Two-way binding in AvalonEdit doesn't work

Question

I have used AvalonEdit in my project that is based on WPF and MVVM. After reading this post I created the following class:

public class MvvmTextEditor : TextEditor, INotifyPropertyChanged
{
    public static DependencyProperty DocumentTextProperty =
        DependencyProperty.Register("DocumentText", 
                                    typeof(string), typeof(MvvmTextEditor),
        new PropertyMetadata((obj, args) =>
        {
            MvvmTextEditor target = (MvvmTextEditor)obj;
            target.DocumentText = (string)args.NewValue;
        })
    );

    public string DocumentText
    {
        get { return base.Text; }
        set { base.Text = value; }
    }

    protected override void OnTextChanged(EventArgs e)
    {
        RaisePropertyChanged("DocumentText");
        base.OnTextChanged(e);
    }

    public event PropertyChangedEventHandler PropertyChanged;
    public void RaisePropertyChanged(string info)
    {
        if (PropertyChanged != null)
        {
            PropertyChanged(this, new PropertyChangedEventArgs(info));
        }
    }
}

and used the following XAML to use this control:

<avalonedit:MvvmTextEditor x:Name="xmlMessage">
   <avalonedit:MvvmTextEditor.DocumentText>
      <Binding Path ="MessageXml" Mode="TwoWay" 
               UpdateSourceTrigger="PropertyChanged">
         <Binding.ValidationRules>
            <local:XMLMessageValidationRule />
          </Binding.ValidationRules>
      </Binding>
   </avalonedit:MvvmTextEditor.DocumentText>
</avalonedit:MvvmTextEditor>

but the binding works OneWay and doesn't update my string property nor run validation rule.

How can I fix binding to work as expected TwoWay?