 BChls B850) using Mathematica. You will also compute the excitation transfer rate from BChl B800 to BChl B850a and the reverse transfer rate from BChl B850a to BChl B800.

First, you need to calculate the directional vectors of the BChl transition dipole moments $\vec{d}_1$, $\vec{d}_3$ and the relative position vector $\vec{r}_{13}$ (both magnitude $r_{13}$ and direction $\hat{r}_{13}$) between the two BChls from segments BCA1 and BCA2 in the LH-II coordinate file lh2.pdb. You will use these information to compute the induced dipole-induced dipole interactions term $U_{13}$ between them. Then you will compute the transfer rate $k_{13}$ between two BChls B850a. Finally, you will determine the excitation transfer rate $k_f$ from BChl B800 to BChl B850a and the reverse transfer rate $k_r$.

Figure 6: Mathematica Graphics User Interface for the file transfer.nb

6

Open a Mathematica notebook file transfer.nb. Open a terminal window and change directory to your working directory for this tutorial.

tbss> cd $\sim$/tbss.work/photo-tutorial-files$/$1-excitation

Open the mathematica file transfer.nb.

tbss> mathematica transfer.nb

This should open the mathematica file as a window shown in Figure 6.

STEP 1: Execute commands to define constants used and load a Vector Analysis package. Click anywhere within the cell covered by a blue bracket on the right which has a semi-arrow. Then hold down the Shift key and press Enter. This will execute the command lines in the cell which define some physical constants to be used in the calculation and load a package called Vector Analysis into the program.

NOTE: Please ignore "Possible spelling error" message due to the naming of variables used.

STEP 2: Calculate the directional vector $\hat{d}_1$ of the transition dipole moment $\vec{d}_1$ of the BChl B850a in segment BCA1. In the Mathematica window, replace the text {xNB1, yNB1, zNB1} and {xND1, yND1, zND1} by the actual coordinates of nitrogen atoms NB and ND of the BChl B850a in segment BCA1 read from the previous subsection in the same format e.g. {1.23, 4.56, 7.89}. Then execute the commands to calculate the directional vector $\hat{d}_1 = \frac {\vec{r}_{ND1}- \vec{r}_{NB1}} { \left\vert \vec{r}_{ND1}- \vec{r}_{NB1} \right\vert}$ by holding down the Shift key and then press Enter.

STEP 3: Calculate the directional vector $\hat{d}_3$ of the transition dipole moment $\vec{d}_3$ of the BChls B850a in segment BCA2. Repeat the previous step by replacing the text {xNB3, yNB3, zNB3} and {xND3, yND3, zND3} with the actual coordinates of nitrogen atoms NB and ND of the BChl B850a in segment BCA3 read from the previous subsection in the same format e.g. {1.23, 4.56, 7.89}. Then execute the commands to calculate the directional vector $\hat{d}_3 = \frac { \vec{r}_{ND3}- \vec{r}_{NB3} } { \left\vert \vec{r}_{ND3}- \vec{r}_{NB3} \right\vert }$ by holding down the Shift key and then press Enter.

STEP 4: Compute the separation vector $\vec{r}_{13}$ and the distance $r_{13}$ (Å) between the the central Mg$^{2+}$ ions of the two BChls B850a. (NOTE: In the mathematica script, the distance $r_{13}$ is denoted as L13). Replace the text {xMG1, yMG1, zMG1} and {xMG3, yMG3, zMG3} by the actual coordinates of the two Mg$^{2+}$ ions, named MG , on the two BChls B850a read from the previous subsection in the same format e.g. {1.23, 4.56, 7.89}. Then execute the commands to calculate the relative position vector of $\vec{r}_{13} = \vec{r}_1 - \vec{r}_3$ and the separation distance $r_{13} = \sqrt{\vec{r}_{13} . \vec{r}_{13}}$ (measured in Å) by holding down the Shift key and then press Enter.

Q1: What is the separation distance between two BChls B850a? How large is it compare with the size of a BChl? Why is this important?

STEP 5: Calculate the directional vector $\hat{r}_{13}$ of the relative position vector $\vec{r}_{13}$. Execute the commands to calculate the directional vector $\hat{r}_{13}$ of the relative position vector $\vec{r}_{13}$ i.e. $\hat{r}_{13} = \frac {\vec{r}_{13}} { r_{13} }$ by holding down the Shift key and then press Enter.

STEP 6: Calculate the induced dipole-induced dipole interaction term $U_{13}$. (cm$^{-1}$) Execute the commands to compute the induced dipole-induced dipole interaction term $U_{13}$ calculated according to the second equation above by holding down the Shift key and then press Enter.

STEP 7: Calculate the excitation transfer rate $k_{13}$ (per ps). Execute this section to compute the the excitation transfer rate $k_{13}$ according to the first equation above by holding down the Shift key and then press Enter.

Q2: What is the rate of excitation transfer between two BChls B850a?

Now let's calculate the excitation transfer rate $k_f$ from BChl B800 to Bchl B850a and the reverse rate $k_r$.

STEP 8: Calculate the forward rate $k_f$ of excitation transfer from a BChl B800 to its nearest BChl B850a. Up to here we have determined all the quantities required for the calculation of the transfer rate from the BChl B800 of segment BCA1 to its nearest BChl B850a of the same segment for you. You just need to execute the commands to calculate the rate of forward energy transfer by holding down the Shift key and then press Enter.

STEP 9: Calculate the rate $k_r$ of excitation transfer from a BChl B850a to BChl B800. Similarly in this step, we have determined all the quantities required for the calculation of the transfer rate from the BChl B850a of segment BCA1 to its nearest BChl B800 of the same segment for you. You just need to execute the commands to calculate the backward transfer rate by holding down the Shift key and then press Enter.

Q3: Compare the forward excitation transfer rate $k_f$ with the reverse transfer rate $k_r$. Which direction of excitation transfer is more favorable?

You can see that the excitation transfer rate from BChl B800 to BChl B850a is six orders of magnitude higher than that of the reverse direction. So it is much more favorable for excitation energy to be transferred from a ring of 8 BChls B800 to a ring of 16 BChls B850a within an LH-II. The factor that directly affects the forward and reverse rates is the overlap integral $J_{ij}$ which depends on the energies of the emission peak $E_i$ and absorption peak $E_j$. When $E_i$ is more than $E_j$ you get large value of overlap integral and vice versa.