Disordered regions and intrinsically disordered proteins

TOPO can treat part of a chain as an intrinsically disordered region (IDR) or an entire chain as an intrinsically disordered protein (IDP). An IDR is declared in the optional disordered: section of the same domain-definition YAML file supplied through domain_def. No separate file, argument, or INI key is required.

This page explains:

  • how to declare an IDR;

  • what changes in the coarse-grained model;

  • why TOPO uses an Ashbaugh–Hatch 12–6 interaction for IDR-involving pairs;

  • how idr_scale and eps_ev_kj affect the ensemble;

  • how IDRs affect native-contact analysis, stability optimization, and continuous synthesis.

Note

Scope. This treatment applies only to TOPO’s Cα model. It is implemented by topo.utils.nonbonded.apply_disorder() at the end of the nonbonded build. The same disordered: declaration is used automatically by isolated-protein simulations, native-contact (Q) analysis, the nscale optimizer, and continuous synthesis (CSP).

Quick start

Add a disordered: block to the domain-definition file:

n_residues: 283

intra_domains:
  A: {residues: [25-149], nscale: 1.0}
  B: {residues: [150-283], nscale: 1.0}
inter_domains:
  A-B: 0.5

disordered:
  residues: [1-24, 150-165]
  idr_scale: 0.10
  eps_ev_kj: 0.8368

The defaults are:

idr_scale: 0.10
eps_ev_kj: 0.8368

Use these defaults unless you have experimental information that supports recalibration for your system.

Declaring a residue disordered has one central consequence:

Its native Gō contacts are removed and replaced by weak, sequence-dependent, non-native interactions, while excluded volume and the transferable backbone remain active.

Bonds, angles, dihedrals, and electrostatics are not changed.

Physical picture

An IDR model must represent two distinct physical properties:

  1. Excluded volume: every residue has a finite size, so two beads cannot occupy the same space.

  2. Residue-dependent attraction: different residue pairs can have different weak tendencies to associate.

A useful analogy is a solid ball with an adjustable sticky surface. The hardness and size of the ball determine whether particles can overlap; the surface stickiness determines how strongly they attract after approaching one another.

For IDRs, these properties should be tunable independently. A polar pair may be only weakly attractive, but its beads must not become smaller or easier to overlap. Conversely, strengthening hydrophobic attraction should not simultaneously inflate the repulsive core.

This separation is particularly important for IDPs. Unlike a folded protein, an IDP has no single native structure stabilized by a fixed contact network. Its dimensions emerge from a delicate balance among:

  • excluded volume;

  • weak residue–residue attraction;

  • electrostatic attraction and repulsion;

  • backbone conformational entropy;

  • solvent-mediated effects.

Small changes in that balance can move an ensemble from expanded to compact.

What marking a region as disordered changes

TOPO already uses a transferable, non-Gō local backbone for every residue:

  • 3.81 Å bonds;

  • a double-well transferable angle potential;

  • the Karanicolas transferable dihedral potential.

Folded structure is encoded through native Gō contacts. Marking a region disordered therefore removes every native contact involving that region and replaces it with a weak interaction that does not encode a particular fold.

Every nonlocal residue pair belongs to one of three classes:

Pair class

Nonbonded potential

Native contacts

Well depth

Well position

Folded–folded

Gō 12–10–6

Retained

Existing native-contact energy, or the non-native floor

Native Cα distance, or the Karanicolas–Brooks sum rule

IDR–IDR

Ashbaugh–Hatch 12–6

Removed

max(εNN, sIDR εBT(i,j))

Sum of the two residue radii

Folded–IDR

Ashbaugh–Hatch 12–6

Removed

Same rule as IDR–IDR

Sum of the folded and IDR residue radii

The two potentials act on disjoint OpenMM interaction groups:

  • the 12–10–6 force evaluates {folded} × {folded};

  • the Ashbaugh–Hatch force evaluates {idr} × {idr} and {idr} × {folded}.

Consequently, each pair is evaluated exactly once. This is important because OpenMM takes the union of a force’s interaction groups. If the same pair were admitted to both forces, OpenMM would add both potentials without raising an error.

How IDR-involving pairs interact: the current AH–LJ form

Define the dimensionless Lennard–Jones shape

\[L(r)=4\left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right],\]

with

\[\sigma=2^{-1/6}R_{ij}.\]

The minimum of \(L(r)\) is therefore at \(r=R_{ij}\), where \(L(R_{ij})=-1\). TOPO evaluates every IDR–IDR and folded–IDR pair with

\[\begin{split}U_{ij}(r)= \begin{cases} \varepsilon_{\mathrm{EV}}L(r) +\left(\varepsilon_{\mathrm{EV}}-\varepsilon_{ij}\right), & r\le R_{ij},\\[4pt] \varepsilon_{ij}L(r), & r>R_{ij}. \end{cases}\end{split}\]

The three quantities have separate roles:

  • \(R_{ij}\) sets the geometric length scale and the minimum-energy separation;

  • \(\varepsilon_{\mathrm{EV}}\) (eps_ev_kj) controls the energetic hardness of the repulsive core—that is, the energy penalty for bead overlap;

  • \(\varepsilon_{ij}\) controls the residue-pair-specific attractive well depth, or bead “stickiness.”

The additive term \(\left(\varepsilon_{\mathrm{EV}}-\varepsilon_{ij}\right)\) in the repulsive branch makes the potential continuous at \(R_{ij}\). Because \(L(R_{ij})=-1\),

\[\begin{split}\begin{aligned} U_{ij}(R_{ij}^{-}) &=-\varepsilon_{\mathrm{EV}} +\left(\varepsilon_{\mathrm{EV}}-\varepsilon_{ij}\right) =-\varepsilon_{ij},\\ U_{ij}(R_{ij}^{+}) &=-\varepsilon_{ij}. \end{aligned}\end{split}\]

Without this additive term, the two branches would meet at \(-\varepsilon_{\mathrm{EV}}\) and \(-\varepsilon_{ij}\) and would generally be discontinuous. The force is also continuous: \(R_{ij}\) is the minimum of \(L(r)\), so \(\mathrm{d}L/\mathrm{d}r=0\) there, and the additive constant has zero derivative.

This construction largely decouples bead size and hardness from bead attraction. \(R_{ij}\) sets where the core is located, \(\varepsilon_{\mathrm{EV}}\) sets how energetically difficult it is to push two beads into that core, and \(\varepsilon_{ij}\) sets how strongly the beads attract outside the minimum. Thus, residue-pair attraction can be tuned without using the same parameter to rescale the repulsive wall. The decoupling is not mathematically perfect if bead size is defined through an energy-dependent effective-core criterion, but it is the main practical advantage of the AH split.

Thus, marking a residue as disordered gives every nonlocal pair involving that residue a finite excluded-volume core and a sequence-dependent attractive well, without encoding a native contact.

Why Ashbaugh–Hatch LJ is appropriate for IDRs

The AH construction splits the potential at its minimum. The repulsive branch uses \(\varepsilon_{\mathrm{EV}}\), whereas the attractive branch uses the pair-specific \(\varepsilon_{ij}\). Consequently, TOPO can change residue stickiness without using the same parameter to rescale the repulsive wall.

In conventional Ashbaugh–Hatch notation, the pair stickiness can be written as

\[\lambda_{ij}=\frac{\varepsilon_{ij}} {\varepsilon_{\mathrm{EV}}}.\]

Two useful limits follow:

  • If \(\varepsilon_{ij}=0\), the mathematical expression becomes a WCA-like repulsive core with zero energy for \(r>R_{ij}\).

  • If \(\varepsilon_{ij}=\varepsilon_{\mathrm{EV}}\), it becomes ordinary LJ 12–6.

In the actual TOPO parameterization, the non-native energy floor means that idr_scale: 0 leaves a very small residual well rather than producing the exact \(\varepsilon_{ij}=0\) mathematical limit. It is therefore best described as an approximately self-avoiding or WCA-like reference, not a strictly pure WCA chain.

Under this split, varying \(\varepsilon_{ij}\) from 0 to 2 kJ/mol moves the \(U(r)=k_BT\) core only from approximately \(0.846R\) to \(0.819R\). The attraction changes strongly while the effective bead size changes by only about 3.2%.

Why IDR pairs do not use TOPO’s Gō 12–10–6 potential

TOPO’s folded-contact potential is

\[U_{12-10-6}(r) = \varepsilon\left[ 13\left(\frac{R}{r}\right)^{12} -18\left(\frac{R}{r}\right)^{10} +4\left(\frac{R}{r}\right)^6 \right].\]

It has a minimum of \(-\varepsilon\) at \(r=R\). This form remains appropriate for TOPO’s folded native-contact model, but it has two undesirable properties when applied to every nonlocal IDR pair.

1. The repulsive core and attractive well are coupled

The same parameter \(\varepsilon\) multiplies the entire potential. Increasing \(\varepsilon\) therefore changes both:

  • the attractive well;

  • the energetic repulsive wall.

Thus, the model cannot request “stronger attraction with the same excluded volume.” A parameter intended to describe residue stickiness would also alter the distance at which the repulsive energy becomes comparable with \(k_BT\). This makes it difficult to interpret that parameter as a clean solvent-quality control: an observed change in chain dimensions could arise from stronger attraction, altered excluded volume, or both.

2. This particular 12–10–6 form has an outer repulsive barrier

The coefficients \((13,-18,+4)\) produce more than a contact minimum. They also produce a positive bump outside the minimum:

  • the potential crosses zero near \(1.25R\);

  • it reaches a local maximum near \(1.45R\);

  • the maximum is approximately \(+0.143\varepsilon\).

A pair approaching from large separation must cross this bump before entering the contact well.

Such a barrier can be useful as an effective description within a native-contact model. However, it is not automatically justified as the same isotropic barrier for every pair involving an IDR residue. IDR residues still reorganize and displace solvent when they associate; the narrower claim is that they do not possess the predefined native-contact event for which TOPO’s Gō barrier was parameterized.

The barrier is also important thermodynamically because it occupies a larger shell of space than the short-range contact well. It can therefore strongly affect the net balance between attraction and repulsion, as described below using the second virial coefficient.

The name “12–10–6” does not itself guarantee a barrier.

The barrier arises from the particular signs and coefficients used by TOPO. Other potentials described as 12–10–6 may have different shapes.

Why ordinary Lennard–Jones 12–6 is not sufficient

The ordinary Lennard–Jones potential is

\[U_{\mathrm{LJ}}(r) = 4\varepsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right].\]

It has no outer barrier. Its minimum occurs at

\[r_{\min}=2^{1/6}\sigma.\]

Replacing the 12–10–6 potential with ordinary LJ would remove the barrier, but it would not solve the coupling problem. The same \(\varepsilon\) scales both the repulsive term, \(4\varepsilon(\sigma/r)^{12}\), and the attractive term, \(-4\varepsilon(\sigma/r)^6\). Therefore, making a residue pair more attractive by increasing \(\varepsilon\) simultaneously makes close bead overlap more energetically costly. Although \(\sigma\) and the zero-crossing distance remain fixed, the distance at which the repulsive energy reaches a thermal scale such as \(k_BT\) changes. Ordinary LJ therefore does not provide an independent “stickiness” parameter at fixed energetic core hardness.

AH-LJ supplies this missing control by using two energy scales. The inner branch is governed primarily by \(\varepsilon_{\mathrm{EV}}\), whereas the attractive well depth is \(\varepsilon_{ij}\). Consequently, the residue-specific \(\varepsilon_{ij}\) values can tune solvent quality and sequence-dependent attraction without proportionally rescaling the repulsive wall. Conversely, \(\varepsilon_{\mathrm{EV}}\) can be chosen to prevent excessive bead overlap without forcing every residue pair to have the same attraction strength. This separation is especially useful for IDRs because their sequence-dependent interactions must vary among residue pairs while their beads should retain a comparable excluded-volume core.

The separation is best described as practical or approximate decoupling: \(R_{ij}\) fixes the geometric contact scale, but an energy-defined effective core can still move slightly when \(\varepsilon_{ij}\) changes. As shown above, that movement is small for the parameter range used here.

The progression is therefore:

TOPO Gō 12–10–6
    ├── couples the repulsive core to the well depth
    └── contains an outer repulsive barrier
                    │
                    │ remove the barrier
                    ▼
Standard LJ 12–6
    └── still couples the repulsive core to the well depth
                    │
                    │ separate the two energy scales
                    ▼
Ashbaugh–Hatch LJ

Thermodynamic interpretation: the second virial coefficient

Well depth alone does not tell us whether a potential is effectively attractive or repulsive. The width and position of every attractive or repulsive region also matter. The second virial coefficient, \(B_2\), summarizes this net pairwise effect.

At low concentration, the osmotic pressure can be expanded as

\[\frac{\Pi}{k_BT}=\rho+B_2\rho^2+B_3\rho^3+\cdots,\]

where \(\rho\) is the particle number density. An ideal gas has only the first term. The \(B_2\rho^2\) term is the leading correction caused by interactions between pairs.

Begin with the non-interacting reference

For completely non-interacting particles,

\[U(r)=0 \qquad\text{at every }r.\]

Their osmotic pressure is exactly the ideal-gas result,

\[\frac{\Pi}{k_BT}=\rho,\]

and there is no pair-interaction correction. Therefore,

\[B_2=0.\]

This can also be seen directly from the integral below. If \(U(r)=0\), then \(e^{-U(r)/(k_BT)}=1\), so the integrand is zero at every distance.

For a spherically symmetric pair potential,

\[B_2 = 2\pi\int_0^\infty \left[1-e^{-U(r)/(k_BT)}\right]r^2\,dr.\]

The sign follows directly from the integrand:

  • If \(U(r)>0\), the region is repulsive and contributes positively to \(B_2\).

  • If \(U(r)<0\), the region is attractive and contributes negatively to \(B_2\).

Therefore:

Microscopic situation

\(B_2\)

Solvent regime

Net pair behavior

Expected polymer tendency

\(U(r)=0\) everywhere

Exactly zero

Non-interacting ideal reference

No pair interaction

Ideal random-walk chain, \(\nu=1/2\)

Repulsion dominates

Positive

Good solvent

Effective excluded volume

Expanded, self-avoiding chain, \(\nu\approx0.588\)

Repulsion and attraction cancel

Approximately zero

Theta solvent

Net pair interaction cancels

Ideal-like large-scale dimensions, \(\nu\approx1/2\)

Attraction dominates

Negative

Poor solvent

Net attraction

Compact chain; approaching \(\nu\approx1/3\) in the dense-globule limit

The non-interacting and theta cases both have \(B_2=0\), but they are not microscopically identical:

  • In a non-interacting system, \(U(r)=0\) and every distance contributes zero.

  • At the theta condition, repulsive regions contribute positively and attractive regions contribute negatively, but their total contributions cancel.

The theta chain therefore looks ideal at sufficiently large length scales even though its residues still interact locally.

The factor \(r^2\) counts the volume of the spherical shell at distance \(r\). Consequently, a modest repulsive barrier at a relatively large distance may contribute strongly. Likewise, two potentials with the same minimum depth need not have the same \(B_2\): a broad shallow attractive well can contribute more than a narrow deep well.

For TOPO’s current AH 12–6 parameterization, the attractive energy in this calculation is the already defined pair well depth \(\varepsilon_{ij}\):

\[\varepsilon_{ij} = \max\left( \varepsilon_{\mathrm{NN}}, s_{\mathrm{IDR}}\varepsilon_{\mathrm{BT}}(i,j) \right).\]

It is the depth of the AH minimum, \(U_{ij}(R_{ij})=-\varepsilon_{ij}\); it is not an additional model parameter.

The following values were calculated for actual residue pairs from bt_potential.csv, using idr_scale = 0.10, \(\varepsilon_{\mathrm{EV}}=0.8368\) kJ/mol, and \(T=300\) K. The pair distance is \(R_{ij}=R_{\min/2,i}+R_{\min/2,j}\), and the integral includes TOPO’s switching region from 1.8 to 2.0 nm. These values describe the AH non-electrostatic term only; Debye–Hückel electrostatics are not included.

System or pair

Pair well depth \(\varepsilon_{ij}\) (kJ/mol)

\(R_{ij}\) (nm)

\(B_2\) (nm³)

Interpretation

Non-interacting reference, \(U(r)=0\)

—

—

0

No pair interaction

CYS–CYS

0.81170

0.60613

−0.01890

Slight net attraction

PHE–MET

0.62342

0.69593

+0.09878

Repulsion still dominates

TRP–MET

0.64434

0.72399

+0.09907

Repulsion still dominates

LEU–ASP

0.00837

0.65664

+0.38327

Very weak attraction; excluded volume dominates

ARG–LYS

0.04184

0.72399

+0.49337

Largest positive non-electrostatic \(B_2\)

CYS–CYS is the strongest pair because its raw BT value is −1.34 kcal/mol:

\[\varepsilon_{\mathrm{BT}}(\mathrm{CYS,CYS}) =4.184\left|-1.34-0.6\right| =8.11696\ \mathrm{kJ/mol},\]

and therefore

\[\varepsilon_{\mathrm{CYS,CYS}} =0.10\times8.11696 =0.811696\ \mathrm{kJ/mol}.\]

Note

Pairwise behavior at the current default parameters.

With \(\varepsilon_{ij}=\max(\varepsilon_{\mathrm{NN}}, 0.10\,\varepsilon_{\mathrm{BT}}(i,j))\) and \(\varepsilon_{\mathrm{EV}}=0.8368\) kJ/mol, 209 of the 210 unique amino-acid pair types have positive second virial coefficients. Their values range from \(+0.09878\) nm³ for PHE–MET to \(+0.49337\) nm³ for ARG–LYS. CYS–CYS is the only negative pair, with \(B_2=-0.01890\) nm³.

Thus, the default does not make every pair attractive. Most pairs remain excluded-volume dominated, while the strongest pair is only slightly net attractive. The overall behavior of an IDP still depends on the frequencies and sequence arrangement of all pair types, electrostatics, bonded terms, and collective conformational sampling.

As the AH attraction increases, \(B_2\) decreases. The interaction therefore moves in the expected thermodynamic direction: from excluded-volume-dominated behavior, through the theta condition, and then toward net attraction. The simulated chain dimensions follow the same trend: \(\nu\) decreases from 0.637 at idr_scale: 0 to 0.276 at idr_scale: 0.30, passing through experimentally relevant IDP dimensions before reaching the collapsed regime.

Sequence-dependent well depth

Every nonlocal pair touching a disordered residue receives

\[\varepsilon_{ij}^{\mathrm{IDR}} = \max\left( \varepsilon_{\mathrm{NN}}, s_{\mathrm{IDR}}\varepsilon_{\mathrm{BT}}(i,j) \right),\]

where:

  • \(s_{\mathrm{IDR}}\) is idr_scale;

  • \(\varepsilon_{\mathrm{NN}}\) is the very small non-native floor;

  • \(\varepsilon_{\mathrm{BT}}(i,j)\) is the sidechain–sidechain BT energy for residue types \(i\) and \(j\).

The BT energy is

\[\varepsilon_{\mathrm{BT}}(i,j) = 4.184\left|\operatorname{raw}(i,j)-0.6\right| \quad\text{[kJ/mol]}.\]

This is the value returned by topo.utils.nonbonded.get_ss_interaction_energy: the raw bt_potential.csv value is shifted by the 0.6 kcal/mol reference, converted to an absolute magnitude, and converted from kcal/mol to kJ/mol.

The interaction is:

  • nonspecific in coverage, because it acts on every eligible nonlocal pair rather than only native contacts;

  • chemically heterogeneous in depth, because the BT energy depends on the two residue types.

Thus, the model can favor hydrophobic contacts more strongly without assigning the IDR a predetermined fold.

Folded–IDR cross interactions

The same depth rule is used for IDR–IDR and folded–IDR pairs. A disordered tail can therefore form transient, sequence-weighted contacts with its folded domain and may adsorb onto its surface.

This is a modeling choice rather than a result established by the calibration. The default idr_scale = 0.10 was fitted using fully disordered proteins, for which every relevant pair was IDR–IDR. Folded–IDR pairs were not represented in that benchmark. Excessive adsorption of an IDR onto its folded domain should therefore be treated as a model-sensitivity question.

No nscale factor is applied

The IDR well depth uses the unscaled BT energy matrix. It does not inherit a folded domain’s nscale because nscale is a folding-stability parameter and an IDR has no native fold to stabilize.

At idr_scale: 1.0, the IDR attraction equals the unscaled sidechain–sidechain BT energy for the same residue-type pair. The mean BT energy is approximately 2.36 kJ/mol, which is well beyond the AH theta point. The calibrated default idr_scale: 0.10 is therefore approximately 10% of an unscaled native sidechain-contact energy.

Pair distance and radius convention

For an IDR-involving pair,

\[R_{ij}=R_{\min/2,i}+R_{\min/2,j}.\]

An IDR residue uses the transferable amino-acid-specific \(R_{\min}/2\) from the parameter table. A folded residue keeps its structure-derived Karanicolas–Brooks \(R_{\min}/2\), including when it interacts with an IDR residue.

The radius is an \(R_{\min}/2\) value, not a \(\sigma/2\) value. TOPO performs the conversion inside the AH force:

\[\sigma_{ij}=2^{-1/6}R_{ij}.\]

No additional \(2^{1/6}\) conversion should be applied when populating the per-residue radius array.

Overriding the per-residue radius, rather than only the pair matrix, also keeps the nascent-chain–nascent-chain and nascent-chain–ribosome excluded-volume channels consistent during continuous synthesis.

Configuration reference

All three top-level sections—intra_domains, inter_domains, and disordered—are optional. Only n_residues is required.

Key

Required?

Type and default

Meaning

disordered

No

Mapping; absent

Enables the IDR treatment when present.

disordered.residues

Yes, if disordered is present

List

Residues to treat as disordered. Native contacts involving these residues are removed.

disordered.idr_scale

No

Float; 0.10

Scales the sequence-dependent BT attraction for IDR–IDR and folded–IDR pairs. Increase to favor compaction; decrease to favor expansion.

disordered.eps_ev_kj

No

Float; 0.8368

Sets the AH repulsive-core energy in kJ/mol, independently of the pair well depth.

Residue numbering is one-based and must match the input PDB. Lists may contain inclusive ranges, individual integers, or both:

disordered:
  residues: [1, 2, 5-10, 150-165]

TOPO applies exactly the residues supplied by the user. It does not predict disorder. MobiDB annotations, experimental information, and low AlphaFold pLDDT may help identify candidate regions, but the final residue definition remains a modeling decision.

Low AlphaFold confidence should not be treated as proof of disorder. When used as a practical screening rule, pLDDT below 70 has been used as an IDR proxy at proteome scale [1].

Overlap with folded domains: disorder wins

A disordered: range may overlap an intra_domains range. The disorder transformation runs after the folded nonbonded build. For every pair touching a disordered residue, the domain scaling computed earlier in the build is discarded and the IDR rule is applied.

n_residues: 100
intra_domains:
  A: {residues: [1-100], nscale: 1.6871}
disordered:
  residues: [40-50]

Here, residues 40–50 are disordered and their nscale has no effect. The remaining portions of domain A still belong to one domain. The reader prints an informational message listing overlapping residues so accidental overlap remains visible.

Choosing and tuning idr_scale

The calibrated default is

idr_scale: 0.10

Use it unless system-specific experimental data support another value.

  • Increasing idr_scale strengthens the sequence-dependent attractive well and generally lowers \(\nu\), favoring compaction.

  • Decreasing idr_scale favors expansion.

  • idr_scale: 0 removes the BT-scaled attraction but retains the tiny non-native floor and the excluded-volume core. It is an approximately self-avoiding reference, not a noninteracting ghost chain.

  • eps_ev_kj is primarily a repulsive-core parameter, not the preferred solvent- quality dial.

The benchmark theta point occurs near idr_scale ≈ 0.32. Values much above 0.3 therefore enter the collapsed regime for that benchmark and should be justified for the system being studied.

A self-avoiding reference may be appropriate for:

  • an entropic linker whose reach is more important than its compaction;

  • a strongly charged, highly expanded sequence;

  • a deliberately minimal reference ensemble.

If SAXS, smFRET, or other ensemble measurements are available for the system of interest, treat idr_scale as a parameter to recalibrate rather than assuming that the global default is optimal.

Electrostatics must be interpreted separately. TOPO’s Debye–Hückel term can favor expansion or compaction depending on charge composition and sequence patterning; it is not universally repulsive.

Validation against SAXS radii of gyration

The default was evaluated on 24 proteins of 24–273 residues with published SAXS radii of gyration. Eighteen IDPs were used for calibration, and six foldable globular proteins were reported separately as controls.

The excluded control proteins were CspTm, R15, R17, hCyp, Protein-L, and sNase. Their published \(R_g\) values describe folded states, so comparing those values with fully disordered simulations does not test the IDR model. This classification was made from protein identity rather than a label stored in the benchmark dataset and should remain visible as a judgment call.

Each protein was simulated as a fully disordered chain for 90 ns of Langevin dynamics at 300 K, starting from an expanded coil. The first 15 ns were discarded. The reported value was the mass-weighted ensemble average

\[\sqrt{\left\langle R_g^2\right\rangle}\]

over the remaining 75 ns.

TOPO and HPS-Urry radii of gyration versus SAXS measurements for 18 intrinsically disordered proteins

Left: TOPO’s Cα IDR model at the calibrated defaults idr_scale = 0.10 and eps_ev_kj = 0.8368 kJ/mol. Right: the HPS-Urry force field evaluated for the same 18 proteins as an external reference. The dashed line is \(y=x\), the green line is the ordinary-least-squares fit, and point color represents fractional deviation. TOPO follows \(y=x\) more closely across the measured range; its fitted slope is 0.81, compared with 0.68 for HPS-Urry.

For the 18-IDP calibration set:

Model

\(\nu\)

\(R_0\)

RMS fractional error

Pearson \(r\)

OLS slope

TOPO Cα IDR

0.566

0.223

12.0%

0.89

0.81

HPS-Urry reference

0.490

0.301

19.7%

0.70

0.68

Experiment

0.551

0.244

—

—

—

The scaling relation was

\[R_g=R_0N^\nu.\]

The RMS value is the root-mean-square fractional deviation:

\[\sqrt{ \frac{1}{N} \sum_i \left( \frac{R_{g,i}^{\mathrm{sim}}-R_{g,i}^{\mathrm{exp}}} {R_{g,i}^{\mathrm{exp}}} \right)^2 }.\]

At idr_scale: 0.10, TOPO gave \(\nu=0.566\). A three-seed check at idr_scale: 0.12 gave \(\nu=0.559\pm0.009\).

Important limitation

The 12.0% RMS deviation does not demonstrate that the model captures detailed sequence-specific differences. A power law fitted directly to the same experimental data already leaves a 9.5% residual. In addition, three chains of length 185 span 36% in experimental \(R_g\) but only 7% in the model, in the wrong rank order.

The model is therefore supported for approximate ensemble dimensions across different chain lengths. It should not currently be used to confidently rank the dimensions of two IDPs of similar length. The remaining discrepancy may involve bonded terms, charge patterning, or other omitted interactions rather than only the contact potential.

Effect on native-contact analysis and stability optimization

The IDR mask is applied consistently to the energy function and native-contact definitions.

Native-contact Q analysis

Any native contact touching a disordered residue is removed from:

  • Q_protein;

  • every Q_domain;

  • interface-Q calculations.

Otherwise, contacts that cannot form would remain permanently in the denominator and artificially lower Q.

Effective domain membership is therefore

\[\text{effective domain}=\text{declared domain}-\text{disordered residues}.\]

The nscale optimizer

The optimizer evaluates folded domains and their interfaces. The IDR remains present and physically active in every simulation round, but it is not a scoring unit, does not enter the convergence check, and is never assigned an nscale.

Always run the optimizer using the domain-definition file that already contains the disordered: section. Declaring an IDR removes cross-boundary native contacts and can therefore alter the stability of the remaining folded domain. Optimizing the fully folded model and adding the IDR afterward would optimize a different energy function.

Continuous synthesis

CSP uses the same disordered: section without additional configuration. At each nascent-chain length, the full disorder mask is restricted to the residues that have emerged.

  • Before any folded residue emerges, the folded 12–10–6 interaction group is empty.

  • After folded residues emerge, the AH force evaluates IDR–IDR and folded–IDR pairs, while the Gō force evaluates folded–folded pairs.

  • The per-residue radius array is shared with the nascent-chain–ribosome excluded-volume construction.

The separate nascent-chain–ribosome interaction remains the existing 12–10–6 form at RIBO_NC_EPS_KJ for every nascent bead, including disordered beads. This preserves the O’Brien ribosome-interaction parameterization. It also means that an IDR bead uses different effective interactions toward the ribosome and toward the protein chain; users should keep this distinction in mind when interpreting nascent-chain behavior.

Fully disordered proteins

For a fully disordered protein, list every residue and omit intra_domains:

n_residues: 92
disordered:
  residues: [1-92]

This is the configuration used to calibrate the defaults. All native contacts are removed, and all eligible nonlocal pairs use the AH interaction.

The Q analysis returns an empty contact list and Q is NaN. The nscale optimizer detects that no foldable native contacts remain and exits with a “nothing to optimize” message.

Do not set both idr_scale and eps_ev_kj to zero as a way to create a self-avoiding chain. Setting eps_ev_kj: 0 removes the intended AH repulsive core. To obtain the approximately self-avoiding reference while retaining physical bead size, use:

disordered:
  residues: [1-92]
  idr_scale: 0
  eps_ev_kj: 0.8368

Starting structures and equilibration

The starting coordinates do not determine the equilibrium ensemble, provided the simulation samples the equilibrium distribution adequately. An IDR that starts from a compact or folded-looking structure can relax after its native contacts are removed, so an extended-chain PDB is not strictly required.

Starting coordinates can nevertheless affect equilibration time. Discard the initial relaxation period, check convergence, and consider simulations from both compact and expanded starting structures when slow collapse, adsorption, or barrier crossing is plausible.

Common pitfalls

  • idr_scale: 0 does not remove excluded volume. It retains the core set by eps_ev_kj and also retains TOPO’s minute non-native attraction floor.

  • eps_ev_kj: 0 removes the intended repulsive core. Do not use it to define a self-avoiding reference.

  • Numbering must match the input structure. Residue numbering is one-based; an incorrect number disorders the wrong residue.

  • Domain overlap is legal. If a residue appears in both a domain and the disorder mask, disorder wins and TOPO prints an informational message.

  • The current disorder mask is system-wide. It is a flat residue set and has no per-chain qualifier.

  • Declaring an IDR weakens the original Gō network. Every native contact with at least one disordered endpoint is deleted. Re-optimize folded-domain stability with the IDR declaration already present.

  • Use spaces in YAML. Do not use tab indentation, and include a space after each colon.

Reference

  1. Tesei G. et al. Conformational ensembles of the human intrinsically disordered proteome. Nature. 2024;626:897–904. https://doi.org/10.1038/s41586-023-07004-5