Intrinsic stiffness and <i>Θ</i> -solvent regime in intrinsically disordered proteins: Implications for liquid–liquid phase separation

Liquid-liquid phase separation (LLPS) exhibited by intrinsically disordered proteins (IDPs) depends on the solvation state around the <i>Θ</i>-regime, which separates good from poor solvent. Experimentally, the <i>Θ</i>-solvent regime of the finite length (<i>N</i>) IDPs, as probed by small angle X-ray scattering (SAXS) and single molecular fluorescence resonance energy transfer (smFRET), is in disagreement. Using computer simulations of a coarse-grained IDP model, we address the effect of chain length on the <i>Θ</i>-regime of IDPs with polar side chains (polyglutamine) and hydrophobic side chains (polyleucine) subject to varying concentrations of cosolvents <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>[</mml:mo> <mml:mi>C</mml:mi> <mml:mo>]</mml:mo></mml:math> , urea (denaturant) or trimethylamine N-oxide (protective osmolyte) in water. Due to their intrinsic stiffness, these IDPs are always expanded on short-length scales, independent of the solvent quality. As a result, for short IDP sequences ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>≈</mml:mo> <mml:mn>10</mml:mn></mml:math> to 25 residues), their propensity to exhibit LLPS cannot be inferred from single-chain properties. Further, for finite-size IDPs, the cosolvent concentration to attain the <i>Θ</i>-regime ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>[</mml:mo> <mml:msub><mml:mi>C</mml:mi> <mml:mi>Θ</mml:mi></mml:msub> <mml:mo>]</mml:mo></mml:math> ) extracted from the structure factor emulating SAXS and pair distances mimicking smFRET differs. They converge to the same cosolvent concentration only at large <i>N</i>, indicating that finite size corrections vary for different IDP properties. We show that the radius of gyration ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mi>g</mml:mi></mml:msub> </mml:math> ) of the IDPs in the <i>Θ</i>-solvent regime satisfies the scaling relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi> <mml:mi>g</mml:mi> <mml:mn>2</mml:mn></mml:msubsup> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>c</mml:mi> <mml:msqrt><mml:mi>N</mml:mi></mml:msqrt> <mml:mo>)</mml:mo></mml:math> , which can be exploited to accurately extract <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>[</mml:mo> <mml:msub><mml:mi>C</mml:mi> <mml:mi>Θ</mml:mi></mml:msub> <mml:mo>]</mml:mo></mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mo>[</mml:mo> <mml:mi>C</mml:mi> <mml:mo>]</mml:mo> <mml:mo>/</mml:mo> <mml:mo>[</mml:mo> <mml:msub><mml:mi>C</mml:mi> <mml:mi>Θ</mml:mi></mml:msub> <mml:mo>]</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo></mml:math> ). We demonstrate the importance of finite size aspects originating from the chain stiffness and thermal blob size in analyzing IDP properties to identify the <i>Θ</i>-solvent regime.