About this App (version 1.0)

The C_eff Calculator is a platform-independent browser-based interface that facilitates the analysis of flexible linker properties by applying the worm like chain model (WLC model). The application allows the analysis of effective concentrations enforced by linkers and of linker dimensions represented as the probability distribution of end-to-end distances. The application requires the user to define the system specific parameters and allows an easy access to important parameters such as the optimal linker length for maximal enhancement of avidity and the effective concentration at a specific linker length. These features help in protein design for biotechnology applications and also in the analysis of biological systems.

The Ceff Calculator was developed by the @ChemesLab. The application is based on the Shiny package and it was coded by Juliana Glavina, PhD, with server implementation by Cesar Leonetti.

Please, report problems, questions and feedback on the app to support@chemeslab.org.

The use of the Ceff Calculator is described in detail in Kjaergaard M, Glavina J and Chemes LB. Predicting the effect of disordered linkers on effective concentrations and avidity with the “Ceff calculator” app (see References tab). This page briefly outlines how to use this app and the theory behind the calculations. It's organized in the following main sections:

• Before you start
• C_eff Plot Tab
  
• P(r) Plot Tab
  
• The theory behind the app: The Worm-Like Chain model (WLC) for flexible polymers
  

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Before you start

The first step towards calculating effective concentrations is to determine the length of the linker and the spacing between binding sites in your system. Here, we briefly outline how to identify these two parameters. For further details, see Kjaergaard M et al. 2020 in the References tab.

Mapping linker boundaries

The mapping of linker boundaries depends on what the linker is connected to. For folded domains with a known 3D structure the linker starts at the first or last residue that is not resolved in the structure. If there is no structure, you can predict conserved domains using CDD or PFAM in combination with a disorder predictor such as MobiDB, which collates the consensus of different predictors, or a single predictor such as MFDp2 to define linker boundaries. If the linker tethers a short linear motif, the linker can usually be assumed to start after the last residue of the motif, or end the first residue before the start of the motif.

Measuring the distance between sites

Intramolecular and multivalent interactions are also modulated by the distance separating the attachment sites of the linker, named r_o in the model. For intramolecular binding, the distance between contact sites is that between the linker attachment site and the binding site. In a bivalent interaction, the relevant distance is the distance between binding sites. If the 3D structure of the complex is available from the Protein Data Bank, you can use a program for visualization and analysis of protein structures, such as Chimera and PyMol and measure the distance between contact sites.

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C_eff Plot Tab

The C_eff Plot Tab (FIG. 1) allows the visualization of effective concentration (C_eff) as a function of linker length for a thethered system.

The Sidebar Panel

The sidebar panel on the left has two main sections, the first one allows the input of basic parameters and download of the data and the second one allows the customization of the X axis.

1. Basic parameters

In this section it is possible to input the parameters that define the system under study. We separate the specific system parameters from those that are generic to the WLC model.

a. Specific system parameters

To calculate the effective concentration enforced by a specific linker on a tethered system, you will need to determine two specific system parameters: the Linker length (N_res) and the separation between binding sites (r_o) as explained in the Before you start section. This section (FIG. 2) allows the input of these values:

• Distance between contact sites (r_o): you should complete this field with the measured distance between two binding sites or between a binding site and the site of tethering
• Specific Linker length (N_res): you should complete this field with your linker’s length for the calculation of a specific C_eff.

Both values can be typed, or modified by steps of 0.1 using the arrows that appear on the right side of the input field when placing the pointer over the field. No plot is produced if negative values are used.

b. WLC parameters

You will also have to choose the WLC parameters. This section (FIG. 3) allows the input of the persistence length (L_p) and the size of one amino acid (b). The default values are set to the standard parameters that work for most protein linkers. Please use reference values here except for advanced use. For more detail on advanced use see Kjaergaard M et al. 2020 (see References tab).

• L_p or persistence length: This value is set to a reference value of 4 Å and will work well for most protein linkers. Only under special circumstances you may choose to change this value.

  Changing the L_p value: Most natural linkers can be well described by using persistence lengths of 3 Å or 4 Å, and we recommend the use of these parameters unless you have a good reason to change it. Some good reasons might be if the linker has a very high proline content (>20%), in which case higher values (~ 9-12 Å) can be tried. If the linker is highly flexible, L_p = 3 Å might be chosen.

• b or size of one amino acid: This value is set to a reference value of 3.8 Å and will work well for all protein linkers. This parameter is only to be changed when modeling non-peptide linkers.

  Changing the b value: The value of b = 3.8 Å represents the average bond length of one amino acid, and should not be changed while working with protein linkers. It might be changed, only when working with other types of polymers.

Both values can be typed or modified by steps of 0.1 using the arrows that appear on the right side of the input field when placing the pointer over the field. No plot is produced if negative values are used.

At the bottom of this section the Download button allows to download the C_eff plot values as a tsv file (tab separated values) to plot with the desired software.

2. Customization

This section (FIG. 4) allows the input of custom limits for the x axis. The minimum and maximum values can be typed or modified by steps of 0.1 using the arrows that appear on the right side of the input field when placing the pointer over the field.

The Plot Panel

The plot panel (FIG. 5) shows the effective concentration as a function of linker length for a given spacing between linker attachment sites (r_o) provided in the Specific system parameters section using the persistence length (L_p) and the size of one amino acid (b) from the WLC parameters sections.

The bar at the top right corner of the plot tab offers different functionalities:

1. Download the plot

The camera button allows to download the plot as a png file.

2. Zoom and Pan buttons

The zoom button allows to zoom in on a region of the graph by clicking and holding the mouse and moving across the region. To zoom out, double-click anywhere on the plot.

The pan button allows to explore the graph by clicking and holding the mouse and dragging the graph. To return to the original view, double click anywhere on the plot.

The + and - buttons allow to zoom in and out.

3. Autoscale and Reset Axes

Once the default view is change, the Autoscale button zooms to a setting optimized to include a wide range of data. To zoom out, double-click anywhere on the plot.

The reset axes button zooms the plot back to a setting that is optimized to show the viewable data.

It is also possible to pan along axes by using the double arrow that appears over the axes and by clicking and holding the mouse and dragging the axis. The axis can be zoom out by using the arrow-to-top/bottom or arrow-to-left/right that appears over the y or x axis by clicking and holding the mouse. Last, axis can also be autoscale by double-clicking on the axis.

4. Toggle and Hover

The toggle spikes button creates two dotted lines that extend the dot over the curve to both axis.

The next two buttons are hovering options. The first button ‘Show closest data on hover’ is selected at all times displays the data for just the one point under the cursor. The second button ‘Compare data on hover’ will show the data for all points with the same x-value (just one value in this case).

The Output Panel

The output panel (FIG. 6) displays the following information based on the parameters defined in the Basic parameters section (see System specific parameters and WLC parameters):

• The Optimal linker length: this is the linker length that produces an optimal affinity enhancement for a given separation between linker attachment sites. This is often desired in biotechnological applications involving linker design or when evaluating whether natural systems are evolutionarily optimized for tight binding.
• The maximal C_eff for optimal linker: This is the C_eff value for the optimal linker length (maximum of the C_eff plot).
• The Results of C_eff calculation for a specific linker length: As many users will focus on a specific system with a defined linker length, the Ceff calculator takes a linker length parameter input in the System specific parameters section and calculates the effective concentration for this linker.

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P(r) Plot Tab

The P(r) Plot tab calculates the distance probability distribution p(r) for a linker of defined length (FIG. 7).

The Sidebar Panel

As in the Ceff plot tab, the sidebar panel on the left has two sections:

1. Basic parameters

In this section it is possible to modify the parameters that define the system. We separate the specific system parameters from those that are generic to the WLC model.

a. Specific system parameters

This section (FIG. 8) allows the input of the linker length (N_res) for plotting the probability distribution of the end-to-end distance of the linker, (p(r)).

This value can be typed or modified by steps of 0.1 using the arrows that appear on the right side of the input field when placing the pointer over the field.

2. WLC parameters

This section is equivalent to the WLC parameters section in the C_eff Plot Tab. The default values are set to standard parameters that work for most protein linkers. The values can be typed or modified by steps of 0.1 using the arrows that appear on the right side of the input field when placing the pointer over the field.

At the bottom of this section the Download button allows to download the P(r) plot values as a tsv file (tab separated values) to pĺot with the desired software.

3. Customization

This section is equivalent to the Customization section in the C_eff Plot Tab. The minimum and maximum values can be typed or modified by steps of 0.1 using the arrows that appear on the right side of the input field when placing the pointer over the field.

The Plot Panel

The plot panel (FIG. 9) shows the probability distribution of the end to end distance, (p(r)) for a linker of defined length defined in the System specific parameters section and the persistence length (L_p) and the size of one amino acid (b) from the WLC parameters section.

The bar at the top right corner of the plot tab offers the same functionalities describe above.

The Output Panel

The output panel (FIG. 10) displays for the system defined in the Specific system parameters and WLC parameters sections:

• The End to end distance (r) with maximal probability: This is the maximum value of the p(r) function"
• The maximal end to end distance: This is the maximal extension that the linker can attain, which is equivalent to its contour length (L_c = b * N_res) calculated as the number of residues multiplied by the size of one aminoacid.

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The theory behind the app: The Worm-Like Chain model (WLC) for flexible polymers

The WLC model describes a linker as a semi-flexible polymer represented as a continuous cylinder with a fixed but randomly directed radius of curvature. The polymer conformation is defined by two variables: the contour length (L_c) and the persistence length (L_p). Here, we present a brief description of the WLC model parameters and functions. References to the WLC model can be found in the References tab.

P(r) calculation from the WLC model

The WLC model provides a mathematical expression for the probability density function p(r), which describes the distribution of end-to-end distances of a polymer. The p(r) can be expressed as:

$$p( r ) = 4 \pi r^2 \left(\dfrac{3}{4\pi L_p L_c}\right)^\frac{3}{2}exp \left(\dfrac{-3r^2}{4 L_p L_c} \right) \zeta \left(r,L_p,L_c\right)$$

Where:

$$\zeta \left(r,L_p,L_c\right) = 1 - \left\{\dfrac{5L_p}{4L_c}-\dfrac{2r^2}{L_c^2}+\dfrac{33r^4}{80 L_p L_c^3}+\dfrac{79 L_p^2}{160 L_c^2}+\dfrac{329 r^2 L_p}{120 L_c^3}-\dfrac{6799r^4}{1600L_c^4}+\dfrac{3441 r^6}{2800 L_p L_c^5}-\dfrac{1089 r^8}{12800 L_p^2 L_c^6}\right\}$$

• L_p is the persistence length and it represents the distance across which the direction of the chain becomes uncorrelated, providing a measurement of the stiffness of the chain.
• L_c is the contour length defined as the length of a fully extended chain (L_c = b * N_res)
• N_res is the number of amino acids in the linker
• b is the average size of an individual amino acid (3.8 Å)

For a linker of defined length, the end to end distances show a bell-shaped distribution that peaks at intermediate extension and is unlikely to be very compact or fully extended (FIG. 11A). On the other hand, increasing the stiffness of a linker of fixed length leads to a more extended ensemble (FIG. 11B).

The Effective Concentration C_eff

The effective concentration, C_eff, represents the concentration of one linker end at a fixed distance from the site of tethering. Once the probability density function p(r) for a linker is known, the effective concentration enforced by the linker when it is restrained to a distance r_o between binding sites is given by:

$$C_{eff} = \dfrac{p(r_{o})}{4\pi r_o^2} \dfrac{10^{27} \unicode{x212B}^3 l^{-1}}{N_A}$$

Where:

• C_eff is the effective concentration (expressed in molar units).
• p(r_o) is the probability density distribution for the linker
• r_o is the distance between binding sites defined for each system
• N_A is Avogadro's constant

For a fixed distance, we can model how C_eff depends on linker length. The C_eff plot shows that C_eff gradually increases with linker length up to a maximal value, and then decreases monotonically but slowly as linker length increases for longer linkers. FIG. 12B and FIG. 12C show how the C_eff value depends on linker length. The optimal C_eff value is reached when the linker length is such that the linker dimensions (average end-to-end distance) place the end of the linker at the secondary binding site. For very short linkers, the extension will rarely reach the secondary binding site, and for longer linkers, a larger space is explored away from the fixed distance, decreasing the C_eff value (right side of the plot in FIG. 12B).

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References

If you use this app please Cite us!

• Kjaergaard M*, Glavina J, Chemes LB*. Predicting the effect of disordered linkers on effective concentrations and avidity with the “Ceff calculator” app. Methods in Enzymology. 2020.
  Submitted

References

If you use this app please Cite us!

• Kjaergaard M*, Glavina J, Chemes LB*. Predicting the effect of disordered linkers on effective concentrations and avidity with the “Ceff calculator” app. Methods in Enzymology. 2020.
  Submitted

Worm Like Chain (WLC) Model

• Zhou HX. Polymer Models of Protein Stability, Folding, and Interactions. Biochemistry. 2004 Mar 2;43(8):2141-54.
  PubMed. DOI: 10.1021/bi036269n

• Zhou HX. Quantitative Account of the Enhanced Affinity of Two Linked scFvs Specific for Different Epitopes on the Same Antigen. J Mol Biol. 2003 May 23;329(1):1-8.
  PubMed. DOI: 10.1016/s0022-2836(03)00372-3

• Zhou, HX. The affinity-enhancing roles of flexible linkers in two-domain DNA-binding proteins. Biochemistry, 2001. 40(50): p. 15069-73.
  PubMed. DOI: 10.1021/bi015795g

• Rubinstein M and Colby RH. Polymer physics. Oxford University Press.2003.
  ISBN: 978-0198520597