<DataGrid HorizontalAlignment="Left" Height="100" Margin="54,25,0,0" VerticalAlignment="Top" Width="264"
    IsSynchronizedWithCurrentItem="True" AutoGenerateColumns="False"
    ItemsSource="{Binding DataSource}" SelectedItem="{Binding SelectedPerson}">
    <DataGrid.Columns>
        <DataGridTextColumn Binding="{Binding FirstName}" />
        <DataGridTextColumn Binding="{Binding LastName}" />
    </DataGrid.Columns>
</DataGrid>

public class Person
{
    public string FirstName { get; set; }
    public string LastName { get; set; }
}

public class VM : Notifier
{
    private ObservableCollection<Person> dataSource;
    public ObservableCollection<Person> DataSource
    {
        get { return dataSource; }
        set
        {
            dataSource = value;
            OnPropertyChanged("DataSource");
        }
    }

    private Person selectedPerson;
    public Person SelectedPerson
    {
        get { return selectedPerson; }
        set
        {
            selectedPerson = value;
            OnPropertyChanged("SelectedPerson");
        }
    }

    private string string1;
    public string String1
    {
        get { return string1; }
        set
        {
            string1 = value;
            OnPropertyChanged(String1);
        }
    }

    public VM()
    {
        String1 = "abc";

        DataSource = new ObservableCollection<Person>();
        DataSource.Add(new Person { FirstName = "Alpha", LastName = "Apple" });
        DataSource.Add(new Person { FirstName = "Beta", LastName = "Banana" });
        DataSource.Add(new Person { FirstName = "Charlie", LastName = "Cucumber" });
    }

    public ICommand SetSelectedCommand => new RelayCommandBase(SetSelected);
    private void SetSelected(object parameter = null)
    {
        SelectedPerson.FirstName = String1;
        SelectedPerson = SelectedPerson;    // force setter to run
        DataSource = DataSource;            // force ObservableCollection setter to run
    }
}