Proton spin from small- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"> <mml:mi>x</mml:mi> </mml:math> with constraints from the valence quark model

We apply the valence quark model recently calculated for polarized proton to constrain the non-perturbative initial condition for the small-$x$ helicity evolution. The remaining free parameters are constrained by performing a global analysis to the available polarized small-$x$ deep inelastic scattering data. A good description of the world data is obtained with only 8 free parameters. The model parameters are tightly constrained by the data, allowing us to predict the proton polarized structure function $g_1^p$ to be negative at small $x$. Furthermore, we obtain the small-$x$ quark and gluon spins to give a contribution $\int_{10^{-5}}^{0.1} dx \left( \frac{1}{2}ΔΣ+ ΔG \right) = 0.76 \pm 0.13$ or $1.70\pm 0.20$ to the proton spin, depending on the applied running coupling prescription.