Place water in a large bowl. Sprinkle in yeast; stir until dissolved. Add bran to yeast mixture; let stand 10 minutes. Add 1 cup all-purpose flour, sugar, and salt. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Stir in whole wheat flour and enough remaining all-purpose flour to make a soft dough. Knead on lightly floured surface until smooth and elastic, about 8 to 10 minutes. Place in greased bowl, turning to grease top. Cover; let rise in warm, draft-free place until doubled in size, about 45 minutes.
Punch dough down; turn out onto lightly floured surface. Divide dough into 12 equal pieces. Roll each piece to 18-inch rope; shape into pretzels. Place on greased baking sheets. Brush lightly with beaten egg; sprinkle with salt.
Bake in preheated 425oF oven for 15 to 20 minutes or until done. Remove from sheet; cool on wire rack.