# 16. Paradoxical regulation in intra- and intercellular circuits¶

**Design principles**

Futile cycles in kinase systems can linear amplification that is robust to total protein copy numbers.

Biphasic control can be accomplished with paradoxical regulation.

**Concepts**

Paradoxical regulation

Cellular circuits are necessarily autocatalytic

**Techniques**

Use of conservation laws can greatly simplify analysis.

Plotting phase portraits with a separatrix

```
[1]:
```

```
import numpy as np
import scipy.integrate
import biocircuits
import bokeh.io
import bokeh.plotting
import panel as pn
pn.extension()
bokeh.io.output_notebook()
```