What you need to decide on is what actually is happening in the video. The are two main things
1. The strands are twisting
2. The two sides of the dna strand are separating after they straighten out.
This said, no single modifier will do this for you so you have to do it in two stages.
You will be using three modifiers here, twist, volume select and morpher.
Here's what I ended up with after doing the below steps:
1. Create your dna strand as one object and name it 'dna_closed', keeping the two halves of the coloured...'legs' as separate elements. Also move the pivot point to one end but in the center of its cross section (half way up, half way accross)
2. Next, clone your strand and name it 'dna_open'. This will your opened out dna strand and act as a morph target.
3. Go into element mode in 'dna_open' and move the two halves away from each other at equal amounts. NOTE: dont just move one side, you must move both side away from the center equally
4. Go back to 'dna_closed' and add a twist modifier then increase the 'Angle' value untill you have your desired twisted...ness. I chose 540°
5. Now, the twist modifier has no 'end' by default, it will twist the object off into infinity unless you tick the 'Limit Effect' box. After ticking this box the dna strand will flatten out again, so enter a value into the 'Upper limit' that is equal or greater than the overall length of the strand. My strand is 30cm long so I just dragged the spinner past the 30cm point.
Now if you select the twist modifier to activate the gizmo, you can move the gizmo up and down to see the twisting/untwisting effect in realtime. Make sure you undo any changes to the gizmo's position. What you have done here is ttell the modifier to only affect the mesh within the limits set, outside of those limits it will have no effect.
You now need to animate the twist modifier moving from fully over the mesh at frame 0, to competly away from the mesh at frame 100 (or what ever time scale you prefer) This will only work if you animate the gizmo moving down the length of the mesh away from the gizmo. (Opposite direction to the way the image shows, sorry about that ;)
6. So now you should be able to slide the time slider and the twisted dna strand should unravel over the length of your animation, the only thing is it doesnt open out.
Next add a 'Volume Select' modifier. Choose 'Vertex' at the top of the modifier oiptions, select the gizmo as you did with the twist modifier and move it down the mesh slightly, then turn on soft selection and adjust the 'falloff' value until you get a slight fade like in my image (I used a falloff value of 6 but its up to you and can be adjusted afterrwards if its not right)
7. Next you are going to animate the gizmo of the 'Volume Select' modifier so that it follows the twist modifier's gizmo. at frame zero, move the gizmo far enough away from the mesh so that it isnt affecting the verts. The gizmo needs to on be the side of the pivot point of the model not the other end. Then animate it over the length of the animation to move over the mesh. At frame 100 the gizmo should be selecting all the verts. If you now drag the time slider the dna strand should twist and then get selected by the voluyme select modifier once it starts to straighten out.
8. Now add a 'Morpher' modifier and click the 'Load Multiple Targets' button, then select 'dna_open' and click 'Load'
9. The first button on the modifier will now be called 'dna_open' and you should turn the value up to 100 so the target is fully affecting the mesh.
This morpher modifier will only affect the verts that are selected in the volume select modifier below it.
So now hopefully your animation will be working properly, you can go back and adjust the volume select modifier's position and soft selection value to suit if the change isnt smooth enough or out of sync, but this is how I did it.
