## Step 1: Define the symbolic representation of the problem
Let's denote the variables as follows:
- $x_1$ for the amount of oats in kilograms
- $x_2$ for the amount of sunflower seeds in kilograms

The objective is to minimize the cost of the mixture. The cost of oats is $50 per kilogram, and the cost of sunflower seeds is $70 per kilogram. Therefore, the objective function can be represented as:
\[ \text{Minimize:} \quad 50x_1 + 70x_2 \]

## Step 2: Define the constraints based on the problem description
The mixture must contain a minimum of 250 units of protein and 400 units of fat.
- Oats contain 5 units of protein and 16 units of fat per kilogram.
- Sunflower seeds contain 10 units of protein and 22 units of fat per kilogram.

Therefore, the constraints can be represented as:
\[ 5x_1 + 10x_2 \geq 250 \] (protein constraint)
\[ 16x_1 + 22x_2 \geq 400 \] (fat constraint)

Additionally, $x_1 \geq 0$ and $x_2 \geq 0$ because the amounts of oats and sunflower seeds cannot be negative.

## 3: Symbolic representation in the required format
```json
{
    'sym_variables': [('x1', 'oats'), ('x2', 'sunflower seeds')],
    'objective_function': '50*x1 + 70*x2',
    'constraints': [
        '5*x1 + 10*x2 >= 250',
        '16*x1 + 22*x2 >= 400',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## 4: Convert the problem into Gurobi code
```python
import gurobi

def solve_animal_feed_mixture():
    # Create a new model
    model = gurobi.Model()

    # Define the variables
    x1 = model.addVar(name="oats", lb=0)  # Amount of oats
    x2 = model.addVar(name="sunflower_seeds", lb=0)  # Amount of sunflower seeds

    # Objective function: Minimize the cost
    model.setObjective(50 * x1 + 70 * x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(5 * x1 + 10 * x2 >= 250, name="protein_constraint")
    model.addConstr(16 * x1 + 22 * x2 >= 400, name="fat_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Amount of oats: {x1.varValue} kg")
        print(f"Amount of sunflower seeds: {x2.varValue} kg")
        print(f"Total cost: ${model.objVal:.2f}")
    else:
        print("No optimal solution found.")

solve_animal_feed_mixture()
```